Selectivity of chemical reactions of fullerenes is a challenge of fullerene science, as there is no universal tool to predict the favorability of the alternative reaction modes within one reactivity index. In the present work, using Bader’s theory Atoms-in-Molecules, we have studied the ellipticities of the C = C bonds in the molecules of fullerene adducts C60Xn (n = 1–3) with symmetric addends X = CH2 and O. We have obtained the correlations between the thermodynamic stability of the fullerene adducts and bond ellipticities in the molecular of their precursors. The found regularity may be further used for predicting the reactivity of fullerene derivatives and estimating of regioselectivity of the cycloaddition reactions.
Research Article
Bader’s bond ellipticity as a reactivity index for stepwise [2+1]-cycloaddition to the C60 fullerene
https://doi.org/10.21203/rs.3.rs-3618335/v1
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ellipticity
AIM analysis
fullerene
cycloadduct
reactivity index
density functional theory
The C60 (Ih) fullerene is a molecular truncated icosahedron consisting of chemically equivalent carbon atoms [1–3]. There are two types of bonds in C60: longer ordinary 5.6-bonds and shorter double 6.6-bonds. Overwhelming majority of cycloaddition reactions proceed via 6.6-bonds, in particular, the formation of [2 + 1]-cycloadducts C60(CR1R2)n that make up the largest class of fullerene derivatives [4, 5].
The preparative chemistry of fullerenes become more complex and the interest in the synthesis and chromatographic separation of bis-, tris- and multiadducts of C60 increases. In addition to fundamental relevance, these studies aim to produce the fullerene derivatives with the tunable chemical and physical properties. For example, growing number of hydrophilic addends decorating the fullerenes cage of methanofullerenes С60[C(COOH)2]n enhances their solubility [6, 7]. Selected fullerene bisadducts exhibit improved device performances when used in organic solar cells [8–15].
Upon decorating the C60 cage with chemical groups, it loses the original symmetry. As consequence, the reaction sites of cycloaddition (chemical bonds) become inequivalent and the number of regioisomers corresponding to C60Xn increases with n [16–18]. It makes both experimental and theoretical chemistry of fullerene multiadducts complicated. Indeed, such multiadducts are usually synthesized as hardy separable mixtures [19, 20] and the works succeeding in the isolation of whole sets of the formed fullerene adducts are rare [21, 22]. In the theoretical aspect, computational description of the reactions leading to C60Xn requires the calculations of numerous heat effects and activation energies [18, 23–27]. The use of the reactivity indices to escape the necessity of the computational thermochemistry seems very attractive in this aspect. The reactivity indices for predicting the chemical properties of fullerenes themselves are well explored (the list of the used indices includes bond orders, curvature, polarizability indices, pyramidality angles, energies of frontier orbitals, topological descriptors etc.) [1, 27–33]. However, the applicability of these indices to fullerene derivatives was limitedly studied in a few works [34–37].
In original C60, all double bonds belong to one type. Monoadducts C60X with symmetric addend X have 9 inequivalent double bonds. In regioisomeric bisadducts C60X2, their number increases. The conventional designations of the bonds in C60X are shown in Fig. 1.
Atoms-in-Molecules (AIM) theory designed by Richard Bader provides powerful quantum-chemistry instruments for calculating molecular properties that could be further used for predicting reactivity [38–41]. According to the Bader’s theory [38, 39], the electron density function ρ(r) and set of critical points (points in which gradient ∇ρ(r) = 0) completely determines the structure of a molecular system. Depending on the sign of major components relating to the electron density’s curvature, there are several types of its critical points. The most important points for analyzing chemical structure lie on the bonding paths connecting atoms in a molecule and are called bond critical points (BCP). In terms of AIM theory, these points are designated as (3, − 1) that means that the corresponding curvature of the electron density has two negative (λ1, λ2 < 0) and one positive (λ3 > 0) eigenvalues. The presence of BCP between two atoms indicates their chemical bonding.
The question about the usability of AIM theory for fullerene multiadduct studies have not been explored. Therefore, in the present work, we have scrutinized the correlations between the bond ellipticities, computed in terms of Bader’s theory, and stabilities of the formed [2 + 1]-adducts. For this purpose, we select model compounds widely used in the theoretical chemistry of fullerenes [35, 42], viz. C60Xn (n = 1–3) whereby X designated divalent oxygen or CH2 moiety (Fig. 2). Compounds C60On and C60(CH2)n are called fullerene epoxides and cyclopropafullerenes, respectively.
Density functional theory method based on the combination of Perdew–Burke–Ernzerhof functional [43] and 3ζ valence-split basis set [44] implemented in program Priroda [45] were used in the study. This method correctly reproduces geometries, energetic and molecular properties of fullerenes, their derivatives, and related carbon-rich molecules [46–55]. The structures of the molecules were completely optimized. All computed molecular geometries correspond to the minima of the potential energy surfaces. The total energy hessians calculated for these compounds do not contain imaginary frequencies.
To predict the thermodynamic favorability of the fullerene core decoration, we analyzed the molecules of adducts C60X and bisadducts C60X2 (X = CH2, O) in terms of AIM theory as implemented in the abovementioned program Priroda. For this purpose, we performed the AIM topographical analysis of electron density of the molecules and used the numerical parameters of BCPs to calculate the ellipticities of chemical bonds. Electron density in each BCP is characterized with three eigenvalues of the hessian, conventionally designated as λ1, λ2, and λ3. The λ3 component is positive and describes the behavior of the electron-density function in the direction of the bonding path whereas λ1 and λ2 are negative and correspond to the directions orthogonal to it. The ellipticity of a chemical bond (ε) is deduced from its parameters |λ2| < | λ1| [56]:
$$\epsilon =\frac{{\lambda }_{1}}{{\lambda }_{2}}-1$$
1
The relative thermodynamic stability of the molecules in the series of regioisomeric fullerene adducts was estimated with the relative energy values Ei calculated as:
$$\varDelta {E}_{i}={E}_{i}-{E}_{\text{m}\text{i}\text{n} }$$
2
where Emin is energy of most stable compound in a series (that is the isomer with the minimal total energy). The quantum chemical calculations of the Ei values were included zero-point vibration contributions. Note that in the С60(СH2)2 and C60O2 series, isomers cis-1 are found most stable that agrees with the previous studies [13, 24, 25].
According to AIM theory, a molecular structure is characterized with the set of critical points. Each critical point is defined with the pair of numbers (ω, λ), where ω is a rank of the matrix of second derivatives ρ(r) computed in a critical point (or number of nonzero λi, i = 1–3), and λ is the signature (algebraic sum of signs λi). Values λ1, λ2, λ3 represent eigenvalues of matrix of second derivatives ρ(r) computed in a critical point.
The presence of critical point (3, − 1) indicates the chemical bonding in the molecular structure. Negative eigenvalues λ1 and λ2 at this point correspond to the eigenvectors, which are perpendicular to the bonding path, and positive eigenvalue λ3 is directed along the path. Accordingly, values λ1 and λ2 characterize a contraction ratio of electron density along the bonding path. The number λ3 determines a degree of displacement of electron density to atomic nuclei. From this point of view, ellipticity indicates the contribution of the π-component to the bonding between atoms, i.e. it is actually an analogue of the bond order [56]. Chemical bonds with cylindrical symmetry are expected to have λ1 ≈ λ2. Ethane, benzene, and ethylene are considered as golden standards in computational chemistry. The ellipticities of their carbon–carbon bonds are 0.0000, 0.2836, and 0.1783, respectively (calculated with the PBE/3ζ method). Thus, chemical bonds with pronounced π-component demonstrate high ε values.
For the 6.6- and 5.6-bonds of the C60 molecule, we obtained the following a sets of eigenvalues λ6.6 = (–0.6242, − 0.5182, 0.3157) and λ5.6 = (–0.5474, − 0.4754, 0.3921). The calculation of the ε values via Eq. (1) gives = 0.20458 and ε5.6 = 0.15134. Ellipticity reflects the partial delocalization of the π-electron density: the ε6.6 value fits into the interval between typical double and aromatic (delocalized) double bonds whereas ε5.6 lies out this range that corresponds to a more pronounced single-bond character. Thus, 6.6 bonds explicit double-bond properties and this agrees with the experimental observations, in which major chemical reactions proceed via attaching addends to the 6.6 bonds of the C60 core. In contrast, the addition to the 5.6-bonds of C60 in most cases does not occur or requires special conditions [4, 5, 20]. Herewith, 5.6 mono- and bisadducts of C60 are unstable and prone to undergo the isomerization to their 6.6-counterparts [20]. This is in good agreement with the Mulliken bond orders: p6.6 = 1.44 and р5.6 = 1.23 (estimated with PBE/3ζ method [23]).
We consider stepwise cycloadditions to the C60 core according to Fig. 2. As follows from the scheme, the number of regioisomers increases in each step being 1, 8, and 45 for C60X, C60X2, and C60X3, respectively (cf. [24]).
In the C60CH2 and С60O molecules, there are 8 inequivalent 6.6 bonds; their ellipticities vary in the narrow ranges: 0.1944…0.2188 and 0.1938…0.2169, respective;y (Table 1). However, these small differences are decisive and the ε values well correlate with the relative energies of regioisomeric С60O2 and C60(CH2)2 (Fig. 3). The larger bond ellipticity in C60X, the more stable the formed bisadduct C60X2.
| Monoadduct | Bond type | λ1 | λ2 | λ3 | ε | ∆E, kJ/mol |
|---|---|---|---|---|---|---|
| C60CH2 | cis-1 | –0.6276 | –0.5149 | 0.3069 | 0.2188 | 0.00 |
| cis-2 | –0.6251 | –0.5234 | 0.3199 | 0.1944 | 22.70 | |
| cis-3 | –0.6182 | –0.5146 | 0.3205 | 0.2012 | 16.08 | |
| e | –0.6233 | –0.5181 | 0.3167 | 0.2031 | 7.60 | |
| trans-1 | –0.6248 | –0.5189 | 0.3154 | 0.2039 | 10.22 | |
| trans-2 | –0.6226 | –0.5172 | 0.3169 | 0.2039 | 8.43 | |
| trans-3 | –0.6232 | –0.5175 | 0.3165 | 0.2043 | 6.99 | |
| trans-4 | –0.6247 | –0.5189 | 0.3157 | 0.2038 | 11.05 | |
| C60O | cis-1 | –0.6329 | –0.5201 | 0.3066 | 0.2169 | 0.00 |
| cis-2 | –0.6242 | –0.5228 | 0.3188 | 0.1938 | 29.44 | |
| cis-3 | –0.6188 | –0.5155 | 0.3199 | 0.2004 | 24.64 | |
| e | –0.6259 | –0.5190 | 0.3154 | 0.2060 | 17.05 | |
| trans-1 | –0.6248 | –0.5188 | 0.3158 | 0.2043 | 19.78 | |
| trans-2 | –0.6230 | –0.5175 | 0.3164 | 0.2038 | 18.66 | |
| trans-3 | –0.6232 | –0.5175 | 0.3169 | 0.2043 | 18.02 | |
| trans-4 | –0.6248 | –0.5189 | 0.3153 | 0.2039 | 19.83 |
According to the calculations, the electronic structure of the C60 core is quite complex. Indeed, there is a pandemonium of the points in the ∆E vs. ε plot at the center of the correlation field corresponding to the bonds e, trans-4, trans-3, trans-2, and trans-1 (Fig. 3). All these bonds are located in the fullerene hemisphere opposite to that one having already functionalized bond. In contrast, the ellipticities of the bonds of the same fullerene hemisphere decisively differ though the bonds lie close to each other. Moreover, the closest bonds cis-1 and cis-2 demonstrate the highest and the lowest reactivity in the whole set: the formed cis-2-C60X2 adducts are > 20 kJ/mol less stable as compared with cis-1-C60X2. Thus, the perturbation of the electronic system induced by the addition of the first addend is more pronounced in the same hemisphere and vice versa.
Note that the higher reactivity of the cis-1 bond in C60X is the experimentally known fact confirmed for various non-bulky X (so that the addition of the X groups to the fullerene core meet no steric hindrances) (cf. [4, 12, 17, 18, 24, 25, 27]).
We tried to obtain a similar plot in the case of the next step of the X addend attachment to C60, i.e. for conversion C60X2 → C60X3. As found, eight bisadducts must be separately treated to obtain the correlations like in the case of the C60X → C60X2 conversion. The calculated Bader’s parameters of the bonds in the C60X2 molecules and relative stabilities of C60X3 are collected in Supplementary materials. The ∆E vs. ε plots are presented in Figs. 4–7. Note that the plots contain different number of points as the number of trisadducts C60X3, formed from the certain C60X2, depends on the its symmetry. The optimized geometries of C60X3 correspond to the following symmetry point groups: Cs symmetry is found for cis-1, cis-2, e and trans-4 isomers; C2 symmetry is inherent in cis-3, trans-2 and trans-3; trans-4 isomers are D2h-symmetric. For example, trans-1-C60X2 (X = O and CH2) molecules have the highest symmetry in the set. Therefore, their molecules have only 5 inequivalent reaction sites, a minimal number among all C60X2 regioisomers.
As in the case of the previous reaction step, the plots corresponding to each transformation C60X2 → C60X3 demonstrate that the relative stabilities of C60X3 and bond ellipticities in the C60X2 molecules are symbatic values. Though the dependences ΔE = f(ε) have the quality of correlations, they allow revealing most and least stable C60X3 isomers formed from the considered C60X2 bisadduct. We have collected the C60O3 “extremal” structures in Table 2. Note that the addition patterns in the case of the most and least stable C60(CH2)3 isomers are the same. As the similar addition patterns are obtained for different addends (polar O and nonpolar CH2), we assume that the found regularity could be used without quantum-chemical calculations in the case of the fullerenes derivatives with other addends.
In the present work, we have calculated the bond ellipticities in the fullerene adduct molecules. The larger ellipticity, the closer properties of the chemical bond to the typical double C = C bond that facilitates the cycloaddition reaction via this reaction site. In the case of the monoadducts C60X, the bonds with strikingly different ellipticities lie in the same hemisphere with the attached addend X. In contrast, the differences between the ε values of the bonds of the “non-functiionalized” part of the fullerene moiety are in the narrow range. This indicates the influence of the attached addends on the reactivity of the 6.6-bonds.
We found a regularity between these estimates and the thermodynamic stability of the fullerene adducts formed at the next stage. The larger bond ellipticity in C60X (C60X2), the higher stability of formed C60X2 (C60X3). The found regularities have the quality of correlations, but they at least allow defining most and least stable regioisomers.
The found regularity may be further used for predicting the reactivity of fullerene derivatives and estimating of regioselectivity of the cycloaddition reactions.
Author Contributions DSS: conceptualization, writing-original draft, analysis of the computed values. AAT: writing-original draft, quantum-chemical computations. ISS: quantum-chemical computations. DSS: conceptualization, writing-original draft, analysis of the computed values, funding acquisition. All authors reviewed the manuscript.
Funding This research was funded by the Russian Science Foundation, project “Information entropy of chemical reactions: A novel methodology for digital organic chemistry”, grant number 22-13-20095.
Conflict of interest The authors declared there were no fnancial interests or individual relationships to impact the published paper.
Supplementary Information The online version contains supplementary material available at REF.
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No competing interests reported.