#### 3.2. Electronic Energies of Reaction in Vacuum

The complete theoretical analysis of the Diels–Alder reaction for the fullerenes studied, i.e., including all the meaningful regioisomers (see

Figure 3 and

Figure 4), was undertaken at the M06-2X/6-31+G(d) level of theory. Additionally, single-point energy calculations in a vacuum were performed on a representative set of the M06-2X/6-31+G(d) optimized geometries using other computational methods: (i) B3LYP/6-31+G(d) to evaluate the effect of changing the DFT functional, and, in particular, to compare between functionals with and without empirical dispersion correction terms; (ii) M06-2X/6-311++G(2d,p) to evaluate the effect of improving on the basis set; and (iii) MP2/cc-pVDZ, and simultaneously HF/cc-pVDZ, to explore the use of ab initio quantum chemistry methods and the contribution of correlation energy for these Diels–Alder reactions. The calculations referred to in (i) and (ii) were made for the reactions involving the following representative set of fullerenes: AC

_{60}MA, the two most stable regioisomers of AC

_{70}MA (isomer

a and isomer

b), and four regioisomers of AIC

_{60}BA (

trans-1,

trans-3

a,

e-face

a, and

cis-2

b), which include the most stable

e-face

a regioisomer and different types of bisadducts (concerning the bond distance between the anthracene and indene groups). The reaction of C

_{60} with indene was also studied using all the DFT methods referred before to evaluate the differences in reactivity and stability due to changing the diene.

The calculated electronic energies of the reaction in a vacuum, at

T = 0 K, ∆

_{r}E_{el,m} (g, 0 K), for the Diels–Alder cycloadditions between the fullerenes studied with two dienes, indene (Ind) and anthracene (Ant), are presented in

Table 1.

Concerning the two dienes, Ind and Ant, the results are not surprising in indicating that the reaction with Ind is more exothermic than with Ant. This agrees with the greater reactivity of Ind as a diene in Diels–Alder cycloadditions [

34]. The reactions of the three dienophiles, C

_{60}, C

_{70}, and IC

_{60}MA with Ant, show similar ∆

_{r}E_{el,m} (g, 0 K), considering the most stable regioisomers of AC

_{70}MA (

isomer a) and AIC

_{60}BA (

e-face

a). Thus, our computational study does not predict a significant difference in reactivity, due to electronic structure (which is the most significant contribution to the enthalpy of the reactions), between the three fullerenes when reacting with Ant.

For C

_{70}, the comparison between the regioisomers clearly reveals an increasing tendency for reaction, more negative ∆

_{r}E_{el,m} (g, 0 K), as the [6,6] additions occur closer to the edges of the fullerene, where it is more similar to C

_{60} in local curvature and bond length. This is also well understood by inspecting the type of [6,6] bonds to where the diene binds. In isomer

a, it binds to an α type and in isomer

b to a β type bond (see

Figure 1a). The much higher (more positive) values of ∆

_{r}E_{el,m} (g, 0 K) for isomers

c and

d are easily explained by noting that these [6,6] bonds, contrary to α and β types, are not linked to two pentagons. In isomer

c, it is linked to one pentagon and in

d to none (the [6,6] bond is surrounded by four hexagons). This bond topology in C

_{70} is very impactful on reaction energetics, making the reaction for isomer

d energetically very disfavored.

For the reaction of IC

_{60}MA with Ant, three groups of regioisomers can be discerned. The

cis-isomers are clearly the least stable, which probably arises from steric repulsions between the two addends. Note that the most unstable regioisomer is

cis-2

a, which is the one (excluding the

cis-1 isomers) where the addends have a more notorious overlap of electron density. The

trans- and equatorial regioisomers fall into two categories of energetic stability. They all have similar energies and thus significant populations at equilibrium, and all should be considered as relevant products in Diels–Alder cycloadditions. The three

e- isomers,

trans-3

a, and

trans 3

b stand out as the most stable in the set. Among these five, the differences in ∆

_{r}E_{el,m} (g, 0 K) are so small they can be neglected, and no isomer deserves to be clearly distinguished as the most stable. The remaining

trans-regioisomers (1, 2

a, 2

b, 4

a, and 4

b) are slightly less stable. Our computational results are in good agreement with the literature concerning both the little differentiation between dienophiles [

22,

23] and the ordering in reactivity [

24,

25,

26].

In fullerenes, reactivity is closely related to double-bond character and curvature; usually, the most reactive bonds are shorter and have higher local curvature [

2,

4]. Although bond length can be a good predictor of reactivity (here, we speak of reactivity in the context of reaction kinetics), its implications on reaction thermodynamics are questionable. In C

_{70}, there is a qualitative relation between the calculated bond lengths and ∆

_{r}E_{el,m} (g, 0 K) for the reaction with Ant. The lengths of the bonds in C

_{70}, in Å, to which Ant binds increase in the order: isomer

b (1.380) < isomer

a (1.390) < isomer

c (1.416) < isomer

d (1.470), which approximately follows the increase (more positive value) in ∆

_{r}E_{el,m} (g, 0 K). In IC

_{60}MA, the bonds leading to the more stable

e- regioisomers are, in fact, slightly shorter than those leading to the

trans- (≈ 1.387 Å vs ≈ 1.389 Å, respectively). On the other hand, the trend in ∆

_{r}E_{el,m} (g, 0 K) among the

trans-isomers has no apparent relation to the calculated bond lengths.

This work is more focused on the thermodynamics of Diels–Alder reactions than on the kinetic reactivity (for which bond length could be a better descriptor). Thermodynamic analysis requires knowledge of both reactants and products. In this way, and given that in this analysis, the reactants are the same, it makes more sense to compare the products (i.e., the AC

_{70}MA and AIC

_{60}BA regioisomers). The trend for AC

_{70}MA highlights the contribution from the local curvature of the fullerene and aromaticity for reaction energetics. ∆

_{r}E_{el,m} (g, 0 K) is more negative for those isomers in which Ant binds to a bond with greater local curvature. Isomers

a and

b show greater local curvature (similar to C

_{60}), but for isomers

c and

d, the reactive zone is more planar. Curvature is intrinsically related to bond strain, with greater curvature being associated with the release of more strain energy upon binding to the diene [

2,

4,

9]. The question of aromaticity in fullerenes is still controversial, but some authors advocate that, contrary to the non-aromatic pentagons, the hexagons can be slightly aromatic [

35,

36]. By this reasoning, the formation of the least stable AC

_{70}MA isomers,

c and

d, implies the disruption of aromaticity in three and four hexagons, respectively, while in isomers,

a and

b, only two hexagons are sacrificed. The differentiation between

trans- and

e- isomers in AIC

_{60}BA, however, is harder to rationalize. All isomers are close in energy, and bond topology in the C

_{60} framework is less diverse than in C

_{70} [

9]. We investigated the possibility that, after the binding of the diene, the four adjacent hexagons (not those directly bonded) become more aromatic. This makes sense, considering that the release of strain upon addition allows for some geometry adjustment of the surrounding fullerene surface [

35]. A quick inspection of the IC

_{60}MA and AC

_{60}MA optimized structures supports this hypothesis, with the four hexagons adjacent to the addend (see

Figure 5) showing more aromatic character, as suggested by shorter bonds (on average) and smaller bond-length alteration, in comparison to C

_{60}. Following this reasoning, we employed the Harmonic Oscillator Model of Aromaticity (HOMA) [

37,

38] on the fullerenes to gain some insight into the aromatic character of the relevant hexagons. The bond lengths used in the calculation of HOMA indices were taken from the M06-2X optimized structures. HOMA indices were calculated using the refined model of HOMA [

39]:

where

n is the number of bonds considered in the summation (

n = 6 for the case of fullerene hexagons),

r_{opt} stands for the optimal value of a specific bond length in a fully aromatic system (for CC bonds

r_{opt} = 1.388 Å) [

38],

r_{av} is the average bond length of the

n bonds,

r_{i} is the bond length of bond i, and α = 257.7 Å

^{−2} is a normalization constant. The term EN is closely related to de-aromatization due to bond energies, and GEO is attributed to de-aromatization due to geometric contributions, namely bond length alternation.

In our HOMA analysis, we have distinguished between three types of hexagons (see

Figure 5): Type-I (those not adjacent to an addend, similar to all hexagons in C

_{60}), Type-II (the four hexagons adjacent to an addend), and Type-III (the superposition of two Type-II hexagons). This distinction is based on the different bond topology and calculated HOMA indices, which increase (higher aromaticity) in the order: Type-I < Type-II < Type-III. The detailed results of our HOMA analysis are presented in

Table 2. In C

_{70}, there are three types of hexagons: Ia, the hexagons in the edges of C

_{70}; Ic, the hexagons in the center of the equatorial belt of C

_{70}; and Ib, the rings adjacent to Ic in the equatorial belt (see

Figure S1 for a visual display). This gives rise to three different Type-II rings in C

_{70} adducts: IIa, IIb, and IIc.

Table 2 shows the average HOMA results for each type of hexagon in the fullerenes studied; the detailed results for each individual hexagon are presented as

Supplementary Materials. We chose to present the average values for a question of simplicity, and because the HOMA values do not vary significantly within the same type of hexagon. Furthermore, it is noteworthy that the addition of the addend only alters the aromaticity of the four adjacent hexagons; all the others remain similar to the Type-I hexagons in the parent C

_{60} or C

_{70}. The results reveal a clear gain in aromaticity in the four hexagons adjacent to the adduct; in the C

_{60} adducts, HOMA increases from ~0.2 (Type-I) to 0.4-0.5 (Type-II). This increase in aromaticity is manifested by a decrease in both the EN and GEO parameters, although according to HOMA, de-aromatization in fullerenes has a stronger contribution from bond length alternation. In the case of the bisadducts, the aromaticity gain is even larger if the ring is simultaneously adjacent to both addends (Type-III), in which case the HOMA value is ~0.7. All three equatorial isomers of AIC

_{60}BA have two such hexagons, which can help explain their higher stability. However, aromaticity changes do not seem enough to explain the relative stability of

trans-3 isomers, which remains to be explained. The same trend upon addend addition is observed in C

_{70}. Although aromaticity contributes to ∆

_{r}E_{el,m}, the differentiation between AC

_{70}MA isomers

a and

b is better explained by bond curvature and release of strain. The gain in aromaticity upon addition probably contributes to the exothermicity of Diels–Alder reactions with fullerenes. According to our HOMA analysis, and as previously reported in the literature [

35,

40,

41], the energetic stability of similar hexakis adducts of C

_{60} will probably benefit from all adjacent hexagons (a total of eight) being Type-III (in fact, these rings can be even more aromatic than Type-III because they result from the superposition of three Type-II hexagons). Another important remark is the difference in aromaticity found for C

_{70} hexagons—while the aromaticity of Type-Ia edge hexagons seems comparable to the C

_{60} hexagons—Type-Ic equatorial hexagons have a much higher HOMA value. On the contrary, Type-Ib hexagons (adjacent to Type-Ic) are probably slightly anti-aromatic.

#### 3.3. Effect of Computational Method and Basis-Set

Next, we analyze the effect of the computational method on the calculated reaction energies (see

Table 1). When compared to the dispersion-corrected M06-2X, the B3LYP functional predicts the fullerenes to be much less reactive towards Diels–Alder addition. In fact, Solà and co-workers have shown that the inclusion of dispersion corrections is essential for the study of the chemical reactivity of fullerenes [

42,

43]. Moreover, as verified experimentally [

12,

13,

14,

15], these reactions are significantly exothermic, thus proving that M06-2X describes energetics better. A great fraction of the B3LYP destabilization is due to the inability of this functional to describe London dispersion forces. B3LYP probably “sees” the interaction between the Ant “wings” with the fullerene surface as highly repulsive. Interestingly, the differences, ∆, between the values of ∆

_{r}E_{el,m} (g, 0 K) calculated using M06-2X and B3LYP (both with 6-31+G(d)) nicely support this reasoning. The smallest difference is observed for R1 (∆ = 86 kJ∙mol

^{−1}), the reaction with Ind, which is smaller than Ant and has only one “wing” to overlap with the C

_{60} surface. For the two AC

_{70}MA isomers, ∆ is greater for isomer

b (113 as compared to 105 kJ∙mol

^{−1} for isomer

a). Moreover, for isomer

a, ∆ is similar to three representative isomers of AIC

_{60}BA (

trans-1,

trans-3

a, and

e-face

a); for

cis-2

b ∆ is slightly larger. This is easily understood by recalling that in isomer

a of AC

_{70}MA, Ant binds to the place where the local curvature is more similar to C

_{60}. In isomer

b of AC

_{70}MA, ∆ is larger because one “wing” of Ant overlaps with the flatter surface of C

_{70} closer to the equatorial belt of hexagons. In the

cis-2

b isomer of AIC

_{60}BA, ∆ is larger because there is an additional intramolecular contact between the two addends.

If using M06-2X with a larger basis set, 6-311++G(2d,p), a less negative ∆

_{r}E_{el,m} (g, 0 K) was found for all fullerenes studied. It is known that using a more complete basis set lowers the energy of a molecular species, approximating it to the variational limit (the calculated electronic energies for each species are compiled in the

Supplementary Materials). However, the electronic energies were lowered more in favor of the reactants, making the reactions less favorable. The differences between the two basis sets are very constant for all reactions (∆ ≈ 10 kJ∙mol

^{−1}). One possible contribution to this difference is the smaller basis set superposition error (BSSE) if using the larger 6-311++G(2d,p) basis [

44]. The stabilizing interaction between the fullerene surface and the addend in adduct molecules is probably overestimated to a larger extent, due to BSSE, by the smaller 6-31+G(d). The MP2/cc-pVDZ result for R2 corroborates the importance of correlation energy in the fullerenes, with the uncorrelated Hartree-Fock (HF) predicting a substantially less negative ∆

_{r}E_{el,m} (g, 0 K). This correlation is manifested in both electronic conjugation effects and London dispersive interactions, although MP2/cc-pVDZ probably overestimates the correlation energy.

Together, these results highlight the importance of intramolecular interactions with a significant contribution from London dispersion, such as π∙∙∙π interactions involving the aromatic addends at the surface of the fullerene.

#### 3.4. Solvation Energies in M-Xylene. Enthalpies of Reaction at T = 298.15 K

The optimized M06-2X/6-31+G(d) geometries in a vacuum were used to calculate the electronic energies of solvation, ∆

_{solv}E_{el,m} (

m−xylene) for each species using SMD [

33]. To economize the computational resources, the geometries were not reoptimized in solution, except for a few representative cases. Given the relatively low flexibility of the molecules studied (e.g., they do not possess low-frequency vibrational modes such as those associated to internal rotation) and the absence of any obvious conformational equilibria, we assume that the preferred geometry in the weakly polar

m-xylene shall not be significantly different from that in the gas phase. In fact, for those molecules that were reoptimized in a solution of

m-xylene using SMD, the calculated electronic energies show variations smaller than 1 kJ∙mol

^{−1}, thus supporting our assumption (details in the

Supplementary Materials). The calculated ∆

_{solv}E_{el,m} (

m−xylene) for all the molecules studied are presented in

Table 3. The results follow the expected trend, with ∆

_{solv}E_{el,m} (

m−xylene) becoming more negative as the molecular surface area increases. There are only slight differences between regioisomers; however, this can alter the order of reactivity if going from gas to solution. To analyze the effect of solvation on the Diels–Alder reactions, we calculated the energies of reaction in solution, ∆

_{r}E_{el,m} (

m−xylene), and the results are presented in

Table 4. The inclusion of solvation makes ∆

_{r}E_{el,m} less negative, meaning that

m-xylene solvates the reagents better than the products. This makes sense, considering the stoichiometry of the reactions and the smaller total surface area of the product in comparison to the sum of the two reactants. The effect of solvation is most pronounced in R2 (C

_{60} + Ant reaction) and less pronounced in R1 (C

_{60} + Ind), although this differentiation among reactions is very faint. The trend in ∆

_{r}E_{el,m} for the reactions of the three dienophiles (C

_{60}, C

_{70}, and IC

_{60}MA), considering the most stable isomers, is tenuous both in a vacuum and in

m-xylene. However, solvation makes it slightly more noticeable in the order of more negative ∆

_{solv}E_{el,m}: C

_{60} < C

_{70} < IC

_{60}MA. In the regioisomers of AIC

_{60}BA, the effect of solvation also shows a small dependence on the addends’ position, slightly favoring the

e-edge isomer. In short, solvation by

m-xylene, as simulated by SMD, makes the reaction ≈ 20 kJ∙mol

^{−1} less exothermic, but this effect is almost independent of the dienophile, diene, and position of the addends.

The former analyses of ∆

_{r}E_{el,m}, at

T = 0 K, ignore the contributions from ZPE and thermal enthalpy. In this work, we also pretend to report reasonable computational predictions of reaction thermodynamics under real experimental conditions. To this end, we have estimated the standard molar enthalpies of reaction in

m-xylene, at

T = 298.15 K,

${\Delta}_{\mathrm{r}}{H}_{\mathrm{m}}^{0}\left(m-\mathrm{xylene},298.15\mathrm{K}\right)$ using the following Equation:

where

${\Delta}_{\mathrm{r}}{E}_{\mathrm{el},\mathrm{m}}^{6-311++\mathrm{G}\left(2\mathrm{d},\mathrm{p}\right)}\left(\mathrm{g},0\mathrm{K}\right)$ is the calculated electronic energy of reaction in a vacuum using M06-2X/6-311++G(2d,p) (

Table 1),

$\Delta \left({\Delta}_{\mathrm{solv}}{E}_{\mathrm{el},\mathrm{m}}^{6-31+\mathrm{G}\left(\mathrm{d}\right)}\right)$ is the calculated solvation energy of the reaction using M06-2X/6-31+G(d) (

Table 3, products minus reactants), and

${\Delta}_{\mathrm{r}}{H}_{\mathrm{thermal},\mathrm{m}}^{6-31+\mathrm{G}\left(\mathrm{d}\right)}$ is the thermal correction to enthalpy (including ZPE and thermal enthalpy from 0 K to 298.15 K), as obtained from the frequencies calculations using M06-2X/6-31+G(d). The term

${\Delta}_{\mathrm{r}}{H}_{\mathrm{thermal},\mathrm{m}}^{6-31+\mathrm{G}\left(\mathrm{d}\right)}$ was not corrected for scaling factors, which is a reasonable approximation considering that their contribution shall partially cancel out in the calculation of enthalpies of reaction. The results are presented in

Table 4 for the selected representative reactions.

(m−xylene, 298.15 K) were estimated using Equation (2). ^{3} The AIC_{60}BA regioisomers are ordered up-to-down by increasing proximity of Ant from Ind.

As can be observed, solvation and the correction for thermal enthalpy cause the reactions to be considerably less exothermic; the contribution of ${\Delta}_{\mathrm{r}}{H}_{\mathrm{thermal},\mathrm{m}}^{6-31+\mathrm{G}\left(\mathrm{d}\right)}$ is ≈ +10 kJ∙mol^{−1}. Given the stoichiometry of the Diels–Alder reactions, it follows that this decrease in exothermicity must come from the vibrational component. If considering the classical approximation to the translational and rotational degrees of freedom, simple calculations indicate that this vibrational contribution (ZPE + thermal) to ${\Delta}_{\mathrm{r}}{H}_{\mathrm{m}}^{0}\left(\mathrm{g},298.15\mathrm{K}\right)$ is of about +20 kJ∙mol^{−1}. This probably results from the lower frequency modes that appear in the adduct associated with single bonds and addend flipping.

Some authors have reported experimental determinations of

${\Delta}_{\mathrm{r}}{H}_{\mathrm{m}}^{0}$ for Diels–Alder reactions with C

_{60} and C

_{70} in solution [

12,

13,

14,

15]. The results from different authors vary significantly. Sarova and Berberan-Santos report a

${\Delta}_{\mathrm{r}}{H}_{\mathrm{m}}^{0}$ = −81 kJ∙mol

^{−1} for the reaction C

_{60} + Ant in toluene at <

T> = 316 K, as measured by UV-Vis spectroscopy [

12]. Saunders and co-workers have studied, by NMR, the similar reactions of the two endohedral derivatives

^{3}[email protected]_{60} and

^{129}[email protected]_{60} with 9,10-dimethylanthracene (DMA), determining

${\Delta}_{\mathrm{r}}{H}_{\mathrm{m}}^{0}$ values, at <

T> ≈ 310 K, of −96 (in 1-methylnaphthalene/CD

_{2}Cl

_{2}) [

13] and −95 kJ∙mol

^{−1} (in

o-dichlorobenzene/C

_{6}D

_{6}) [

14], respectively. In Ref. [

13], the authors also determined similar

${\Delta}_{\mathrm{r}}{H}_{\mathrm{m}}^{0}$ values, under the same experimental conditions, for the reaction of C

_{70} with DMA (−88 kJ∙mol

^{−1}) and for the addition of a second DMA addend to C

_{60} (−94 kJ∙mol

^{−1}). For the reactions of endohedral H

_{2}@C

_{70} and (H

_{2})

_{2}@C

_{70} with DMA in

o-dichlorobenzene-

d_{4} and at <

T> = 313 K, Murata et al. have determined by NMR the

${\Delta}_{\mathrm{r}}{H}_{\mathrm{m}}^{0}$ values of −58 and −56 kJ∙mol

^{−1}, respectively [

15], in good agreement with our theoretical predictions.

Isomerism also has a strong impact on Diels–Alder reactions involving fullerenes. According to our computational results, AIC

_{60}BA has at least five regioisomers (and four of these have an enantiomer) with important populations at equilibrium, both in the gas phase and solution. There are four

trans-3 isomers (two enantiomeric pairs) and four equatorial isomers (

e-face

a,

e-face

b, and the enantiomer pair of

e-edge). This gives a total of nine predominant isomers. Additionally, there are nine other residual isomers: the higher-energy

trans-1, -2, and -4 regioisomers and their enantiomers; this excludes all

cis- isomers, which are practically nonexistent at equilibrium. On the other hand, IC

_{60}MA and AC

_{60}MA do not have isomers, and AC

_{70}MA only has two meaningful isomers (

a and

b). Hence, the addition of the diene to the mono-adduct IC

_{60}MA (R4) is entropically favored in this specific contribution of isomerism relative to the first addition to the original fullerenes (R1 to R3). However, one should not forget the contribution of molecular symmetry to entropy [

45,

46,

47]. Given that C

_{60} and C

_{70} are highly symmetrical molecules and the adducts are not, symmetry can have a significant contribution to reaction thermodynamics. The impact of isomers and molecular symmetry on the thermodynamics and kinetics of fullerene reactions are currently under investigation, by experimental and computational methodologies, in our laboratory.

Although this work is focused on the thermodynamics of the fullerene reactions studied, Diels–Alder cycloadditions can be limited by kinetics. We feel that we could not conclude our discussion without a small remark on the kinetic feasibility of such reactions in the gas phase. The electrophilicity index, ω, of the reactants can be used to predict the viability of Diels–Alder reactions [

48,

49,

50]. Thus, for the dienes Ind and Ant, and for the dienophiles C

_{60}, C

_{70}, and IC

_{60}MA, we have estimated the values of ω according to the equation proposed by Parr and co-workers: ω = [(IP + EA)

^{2}]/[8 ∙ (IP − EA)] [

48], where IP and EA are, respectively, the ionization potential and the electron affinity of the molecules in the gas phase. In this context, dienophiles, being electrophilic in nature, are characterized by high ω values, while a small ω can be related to the small electrophilic tendency and thus, in principle, high nucleophilic character (as expected for dienes) [

49,

50]. The results are presented in

Table 5. As a rough approximation, and for the sake of comparison, we also show the IP, EA, and ω values, as obtained from the HOMO and LUMO energies calculated in this work at the M06-2X/6-31+G(d) level; IP ≈ −

E(HOMO) and EA ≈ −

E(LUMO). Except for IC

_{60}MA, our results show relatively good agreement with the experimental values. Most importantly, the trend in ω is the same for the two approaches: lower ω for the dienes and larger ω for the dienophiles. According to the global electrophilicity scale proposed by Domingo et al., the three fullerenes can be classified as strong electrophiles and the two dienophiles as moderate to marginal electrophiles [

49,

50]. Hence, this analysis provides additional theoretical support for the kinetic feasibility of the Diels–Alder reactions studied in this work.