Polarizability as a landmark property for fullerene chemistry and materials science

Abstract

The review summarizes data on dipole polarizability of fullerenes and their derivatives, covering the most widespread classes of fullerene-containing molecules (fullerenes, fullerene exohedral derivatives, fullerene dimers, endofullerenes, fullerene ions, and derivatives with ionic bonds). These are currently presented by experimental and mainly theoretical works. Particular attention is paid to the analysis of the computational data in terms of additive schemes that assist in understanding the changes in polarizability upon fullerene functionalization and provide a general formula for calculation of polarizability for certain classes of the exohedral derivatives. Additionally, application of polarizability to the physical and chemical problems of fullerene science is discussed. It includes aspects of fullerene reactivity, physicochemical processes in carbon nanostructures (quenching of electronically-excited states, nanocapillarity, etc.) as well as use of fullerene adducts as electron-acceptor materials for organic solar cells and molecular switch devices.

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Denis Sh. Sabirov

Denis Sh. Sabirov was born in 1984 in Magnitogorsk (Russia). In 2009, he received his PhD in the fields of physical, mathematical, and quantum chemistry in Bashkir State University (Ufa, Russia). Currently, he is a senior researcher in the Institute of Petrochemistry and Catalysis of Russian Academy of Sciences. His current interests involve applications of DFT to chemical and physical processes in carbon nanostructures with focus on their polarizability.

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1 Introduction

Fullerenes are undoubtedly the tangible embodiment of abstract beauty. Indeed, researchers working in the field of fullerene science usually point out the perfect shapes of their molecules. However, this is not the only reason why fullerenes attract. Due to their chemical structure, fullerenes and their derivatives have unique properties, promising for diverse applications in materials science, pharmaceutics, and nanotechnology. The astounding history of fullerenes discovery started with theoretical prediction of C₆₀ by Eiji Ōsawa in 1970,¹ semiempirical calculations of C₂₀ and C₆₀ by Bochvar and Galpern in 1974,² and, finally, experimental detection of this compound by Kroto et al. in 1985.³ Thus, from the beginning, fullerene science has been evolving based on relations between theory and experiment.⁴

Theoretical (mainly quantum-chemical) studies currently retain special place in fullerene science. When the buckminsterfullerene was a low available chemical, its properties used to be calculated.⁵ Nowadays, it has become a common compound but not its derivatives or higher/smaller fullerenes. The main difficulty in obtaining pure samples of individual fullerene derivatives deals with non-selectivity of the addition reactions to fullerenes, which is caused by a large number of reactive sites with almost the same reactivity. Therefore, theoretical approaches are developed to preliminarily assess the utility of fullerene derivatives. In this aspect, we should mention computational design of fullerene-based compounds with desirable electronic properties,⁶ static dielectric constants,⁷ polarizability,^8,9 or hyperpolarizability.^10–13 Such approaches allow defining structures of the prospective compounds before the synthetic procedures and focusing on the synthesis of the targeted adducts.

Polarizability (or dipole polarizability) of fullerenes and their derivatives seems to be very important among the mentioned properties because it defines many physical and chemical processes: intermolecular interactions, optical properties (e.g., Kerr effect and Rayleigh light scattering), chemical reactions, and many others.^14–16 Thus, it is highly informative and used in the materials design with advanced properties, e.g., photomodulated systems,¹⁷ molecular functional materials with electron-transfer capability,¹⁸ and compounds with enhanced propensity for supramolecular complexes formation.¹⁹

Studies on fullerene polarizability have started from the first theoretical works of Fowler et al.²⁰ (1990) and Pederson and Quong²¹ (1992). Later, French scientific group has performed the first direct measurements (1999).²² These first works have elucidated that fullerenes are highly polarizable species. It means that analysis of physicochemical processes in fullerene-containing systems should not ignore their polarizability. Currently, the data on the polarizability of diverse fullerenes and their derivatives are presented in periodicals. Experimental studies on C₆₀ polarizability and its clusters have been previously partly reviewed¹⁵ as well as DFT studies on the polarizability of C₆₀ and C₇₀ adducts.⁸ However, many interesting works have remained uncovered. Some of them deal with the relation between the structure and polarizability of fullerene derivatives and its applications to such hot topics of fullerene science as molecular machinery, organic solar cells, and nanomaterials. We think that a review, comprehending these works, should be on time.

The present review is based on two summarizing scientific reports, performed in St. Petersburg (Russia)²³ and Durham (UK),²⁴ and covers experimental and mainly theoretical works in the field of polarizability of fullerenes and their derivatives from 1990s to 2014. It focuses on the numerical data, coupled with the examples of their application. Note that works on the calculation of fullerene polarizability are numerous, so we have tried to include in the review those, which have not lost the relevance by the moment. Thus, the early estimations (e.g., ref. 20, 21 and 25–28) are mentioned here without a detailed consideration. The review is chaptered according to the main types of the fullerene-containing systems (fullerenes, fullerene exohedral derivatives, fullerene dimers, endofullerenes, fullerene ions, and derivatives with ionic bonds) and contains the preceding part where we briefly list necessary basic definitions in the field of polarizability. Additionally, it includes a prospective part, devoted to molecular switch, which can exploit polarizability of different exohedral fullerene derivatives.

2 Basic definitions

Polarizability is a molecular property that describes molecule's ability to acquire induced dipole moment in external electric fields.^14,29 When the molecule is under the electric field E, its induced dipole moment depends on the dipole polarizability α and the high-order polarizabilities (β, γ, etc.):³⁰

In the case of weak fields, μ_ind can be accurately calculated neglecting the high-order terms:

μ_ind = αE (2)

where α is polarizability tensor in arbitrary coordinate system (X, Y, Z):which is symmetric about the main diagonal α_ij = α_ji. It can be considered in the eigen coordinate system (x, y, z) when all the non-diagonal elements become zero:

In classic theory, the values α_xx, α_yy, and α_zz are interpreted as main semiaxes of polarizability ellipsoid of the molecule.²⁹ These are also used for calculation of mean polarizability α and anisotropy of polarizability a²:

These two properties are measurable.^14,29 The trace of tensor α is invariant under coordinate system:

α_xx + α_yy + α_zz = α_XX + α_YY + α_ZZ, (7)

so the diagonal elements of the non-diagonalized tensor (3) are also suitable for calculation of mean polarizability by eqn (5). If the elements from the non-diagonalized tensor are used for calculation of the anisotropy, formula (6) obtains the additional terms:²⁹

Polarizability has the dimension of volume (and expressed in ų or atomic units; 1 a.u. = 0.148 ų) that can be interpreted as a degree of the filling the space by the molecule's electronic cloud. Therefore, molecular systems with a large number of electrons should demonstrate high α values.²⁹

The discussed polarizability is related to the static case of electronic cloud polarization and so-called static electronic polarizability. In the dynamic case, frequency-dependent polarizability α(ω) is considered. Its relation with the static value is roughly described by the following equation:²⁹

where frequency ω₀ characterizes the binding of electrons in the molecule.

Electronic polarizability is directly deducible from experiments if we deal with nonpolar substances. In polar cases, orientational polarizability should be taken into account. The last one depends on the permanent dipole moment of the molecule μ₀ and temperature:

Molecular beam deflection¹⁵ and interferometry^31,32 seem to be the main experimental techniques for determination of polarizability. These methods require significant amounts of substances that is usually impossible in the case of fullerene derivatives, which, in addition, have propensity for dissociation (especially, when irradiated). Two reasons above impose limitations on application of the mentioned experimental techniques to fullerene derivatives. The solution of this problem has come from the computational chemistry (mainly, quantum-chemical methods). Currently, there are many approaches that can be applied to calculation of polarizability of diverse carbon nanostructures. These are based on additivity, dipole interaction, linear response theory, or finite field approach. The overviews of computational techniques, its advantages and disadvantages can be found in ref. 14 and 28. Here, we mention the finite-field approach³³ that has become widespread in fullerene polarizability studies. According to this approach, the elements of the polarizability tensor are calculated as the second order derivatives of the total energy U with respect to the homogenous external electric field E (i.e. the field gradient and higher derivatives are zero):

This approach is well combined with DFT techniques and allows calculating static polarizabilities, which, in general, are enough for chemical and materials science applications.

3 Polarizability of fullerenes and related structures

3.1 Fullerene polarizability

Fullerenes are highly polarizable molecules. This was confirmed by numerous experimental and theoretical methods.^{20–22,34–57} Currently, the experimental data are available only for the C₆₀ and C₇₀ fullerenes whereas polarizabilities of the representatives of the fullerenes family from C₂₀ to C₂₁₆₀ have been calculated in terms of the diverse approaches.^34–57 Among them, DFT-calculations provide the most trustworthy values. We have superposed the most credible data on the selected fullerenes in Table 1. The measured mean polarizability values of C₆₀ and C₇₀ are ∼80 and ∼105 ų, respectively. Note that some experimental works give unreliably high values for C₆₀. For example, such a value (∼1000 ų) has been obtained in ref. 58 and 59 that may be caused by the influence of impurities on the measurements, the unaccounted aggregation of the C₆₀ molecules or possible formation of charge-transfer complexes, which significantly increase polarizability.⁶⁰

Table 1 Static polarizabilities of fullerenes, a comparison between theoretical methods and experimental data (ų)^a

Molecule	Theoretical estimations	Experimental data
a Other summarizing tables can be found in ref. 47, 49 and 54.
C60 (Ih)	78.8 (CPHF/(7s4p)[3s2p]),^35 75.1 (HF/6-31 ++G),^36 83.0 (topological model),^39 77.5 (point dipole interaction model),^40 75.7 (bond order model),^41 82.7 (PBE/3ζ),^45 82.1 (PBE/NRLMOL),^46 82.9 (QSRP model),^49 81.6 (PBE0/SVPD),^51 78.4 (VWN/DZVP/GEN-A2),^52 80.3 (B3LYP/Λ1),^53 78.4 (M06-2X/6-31+G(d,p),^55 71.7 (B3LYP/6-31G(d))^56	76.5 ± 8.0 (molecular beam deflection),^22,38 79.0 ± 6.0 (time-of-flight technique),^37 88.9 ± 6.0 (interferometry)^42
C70 (D5h)	93.2 (CPHF/(7s4p)[3s2p]),^35 89.8 (HF/6-31 ++G),^36 88.3 (bond order model),^41 103.0 (PBE/NRLMOL),^43 102.7 (PBE/3ζ),^44 93.6 (QSRP model),^49 97.8 (VWN/DZVP/GEN-A2),^52 100.7 (B3LYP/Λ1)^53	101.9 ± 13.9 (molecular beam deflection),^38 108.5 ± 8.2 (interferometry)^42
C76 (D2)	112.3 (PBE/3ζ)^50	—
C78 (D3)	115.3 (PBE/3ζ)^50	—
C80 (Ih)	129.4 (PBE/3ζ)^57	—
C84 (D2d)	124.1 (PBE/3ζ),^50 113.3 (CPHF/(7s4p)[3s2p])^35	—
C90 (C2v)	135.8 (PBE/3ζ)^57	—
C100 (D5)	169.0 (PBE/3ζ)^57	—
C120 (Td)	189.8 (PBE/3ζ)^57	—
C240 (Ih)	441.0 (PBE/NRLMOL)^48	—
C540 (Ih)	1192.6 (PBE/NRLMOL),^46 1243.8 (PBE0/SVPD),^51 1254.0 (VWN/DZVP/GEN-A2)^52	—

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Comparison of the most widespread density functionals PBE and B3LYP, applied to the fullerenes, demonstrates the higher efficiency of the first one for calculating α because the B3LYP-based methods underestimate the C₆₀ polarizability (see, e.g., ref. 53 and 56) (Table 1). This is also true for M06-2X density functional and the long-range corrected schemes (e.g., LC-wPBE).⁶¹ The use of the mentioned functionals with the extended basis sets allows avoiding the underestimation. For example, in the case of M06-2X density functional, the extension of the basis set from 6-31G(d) to 6-31+G(d,p) leads to the increase in the calculated value of the C₆₀ mean polarizability from 67.3 (ref. 61) to 78.4 (ref. 55) ų, respectively. In our studies, we prefer the PBE/3ζ method that quantitatively reproduces experimental values of the polarizability of fullerenes, as well as the spectral data (IR and NMR) of C₆₀/C₇₀ and their derivatives.^45,62–65 In addition, semiempirical methods, specially parameterized to reproduce molecular properties of polycyclic hydrocarbons and fullerenes, are developed.⁴⁷

For example, PM6 calculations⁴⁷ have been used to obtain good correlations of dynamic polarizabilities of C₆₀ and C₇₀ with ω (all values in a.u.):

α_C60 = 530(1 + 13ω² + 1020ω⁴) (12)

α_C70 = 645(1 + 19ω² + 5832ω⁴) (13)

High mean polarizabilities of fullerenes are obviously caused by their rich π-electronic systems. Polarizability nonlinearly increases with the number of carbon atoms in the fullerene molecule and, consequently, with the number of readily polarizable π-bonds. This can be clearly demonstrated by polarizability per atom values α/N_C that grow up from 1.368 and 1.471 ų for C₆₀ and C₇₀ to 2.209 ų for C₅₄₀ (mean polarizabilities calculated with PBE/NRLMOL from ref. 43 and 46). It is noteworthy that the analogous nonlinear enhancement of α is typical for one-dimensional and two-dimensional π-conjugated systems such as polyenes,⁶⁶ polyynes,⁶⁷ oligo[n]acenes,⁶⁸ and some conjugated oligomers.⁶⁹ Thus, fullerenes behave similar to the conjugated hydrocarbons in the context of the mean polarizability. To avoid misunderstanding, we should also mention a theoretical work that declares quantum-size effects for polarizability of the fullerenes family.²⁷ This work deals with the relation of two semiempirical estimations of polarizability, which ratio grows up to the critical size of the fullerene molecule and then diminishes. It does not mean that polarizability analogously behaves as it permanently increases with the fullerene size.

In addition, the fullerenes family has clear dependence of mean polarizability per atom on the size. It makes fullerenes outstanding among the other nanostructures, which usually do not demonstrate such trends (see, e.g., DFT study on the gallium arsenide clusters⁷⁰).

As predicted by the point dipole interaction model,⁷¹ the dependence of carbon nanotube polarizability on its size has the saturation length. When the saturation value is achieved, no significant changes of mean polarizability and the longitudinal polarizability of the nanotube are observed. This differs from the case of fullerenes and polycyclic aromatic hydrocarbons though carbon nanotubes have almost the same structure.^66–69 To be strictly stated, this difference between fullerenes and nanotubes should be studied by higher level theoretical (e.g., DFT) or experimental methods.

3.2 Fullerene polarizability and physicochemical processes

Estimation of intermolecular interactions is the nearest application of the measured and calculated polarizabilities of fullerenes. For example, the dispersion interaction is a universal interaction that arise between the molecules. Its energy can be calculated as:where C₆ is a dispersion interaction constant.^14,29 It is calculable in terms of different approaches. Most of them state the dependence of C₆ on the polarizabilities of the participants of interaction (C₆ ∼ α₁, α₂). Thus, the accurate calculation of C_disp has been performed by Kumar and Thakkar⁷² via Casimir–Polder equation (Table 2), in which the imaginary parts of polarizability (i² = −1) are used:

Table 2 Dispersion interaction coefficients C₆ for C₆₀⋯X interactions, a.u. Reprinted with permission from ref. 72© 2011 Elsevier

X	C6	X	C6
a The average of two estimations is given.
H	801.7	Propanol-1	9841
He	364.7	H2CO	4052
Ne	737.3	CH3CHO	6321
Ar	2511	(CH3)2CO	8885
Kr	3592	SF6	7343
Xe	5362	SiH4	5850
Li	8066	SiF4	5561
H2	1098	NH2CH3	5499
N2	2674	NH(CH3)2	8031
O2	2434	N(CH3)3	1.029 × 10^4
Cl2	6230	C2H4	5479
HF	1341	Propene	8135
HCl	3604	Butene-1	1.063 × 10^4
HBr	4654	CCl4	1.421 × 10^4
CO	2834	CH4	3593
CO2	3938	C2H6	6165
NO	2605	C3H8	8745
N2O	4269	n-C4H10	1.124 × 10^4
C2H2	4519	n-C5H12	1.377 × 10^4
O3	4111^a	n-C6H14	1.624 × 10^4
SO2	5399	n-C7H16	1.873 × 10^4
CS2	9300	n-C8H18	2.122 × 10^4
SCO	6347	O(CH3)2	7284
H2S	4652	CH3C3H7	1.248 × 10^4
H2O	2110	O(C2H5)2	1.247 × 10^4
NH3	2982	C6H6	1.313 × 10^4
CH3OH	4690	C60	1.003 × 10^5
C2H5OH	7290

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The obtained numerical data⁷² make up valuable source for comparative estimation of van der Waals interactions of the buckminsterfullerene.

High α values of fullerenes have been used to qualitatively explain the distinctiveness of physicochemical processes in fullerene-containing systems such as the anomalously effective quenching of electronically-excited states of organic compounds by C₆₀ and C₇₀,⁴⁴ propensity of fullerenes for aggregation,⁷³ formation of the donor–acceptor complexes,⁷⁴ and behavior of the atoms, encapsulated by fullerene cages⁷⁵ (including their photoionization⁷⁶). For example, Bulgakov and Galimov have discovered that C₇₀ is substantially more effective quencher from electronically excited states owing to energy transfer than C₆₀.⁴⁴ In these processes, C₆₀ and C₇₀ accept energies of the excited states of organic molecules according to inductive resonant mechanism:

C₆₀ + D^* → [C₆₀⋯D]^* → C^*₆₀ + D (16)

C₇₀ + D^* → [C₇₀⋯D]^* → C^*₇₀ + D (17)

According to the conventional notions,⁷⁷ each of the interactions above is considered as a resonant interaction of two oscillators (energy donor and energy acceptor). Donor and acceptor perturb the electronic structures of each other. Therefore, the different deactivating capabilities of C₆₀ and C₇₀ are explained with the unequal energies of the dipole–dipole interaction W:

where μ_D and μ_A are dipole moments of the donor D and acceptor A molecules and R is intermolecular distances. In comparative experiments,⁴⁴ the donors (e.g., polycyclic hydrocarbons) were the same in the pairs C₆₀⋯D and C₇₀⋯D; i.e. μ_D and R were constant in eqn (18). Thus, the difference in quenching is a consequence of the different dipole moments of C₆₀ and C₇₀. These are both equal to zero in the ground state. However, in eqn (18), we should operate with the dipole moments of the excited states, which are hardly computable for such large molecules as fullerenes. We have proposed that the excited state dipoles are correlated with the dipoles, induced by external electric fields. The last ones, according to eqn (2), depend on polarizability. Numerous theoretical and experimental works demonstrate that C₇₀ is more polarizable than C₆₀. Thus, the larger efficiency of the C₇₀ fullerene as a quencher is attributed to the higher mean polarizability of its molecule.⁴⁴ We can expect that the larger fullerenes, having higher mean polarizabilities, may exhibit higher capabilities for quenching.

Polarizability is also useful for estimation of the conditions for ordering carbon nanostructures (higher fullerenes and nanotubes) in a polymer matrix under external electric fields.⁷⁸ The authors⁷⁸ have deduced formulae for estimation of time t_F, required for rotation of carbon nanostructure on the angle ϑ in the field E:

and the condition for the electric field magnitude, required to perform ordering:where r is radius of the sphere with the volume, equal to the volume of the considered carbon nanostructure, and η is viscosity of the medium. Use of these conditions demonstrated that polarizabilities of C₇₀ and C₈₂ are insufficient to perform their ordering in low electric fields. In the case of nanotubes, which are more polarizable than fullerenes, the field 10⁵ V cm⁻¹ is able to rotate them on the angle ϑ = 60° in polymer medium.⁷⁸

3.3 Fullerene polarizability versus fullerene reactivity

Analysis of reactivity of the fullerene molecules makes up another application of their polarizabilities. Two reasons underlie the usefulness of polarizability for this purpose.⁷⁹ First, the buckminsterfullerene readily forms various molecular complexes with⁷⁴ or without⁸⁰ charge transfer. In addition, quantum-chemical studies of potential energy surfaces of 1,3-dipolar cycloaddition to C₆₀ show that it occurs through the pre-reactionary complexes (for example, in the case of ozone addition to C₆₀ (ref. 45, 81 and 82) and C₇₀ (ref. 45)). Moreover, a key role of dispersion interaction for C₆₀⋯O₃ stabilization has been demonstrated.⁸³ High polarizability of C₆₀ also defines stability of analogous complexes with other species. Moreover, the enhanced stability of van der Waals complexes is able to prevent the further chemical interaction when both of the interacting molecules are highly polarizable (e.g., C₆₀–iodine⁸⁴ and C₆₀–sulphur complexes⁸⁵).

Second, some classic concepts consider eigenvalues the polarizability tensor as semiaxes of polarizability ellipsoid, which covers the molecule and roughly represents its electronic cloud.²⁹ In the case of fullerenes, such an ellipsoid replicates the shape of a fullerene molecule.⁸⁶ To use polarizability for theoretical study of chemical properties, we have considered the fullerene molecule and its polarizability ellipsoid together in a polar coordinate system (with the origin at the center of mass of the fullerene) (Fig. 1). This allowed assigning a point on the ellipsoid and ascribing the index ξ to each reaction site of the molecule:^50,86

where ξ is the polarizability in the direction of the reaction site with polar coordinates ψ and φ. In the case of the highly symmetric C₆₀ molecule, all the indices are the same, being equal to its mean polarizability.

Fig. 1 Polarizability ellipsoid of the fullerene molecule. O is a center of mass, C is an atom on the fullerene surface, ξ is a point on the polarizability ellipsoid.⁸⁶

This approach have become useful for understanding the modes of ozone and diazomethane addition to higher fullerenes C₇₀ (D_5h), C₇₆ (D₂), and C₇₈ (C_2v),⁵⁰ which have unequivalent double bonds in their structures in contrast to C₆₀. As is known, dipole molecules O₃ and CH₂N₂ react with 6.6 bonds of higher fullerenes that results in the formation of fullereno-1,2,3-trioxolanes and fulleropyrazolines (Fig. 2).^87–90 The ξ indices are characterized atoms in the fullerene molecules. Additions of O₃ and CH₂N₂ to alkenes concertedly occur at two reaction sites,⁹¹ so the polarizability indices of the bond Ξ have been calculated as the arithmetic mean:

Ξ = 0.5(ξ₁ + ξ₂) (22)

Fig. 2 Addition of ozone and diazomethane to the most reactive ab bonds of C₇₀.

For all the fullerenes, the calculated Ξ indices are within the range α^(min)_ii ≤ Ξ ≤ α^(max)_ii, characteristic for each fullerene (α^(min)_ii and α^(max)_ii are the smallest and highest eigenvalues of the polarizability tensor). As it turned out, the DFT-calculated heats of reactions of each fullerene increase with the Ξ value. In the case of the C₇₀ fullerene, the ab and cc bonds, located near the poles of the molecule, have the largest Ξ indices (107.6 and 103.0 ų, respectively) (Fig. 3). The heats of the reactions of 1,3-dipolar addition to these bonds are higher than those of the addition to the de and ee bonds. The ab and cc modes of addition are characterized with the lowest activation barriers and the corresponding trioxolanes ab-C₇₀O₃ and cc-C₇₀O₃ have been experimentally detected among the products of the C₇₀ ozonolysis.⁹⁰ It is important that Ξ indices of the ab and cc bonds exceed mean polarizability of C₇₀ whereas in the case of the inert de and ee bonds, these are lower than that. Ξ-criterion of reactivity Ξ > α works well in the case of the other fullerenes.

Fig. 3 Correlations of the heats of addition of diazomethane (1) and ozone (2) to C₇₀ and the polarizability indices of the 6.6 bonds. Adapted with permission from ref. 50© 2009 Springer.

The advantage of this approach is that both theoretical and experimental data on fullerene structure and polarizability are suitable for calculation of polarizability indices.⁷⁹ Later, we have extrapolated this approach to oxidation of the C₆₀ derivatives C₆₀O and C₆₀F₁₈, which also have different bonds in the structure.⁹² Unfortunately, analysis of polarizability tensor is not the optimal way for theoretical studies of chemical properties of fullerenes and their derivatives. The calculated ξ and Ξ indices allow considering the reaction sites of each fullerene separately,⁹³ i.e. these do not provide opportunities to cover the reactivity of the whole fullerenes family by the only correlation (as it is possible in the case of curvature indices and pyramidality angles^{63,79,94–96}). Another disadvantage of this approach is ignoring the type of the adding particle. Thus, the addition of labile intermediates (radicals⁹⁷ and carbenes⁹⁸) to the C₇₀ fullerene takes place through atoms or bonds, which are “disfavored” in terms of polarizability (and curvature).

3.4 Polarizability of the related carbon nanostructures, reactions therein and nanocapillarity

Unusual polarizability of fullerenes has triggered analogous studies of the related carbon nanostructures, which are inorganic fullerenes,⁹⁹ fullerenes with defects,¹⁰⁰ heterofullerenes,^100,101 carbon^71,102–104 and inorganic^105–107 nanotubes. Thus, the brightest examples of the use of their polarizabilities for understanding of physicochemical phenomena should be mentioned in a context of fullerene materials science.

As is known, nanotubes can play role of the thinnest capillaries, which can be filled with guest atoms, ions, or molecules. For example, open carbon nanotubes have been filled with molten AgNO₃ (ref. 108 and 109) using capillarity forces. The process of filling has been analyzed in terms of the approach that links wetting with polarizability.¹¹⁰ If van der Waals forces dominate the interface interaction (chemical interaction and/or charge transfer are absent), the contact angle of wetting θ_c depends on the polarizabilities of the wetting liquid α_L and wetted solid α_S:

Consequently, the condition for wetting θ_c < 90° is achieved when α_L > 2α_S.^108,110 The authors¹⁰⁸ have also provided the expression for estimation of the polarizability of inner cavities α_cav of the curved graphitic surface (it is suitable for carbon nanotubes and may be extrapolated to fullerenes):

α_cav = α_gr(1 − 0.0275θ_P), (24)

where α_cav is the planar graphitic polarizability and θ_P is pyramidality angle, which defines the diameter of the nanotube. Combination of eqn (23) and (24) allows predict the appropriate diameter for nanotube, which may be facilely filled with desirable compounds by capillarity forces.^108,109 Note that Pederson et al.^111,112 have computationally studied polarizabilities of carbon nanotubes and predicted their use as molecular straws before the discussed experiments.

Inner cavities of carbon nanotubes provide additional opportunities of carrying out chemical reactions in the nano-sized reactors.^113–118 When such reactions in the confined spaces are quantum-chemically studied, one should take into account the influence of the carbon framework on the reaction paths. The use of polarizable-continuum models¹¹⁹ is the easiest way to do that. Such models require knowledge of dielectric permittivity of the medium ε, that is, in our case, a nanotube. To perform quantum chemical study on the Menshutkin reaction inside (8,0) and (9,0) carbon nanotubes within this approach, the authors¹¹³ have estimated ε by Clausius–Mossotti formula:

where ρ is a density, which is also a computable property. In addition, the works denote the influence of the polarizability of the nanotube reactor on the chemical behavior of the encapsulated reactants.^113,114,118

Currently, processes of “nano-wetting” and chemical reactions inside fullerenes are mainly hypothetical. However, approaches to make them possible are rapidly developed. For example, molecular surgery¹²⁰ consists of chemical opening the cages, putting a desirable molecule inside, and restoring the initial structure of the cage. Endofullerenes H₂@C₆₀, H₂O@C₆₀, and analogous compounds has been successfully produced by this methodology.^120–122 In this context, carrying out chemical reactions inside higher fullerenes and capillarity-assisted filling them are possible in the nearest future.

4 Polarizability of exohedral derivatives with simple (non-fullerene) addends

4.1 Polarizability of fullerene monoadducts

Polarizability of fullerene derivatives has been studied mainly by theoretical techniques^{8,53,56,92,123–137} because of many reasons such as their possible dissociation under experiments, low solubility, and low availability. The only experimental work in this field deals with the optical polarizabilities of polyfluoro[60]fullerenes measured by Tau–Talbot–Kapitza–Dirac interferometry.¹³⁸ As is known from the theoretical studies, mean polarizabilities of exohedral fullerene derivatives are higher than the polarizability of the original fullerene if its carbon framework does not degrade upon chemical functionalization.^{8,53,56,92,123–137} This is true for both mono- and polyadducts, produced by [1 + 1]-addition or [2 + n]-cycloaddition to C₆₀,^{8,53,56,92,123,125–128,131,132,134,135} C₇₀,^{53,131,133,136,137} C₅₀,¹²⁴ or C₅₆ (ref. 130) fullerenes. Mean polarizabilities of the typical C₆₀ monoadducts are listed in Table 3.

Table 3 Mean polarizabilities of C₆₀ monoadducts, calculated by DFT methods (ų)

Molecule	α (method and reference)
a α(C60) = 71.7 Å^3, calculated with the same method.b Calculated specially for the review with the finite-field methodology, applied to PCBM previously.^8 Chosen for this purpose, the PBE/3ζ method gives heat effect of the transformation PCBM → iso-PCBM equal to −37.4 kJ mol^−1. This is in perfect agreement with the previous estimations of the relative stabilities (iso-PCBM is 41.7–44.5 kJ mol^−1 more stable than PCBM according to the quantum-chemical calculations with DFT and Møller–Plesset perturbation methodologies).^139
R = R′ = H	85.0 (PBE/3ζ)^131
R = R′ = COOH	91.7 (PBE/3ζ)^8
R = C6H5, R′ = (CH2)3COOCH3 (PCBM)	108.4 (PBE/3ζ),^8 85.6 (B3LYP/3-21G*)^126
R = R′ = H	89.0 (PBE/3ζ)^135
R = CH3, R′ = H	77.3 (B3LYP/6-31G*)^125
R = CH3, R′ = –p-C6H4NH2	96.5^a (B3LYP/6-31G(d))^56
R = CH3, R′ = –p-C6H4NO2	100.1^a (B3LYP/6-31G(d))^56
R = CH3, R′ = –p-C6H4NO	100.8^a (B3LYP/6-31G(d))^56
	99.1 (PBE/3ζ)^135
	109.0^b (PBE/3ζ)

---

It is well-known that [2 + 1]-addition to C₆₀ and C₇₀ leads to two types of adducts, viz. 6.6-closed (addition to 6.6 bond) and 5.6-open (addition to 5.6 bond with its simultaneous cleavage) (Fig. 4).⁸⁹ 5.6-Open derivatives are usually formed in a mixture with their 6.6-closed isomers in the same reactions and then convert to them spontaneously or under thermal treating^89,140 (here, we do not consider 6.6-open fullerene adducts; in general, these unexpected products of addition are unstable¹⁴⁰). In the case of C₆₀ adducts, the mean polarizability of a 5.6-open isomer exceeds the polarizability of their 6.6-closed counterpart on ∼0.5 ų (Table 4). It is explained by the contribution of π-electronic system to polarizability of the studied molecules: because all 6.6 double bonds remain unbroken in 5.6-open derivatives (i.e. the initial π-electronic system does not change significantly), these compounds have higher α values than respective 6.6-closed isomers.

Fig. 4 5.6-Open–6.6-closed isomerization of fullerene adducts. The reverse reaction is hypothetical and shown by dashed arrow.

Table 4 Mean polarizabilities and their splits for C₆₀X_n (n = 1 and 6) with 5.6-open and 6.6-closed moieties (in ų; PBE/3ζ calculations)

Fullerene adduct	Mean polarizability		Δα split
	α5.6-open	α6.6-closed
a Taken from ref. 131.b Hexakisadducts with uniform distribution of addends on the fullerene cage. Taken from ref. 9.
C60O^a	83.9	83.2	0.7
C60NH^a	84.8	84.2	0.6
C60CH2^a	85.5	85.0	0.5
C60O6^b	89.3	85.2	4.1
C60(NH)6^b	95.6	90.7	4.9
C60(CH2)6^b	99.3	94.8	4.5

---

This regularity is also typical for 6.6-closed and 5.6-open monoadducts of C₇₀ (Fig. 5). This has been numerically demonstrated on the example of its epoxides (6.6-closed) and oxafullereoids (5.6-open), which can be produced in a mixture at the C₇₀ liquid-phase ozonolysis.⁹⁰ According to PBE/3ζ calculations,¹³¹ all the C₇₀O oxafulleroids have the larger mean polarizabilities (Fig. 5).

Fig. 5 Epoxides and oxidoannulenes, produced by the C₇₀ ozonolysis. Mean polarizabilities are in ų (PBE/3ζ calculation¹³¹).

4.2 Polarizability of fullerene bis- and multiadducts. Influence of isomerism

6.6-Closed–5.6-open isomerism in the case of monoadducts has been considered in the section above. When fullerene adducts with larger number of the attached addends are considered, positional isomerism emerges (Fig. 6). Reactions of addition to fullerenes usually result in the mixture of the adducts C₆₀X_n with variable n.^89,140 For example, the numbers of bisepoxifullerenes C₆₀O₂ and trisepoxifullerenes C₆₀O₃ equal to 8 and 47, respectively.¹⁴¹

Fig. 6 Positional isomerism of C₆₀ bisadducts. Possible positions of the second addend in C₆₀X₂ are denoted.

In our works, we have theoretically investigated mean polarizabilities of the positional isomers of C₆₀X₂ with the simplest addends by PBE/3ζ method.^8,53,132 As it turned out, mean polarizabilities of isomeric bisepoxy-, bisaziridino- and biscyclopropa[60]fullerenes (X = O, NH, and CH₂, respectively) are approximately equal (∼84, ∼86 and ∼87 ų, respectively). This rule holds true for [2 + 3]- and [2 + 4]-adducts of C₆₀ (ref. 135) as well as for multiadducts with greater number of addends.¹³² According to DFT-calculations,¹³² the mean polarizabilities of C₆₀X₆ (X = CH₂ and NH) isomers with compact, focal or uniform distributions of X moieties on a fullerene framework do not differ significantly (Fig. 7).

Fig. 7 Hexakisadducts C₆₀(CH₂)₆ (a–c) and C₆₀(NH)₆ (d–f) with uniform (a and d), focal (b and e), and compact (c and f) distributions of X groups on the fullerene cage and their mean polarizabilities, calculated by the PBE/3ζ method (ų). Reprinted with permission from ref. 132© 2012 Elsevier.

In the case of the substituted cyclopropa[60]fullerenes, mean polarizability remains regardless of positional relationship of the addends attached. We have demonstrated it on the example of two “carboxyfullerenes” – t,t,t- and e,e,e-tris(dicarboxymethano)fullerenes.⁸ These species are produced via reaction of fullerene with malonic ester derivatives¹⁴² and attract a great interest due to their physiological activity, e.g., inhibition activity towards some enzymes¹⁴³. In spite of the different patterns of additions in these compounds, their mean polarizabilities are almost equal (Fig. 8). Their high polarizabilities allow explaining strong propensity for aggregation in solutions, previously studied.¹⁴⁴

Fig. 8 Regioisomeric carboxyfullerenes. Mean polarizabilities, calculated by the PBE/3ζ method,⁸ are shown in ų.

The analogous situation is typical for isomeric halofullerenes C₆₀Hal_n,⁵³ formally considered as [1 + 1] adducts (Table 5). Most of them are produced as mixtures of several isomers. Moreover, isomerization processes are able to take place in halogen-containing fullerene derivatives. For example, a slow room-temperature interconversion of C₁ and C₃ isomers of C₆₀F₃₆ occurs in the presence of ambient atmosphere.¹⁴⁵ Carbon skeletons of the fluorinated matters changes insignificantly upon the isomerization:¹⁴⁶

C₁–C₆₀F₃₆ ↔ C₃–C₆₀F₃₆ (26)

Table 5 Mean polarizabilities of halofullerenes, calculated by the PBE/3ζ method⁵³ (in ų)

Molecule	α	Molecule	α
1,2-C60F2	84.0	1,2-C60Cl2	89.6
Cs-C60F16	87.8	Cs-C60Cl6	100.7
C3v-C60F18	87.4	Th-C60Cl24	140.2
D5d-C60F20	88.4	C1-C60Cl28	149.4
Th-C60F24	84.7	D3d-C60Cl30	161.0
C1-C60F36	88.8	C2-C60Cl30	150.4
C3-C60F36	88.8	1,2-C60Br2	92.9
T-C60F36	89.0	C2v-C60Br6	109.9
D3-C60F48	90.5	Cs-C60Br8	119.6
S6-C60F48	90.4	Th-C60Br24	172.6

---

Isomeric C₆₀Hal_n, differing by the relative Hal positions, are characterized by approximately the same values of mean polarizability, e.g., for all C₆₀F₃₆ isomers α equals to ∼89 ų (Table 5). However, the third isomer T-C₆₀F₃₆ has the carbon skeleton, far from the mentioned isomers, whereas it has the same α. The difference in mean polarizability achieves the highest value for two C₆₀Cl₃₀ isomers (∼10 ų) with dissimilar geometries of carbon frameworks. In addition, the structure of isomer with higher polarizability D_3d-C₆₀Cl₃₀ is characterized by the trannulene equatorial belt, which made up by facilely-polarizable conjugated double bonds (Fig. 9).

Fig. 9 Isomeric C₆₀Cl₃₀ and C₆₀F₃₆ halofullerenes. Trannulene equatorial belt in D_3d-C₆₀Cl₃₀ (a system of π-conjugated bonds) is marked by arrows.

Considering the above, we can conclude that, usually, positional isomerism negligibly effects on mean polarizability of fullerene adducts. However, in rare cases, it can lead to a considerable difference in mean polarizabilities of the positional isomers.

4.3 General formula for calculation of mean polarizability of fullerene exohedral derivatives

Polarizability of diverse multiadducts have been theoretically studied but the question of how polarizability of C₆₀X_n depends on the number of addends n was unanswered for a long time. First, Hu and Ruckenstein estimated mean polarizabilities of the C₆₀ fullerene and its hydrides C₆₀H₂ and C₆₀H₆₀ as 73.8, 74.7, and 77.9 ų, respectively (B3LYP/6-31G(d) calculations).¹²³ However, their study¹²³ was focused on endohedral structures, so they did not pay attention to the striking difference: formation of C₆₀H₂ from C₆₀ increases mean polarizability of 0.9 ų whereas the addition of the next 58 hydrogen atoms leads to the unexpectedly small increase (3.2 ų). Later, Rivelino et al. in the theoretical study of structures, stabilities, and light scattering of fullerenols C₆₀(OH)_n have pointed that the mean polarizability grows up from C₆₀(OH)₂ to C₆₀(OH)₁₈ but then diminishes for C₆₀(OH)₂₄.¹²⁷ This work has explained this falling down by the highly symmetric addition pattern in C₆₀(OH)₂₄ compared to its precursor with 18 hydroxyls. However, as we have shown in Section 4.2, positional isomerism generally does not affect polarizability. Finally, we have found the same situation in the case of epoxy[60]fullerenes C₆₀O_n (ref. 131) (in the set with n up to 30, C₆₀O₁₅ has the maximal α).

To uncover this enigmatic behavior, we have scrutinized the C₆₀X_n polarizabilities with DFT methods and adequate additivity rules. We should mention that the development of the first one usually makes the additive approach unnecessary. Nevertheless, evaluation of additive polarizability has not lost its relevance in structural studies,¹⁴⁷ so we have compared our DFT-calculated polarizabilities within the additive scheme for cyclopropa-C₆₀(CH₂)_n and aziridinofullerenes C₆₀(NH)_n with n up to 30 (which is a number of double bonds in C₆₀ molecule).¹³² For this purpose, the only randomly chosen isomer has been selected for each n as mean polarizability is defined mainly by the number of addends. The following additive scheme has been applied. Each fullerene cycloadduct has been partitioned on (n + 1) subunits of two types: a fullerene cage and n addends attached (Fig. 10). According to this scheme, additive polarizabilities of C₆₀X_n equal to:

α_add(C₆₀X_nmax) = α_C60 + nα_X, (27)

where

α_X = α_C60X − α_C60 (28)

are increments (X are bivalent chemical moieties > CH₂ or > NH). Increments α_X describe the change in polarizability upon the addition of one X fragment, accompanied by disappearance of π-component of one 6.6 bond (α_X > 0).

Fig. 10 Consideration of fullerene adducts in terms of additive scheme, described by eqn (26) and (27).²³

According to eqn (27), additive polarizabilities of C₆₀(CH₂)_n and C₆₀(NH)_n enlarge linearly when n → 30. As DFT calculations show,¹³² the differences between α(C₆₀X_n) and α_add(C₆₀X_n) become greater with n increase and we observe the depression of polarizability Δα, i.e. the negative deviation of α(C₆₀X_n) from α_add(C₆₀X_n):

Δα = α_add(C₆₀X_n) − α(C₆₀X_n) (29)

For both classes of cycloadducts, Δα achieves maximal value at n = 30 (Fig. 11). Based on mathematical induction, the computed data have been analyzed with a fitting function, which unites mean polarizability and number of addends in a fullerene derivative molecule:

where two first terms of the equation make up the additive polarizability and Δα(C₆₀X_nmax) is a depression of polarizability of the totally-functionalized fullerene derivative.¹³² In this formula, the depression of polarizability for fullerene adduct with n addends is:

Fig. 11 Dependences of α on n values, obtained by PBE/3ζ method in terms of additive scheme without (1 – C₆₀(CH₂)_n, 2 – C₆₀(NH)_n) (eqn (27)) and with the correction on the depression of polarizability (3 – C₆₀(CH₂)_n, 4 – C₆₀(NH)_n) according to eqn (30) and (37). Black and white circles correspond to pure quantum-chemically calculated α values of C₆₀(CH₂)_n and C₆₀(NH)_n, respectively. Reprinted with permission from ref. 132© 2012 Elsevier.

We consider that this correction to the additive scheme has physical meaning. Previously,⁸ we have interpreted it as follows. Rewriting eqn (31) as

allows demonstrating that depression of polarizability is proportional to specific depression (depression of the totally-functionalized fullerene derivative per one addend, the first term), degree of functionalization (n/n_max, the second term), and the number of addends attached (the third term).

For this review, we have reanalyzed the computational data, obtained previously,¹³² using combinatorial approach. The number of all possible interactions X⋯X in the C₆₀X_n molecule equals to the number of double combinations:

Then the specific depression of the totally-functionalized fullerene derivative equals to

The depression of polarizability for the arbitrary C₆₀X_n is calculated as

Δα(C₆₀X_n) = C_n²Δα(C₆₀X_nmax)_spec. (35)

Substitution of eqn (33) and (34) in eqn (35) provides the final variant:

The last expression is very close to the mathematically induced eqn (31). However, eqn (36) is more justified. Indeed, considering that the depression of polarizability is caused by the X⋯X interactions, we should obtain Δα = 0 when n = 0 or 1. This is true only when we use eqn (36). Thus, we can finally writing the general equation for mean polarizability of the fullerene derivatives:

Fitting functions (37) (and (30)) render the quantum-chemically obtained values of [2 + 1]-cycloadducts polarizability with high accuracy (Fig. 11). Derived strictly for C₆₀(CH₂)_n and C₆₀(NH)_n cycloadducts, these formulae reproduce well the DFT-calculated mean polarizabilities of other derivatives of C₆₀ (ref. 8 and 53) and C₇₀,^53,133 i.e. it works for the other fullerenes (Fig. 12 and 13) (Table 6).

Fig. 12 Dependence of mean polarizability on the number of the attached atoms in fullerene derivatives (PBE/3ζ calculations). Symbols correspond to the pure quantum-chemically calculated mean polarizabilities; lines show dependences α versus n, obtained by the use of fitting functions (30) or (37). The calculated data have been taken from ref. 53 and 131.

Fig. 13 Dependence of mean polarizability on the number of the attached atoms in fullerene derivatives (PBE/3ζ calculations). Symbols correspond to the pure quantum-chemically calculated mean polarizabilities; lines show dependences α versus n, obtained by the use of fitting functions (30) or (37). The calculated data have been taken from ref. 8, 53 and 133.

Table 6 Parameters of eqn (30) and (37) for calculation of mean polarizability of fullerene adducts

Fullerene adducts	αx (Å^3)	−Δα(C60Xnmax) (Å^3)	nmax
a PBE/3ζ calculations. Taken from ref. 8, 53 and 131–133.b B3LYP/6-31G(d) calculations. Taken from ref. 123.c B3LYP/6-31G(d,p) calculations. Taken from ref. 127.
C60(CH2)n^a	2.31	31.8	30
C60(NH)n^a	1.55	25.2	30
C60On^a	0.50	12.3	30
C60Fn^a	0.65	23.5	48
C60Cln^a	3.45	30.5	30
C60Brn^a	5.12	33.0	24
C60In^a	8.17	47.8	24
C60Hn^a	0.525	17.6	36
C60Hn^b	0.90	49.9	60
C60(OH)n^c	1.26	17.2	24
C70Cln^a	3.78	41.6	28
C70[OOC(CH3)3]n^a	11.51	8.7	10

---

The derived functions (30) and (37) have the respective maxima:

and

It means that the dependence of mean polarizability of fullerene adducts is generally nonmonotonic. This was previously found for C₆₀(OH)_n (ref. 127) and C₆₀O_n.¹³¹

This approach to interpretation of the computational data demonstrate efficiency in the cases of other derivatives of both C₆₀ and C₇₀ (Tables 6 and 7, Fig. 12 and 13). Parameters of the corresponding fitting functions are collected in Table 6. It is noteworthy that formulae (30) and (37) work regardless of the quantum-chemical method used.^53,132 In fact, it is applicable to C₆₀(OH)_n polarizabilities, calculated by Rivelino et al.¹²⁷ with B3LYP/6-31G(d,p) (Fig. 14). In this set, only C₆₀(OH)₁₈ violates the regularity. The reasons for this are addressed to the further studies.

Table 7 Mean polarizability and depression of polarizability (in parentheses) of C₇₀X_n (in ų; PBE/3ζ calculations). Reprinted with permission from ref. 133© 2012 Taylor and Francis

X	C70X8	C70X10
H	101.9 (−1.1)	102.1 (−1.0)
CH3	116.9 (−8.6)	120 (−11.3)
C6H5	196 (−12.4)	217.8 (−17.09)
Cl	123.6 (−9.3)	128 (−12.5)
Br	137.5 (−10.8)	144.8 (−14.9)
OOC(CH3)3	188.1 (−6.6)	209.1 (−8.7)

---

Fig. 14 Mean polarizability of the fullerenols family. Red points correspond to B3LYP/6-31G(d,p) values (taken from ref. 127). Line shows calculation by eqn (37) with parameters, deduced from the mentioned quantum-chemical calculations.¹²⁷

In our works, we pay particular attention to high-polarizability molecules, for example, iodo[60]fullerenes. As known, there are obstacles to synthesize fullerene derivatives with C–I bonds, possibly due to the constraints, arising between the voluminous iodine atoms. It makes the reaction of iodine with fullerene core thermodynamically unfavourable.¹⁴⁸ However, the iodination of C₆₀ fullerene should lead to easily polarizable compounds, which have a strongly expressed response to external electric fields, because C₆₀I_n have the highest mean polarizabilities among the other C₆₀ derivatives. Attempts to the synthesis of the iodinated C₆₀ are being performed.¹⁴⁹

The disadvantage of our explaining the depression of polarizability and the derived general formulae should be noted. One of the parameters, defining Δα value according to eqn (30) and (37), is a maximal number of addends (n_max), which can be attached to fullerene skeleton. The n_max value is difficult to determine exactly (both experimentally and theoretically) whereas it is significant for the effective use of the mentioned formulae. Though the stability of some totally-functionalized fullerene derivatives has been clearly shown (e.g., polyepoxide C₆₀O₃₀¹⁵⁰), n_max = 30 is rather hypothetical value. Nevertheless, the use of both theoretically possible maximal value (n_max = 30) for [2 + 1]-cycloadducts and n_max values for [1 + 1]-adducts, experimentally known at the moment, demonstrates good agreement between our formula and DFT-calculations.

There is the only experimental work where mean polarizabilities of two halofullerenes have been measured.¹³⁸ The experimental technique was based on Kapitza–Dirac–Talbot–Lau interferometry.^31,32 It allowed obtaining experimental data on C₆₀F₃₆ (60.3 ± 7.7 ų) and C₆₀F₄₈ (60.1 ± 7.5 ų), which correspond to the mixtures of isomers.¹³⁸ Nevertheless, as follows from the calculations (Table 5), the structure does not influence on the static mean polarizability of the studied polyfluorofullerenes (though it remains significant for evaluation of dipole moments). Despite the fact that DFT methods, used in ref. 8 and 53, overestimate the respective measured values, both experimental and theoretical studies indicate the equality of C₆₀F₃₆ and C₆₀F₄₈ polarizabilities. A mismatch between the measured and calculated values may occur because the computation represents static polarizabilities while the experiment⁸ yields the optical polarizability at 532 nm laser wavelength. The equality of experimental mean polarizabilities of polyfluorofullerenes with different numbers of addends¹³⁸ confirms our theoretical assumptions about depression polarizability.

4.4 Anisotropy of polarizability and its application to organic solar cells

As mentioned, mean polarizabilities of regioisomeric fullerene bis- and multiadducts are almost the same. Differences are observed for anisotropy of their polarizability (a²). Dependences of a² on the internuclear distance between the central atoms of the attached moieties L for the simplest bisadducts are shown in Fig. 6. Regioisomers are characterized by the different a² values. In the case of X = CH₂ and NH, the highest values of anisotropy are typical for trans-1-C₆₀X₂, and the smallest ones correspond to equatorial bisadducts e-C₆₀X₂ (bisepoxy[60]fullerenes fall out of this trend).⁸

The C₆₀ is initially isotropic and its decoration by functional groups leads to a violation of the isotropy that is reflected by the increase of a² values.⁵³ Upon the functionalization of C₇₀, which is initially anisotropic molecule (a² = 136.89 Å⁶), the changes in anisotropy of polarizability depend on the nature of addends.¹³³ According to PBE/3ζ calculations, hydro[70]fullerenes C₇₀H₈ and C₇₀H₁₀ have the lower a² values (56.13 and 28.26 Å⁶, respectively). Anisotropy enlarges in the case of the other C₇₀ derivatives: the maximal values of anisotropy among the studied compounds characterize C₇₀Ph₈ and C₇₀Ph₁₀ (Fig. 15). Anisotropies of polarizability and calculated increments α_X are cymbate values (Fig. 16).¹³³

Fig. 15 Structures of C₇₀Ph₈ (top) and C₇₀Ph₁₀ (bottom). Polar pentagons, made up by atoms a, are whitened for clarity.

Fig. 16 Dependences of anisotropy of polarizability a² on α_X values for C₇₀ derivatives (PBE/3ζ calculations). Reprinted with permission from ref. 133© 2012 Taylor and Francis.

Thus, isomeric fullerene derivatives demonstrate approximately the same mean polarizabilities and differ by anisotropies. It may be useful for searching of new promising fullerene derivatives for diverse applications. For example, many novel compounds, including fullerene multiadducts, come into use as acceptor materials for organic solar cells.¹⁵¹ Developing structure–property relationships may facilitate the screening of appropriate compounds for this purpose.

Previously, we have found a correlation between the anisotropies of polarizability of dihydronaphthyl-C₆₀ bisadducts (C₆₀dhn₂) and the key output parameters of organic solar cells, based on them.¹³⁵ We have used the output parameters, measured in the same experimental protocols by Kitaura et al. for the isolated and purified derivatives of C₆₀dhn₂.¹⁵² We pay particular attention to this pioneering study¹⁵² because it utilizes the purified bisadducts in contrast to the previous works, which deal with the mixtures of regioisomers. Note that the substituted C₆₀dhn₂ with alkylcarboxy chains (–CO₂C₆H₁₃) in naphthyl moieties were tested in experiments whereas the non-substituted C₆₀dhn₂ were quantum-chemically treated to facilitate the task. This simplification has been justified in the context of experimental study¹⁵³ of fullerene dendrimers, which have the same electrooptical properties regardless the number of generation. The devices with the derivatives of e-, trans-4-, and trans-2-C₆₀dhn₂ as an electron-acceptor material show the highest output solar cells parameters (power conversion efficiency PCE, open circuit voltage V_OC, filling factor, and short-circuit current density) (Fig. 17a). These compounds, as we turned out, are characterized by the lowest values of anisotropy (the most thermodynamically favorable structures for e-, trans-4-, and trans-2-C₆₀dhn₂ are shown in Fig. 18). On the contrary, the devices, utilizing highly anisotropic adducts, show the lowest values of the output parameters (e.g., trans-1-C₆₀dhn₂). Slight discrepancy of data for cis-2- and cis-3-bisadducts with the discussed relationship can be explained by the fact that their mixture was used in devices because of the failed separation.

Fig. 17 Correlation between the output parameters of organic solar cells, based on bis(dihydronaphtho)fullerene derivatives, and calculated anisotropies of regioisomeric C₆₀dhn₂. Panel (a): output parameters PCE and V_OC are taken from the experimental work¹⁵² (reprinted with permission from ref. 135© 2013 American Chemical Society). Panel (b): output parameters PCE and V_OC are taken from the recent experimental work.¹⁵⁵

Fig. 18 The most thermodynamically favorable structures for trans-4-, e-, trans-3-, and trans-2-C₆₀dhn₂.

As known, V_OC values can be accurately predicted with the use of LUMO energies of acceptors.¹⁵⁴ However, in the case of C₆₀dhn₂, we have found a more precise correlation with the anisotropy values than with LUMO levels.¹³⁵ Ordering the donor and acceptor phases is necessary for the transport of charge carriers to the electrodes in solar cells. It would seem that the high anisotropy of polarizability should facilitate the ordering. However, the found correlation is reversed (Fig. 17a). Therefore, we have assumed that the role of disorder in the charge transport process should be reconsidered.

Unfortunately, the situation with application of a² values to fullerene derivatives for organic solar cells is ambiguous. Very recent experimental work¹⁵⁵ declares the enhanced output parameters for trans-3-C₆₀dhn₂. Structurally, the compounds, tested in ref. 155, were exactly the same as in the computational case above. We have superposed experimental data on PCE and V_OC with anisotropies in Fig. 17b. Thus, trans-3-C₆₀dhn₂ is the most anisotropic among the regioisomers, tested in ref. 151. This contradicts to the previous considerations about the enhanced efficiency of the isotropic bisadducts.¹³⁵ However, the correlation field of experimental study¹⁵⁵ is narrow (the measured parameters are close to each other) because a half of possible C₆₀dhn₂ have not been purified and tested. On the other hand, the calculations¹³⁵ did not take into account the alkylcarboxy chains, presented in the compounds from ref. 152. Unfortunately, there are no other experiments, operating with purified fullerene bisadducts. Thus, the further consideration of anisotropy of polarizability in context of organic solar cells demands additional experimental and theoretical studies.

The examples above, despite of the contradictions, may be important for fullerene photovoltaic applications. Here, we just mention three points of why anisotropy of polarizability matters.

The first deals with the dependence of dielectric permittivity on the polarizability, which is described by Clausius–Mossotti equation (eqn (25)).¹⁴ In the recent study,¹⁵⁶ dielectric constant has been considered as a central parameter for efficient solar cells. The authors use it in a scalar form to find out optimal regimes for photovoltaic devices functioning. Based on our studies on anisotropy of polarizability, we propose that tensorial nature of dielectric permittivity and anisotropy of dielectric permittivity should be taken into account to improve the existing models for assessing the efficiency of organic solar cells.

The second point deals with the polarizability of fullerene charge-transfer complexes, which arise when solar cell works. Polarizabilities of the excited states of such complexes may exceed 2000 ų.⁶⁰ Standing this, we can assume that the anisotropy of polarizability affects the charge-transfer complexes decay, required for generating the charge carriers.

At last, the role of anisotropy of polarizability may deal with its influence on wetting processes, which we mention in Section 3.4. If de Gennes equation¹¹⁰ (eqn (23)) is applied to this case, the composite material of organic solar cells may be roughly approximated as nano-droplets of fullerene derivatives (α_L parameter in eqn (23)) wetting the surface of polymer phase (α_S parameter in eqn (23)). This seems reliable in the context of the recent experimental work,¹⁵⁷ in which wetting and surface tension have been efficiently used to explain the observed molecular structure–device function relationship for solar cells utilizing diverse o-xylenyl bisadducts of C₆₀ as electron acceptor materials.

Finally, we mention another value that may be useful in the field under discussion. This is optical anisotropy, defined as:²⁹

As it is seen, in the series above, all the bisadducts were isomeric, i.e. have the same mean polarizabilities. However, comparing fullerene derivatives with more different structures (e.g., with different nature of addends or their unequal numbers) requires a scaling factor, and eqn (40) provides this.

In the regard of recent application of triscyclopropafullerene derivatives to organic solar cells,^158,159 dependence of anisotropy on the average distance between the addends for 47 possible regioisomers C₆₀(CH₂)₃ have been investigated (Fig. 19).¹³⁵ According to calculations, C₃-symmetry isomer has the lowest anisotropy. This isomer is characterized with equidistance of CH₂ and their uniform distribution on the fullerene core. Additionally, the isomer of C₆₀(CH₂)₆ with uniform distribution of addends has been also found isotropic.¹³⁵

Fig. 19 Dependence of anisotropy a² of regioisomeric C₆₀(CH₂)₃ on the average distance between the central atoms of addends. The structural formula of the least anisotropic trisadduct is shown. Reprinted with permission from ref. 135© 2013 American Chemical Society.

Regardless of the application of anisotropy and the physical meaning of the found regularities, anisotropy of polarizability is a good index to describe the structural diversity, emerged upon multiple additions to fullerenes.

4.5 Polarizability and its anisotropy of [60]-PCBM and [70]-PCBM

We separately describe dipole polarizabilities of the substituted cyclopropafullerenes [60]-PCBM and [70]-PCBM because of their high importance for fullerene-based organic photovoltaics.¹⁶⁰ Previously, we have calculated mean polarizability of [60]-PCBM by the PBE/3ζ method (Table 3).⁸ For this review, we have used the same computational methodology to compare polarizability of [60]-PCBM and its isomers in the context of the latest studies. As known, [60]-PCBM has 5.6-open isomer¹⁶¹ and the isomer iso-PCBM with more different structure, which is formed when [60]-PCBM is heated as discovered very recently¹³⁹ (its structural formula is shown in Table 3). Moreover, the last one can play important role in the functioning of PCBM-based organic solar cells because it should accompany “traditional” [60]-PCBM in significant amounts. Thus, [60]-PCBM, open [60]-PCBM, and iso-PCBM have almost the same mean polarizabilities (108.4, 108.9, and 109.0 ų) despite the differences in their structures. [60]-PCBM shows the lowest anisotropy of polarizability than open [60]-PCBM and iso-PCBM (736.08 versus 790.24 and 873.70 Å⁶).

As is known, the synthesized [70]-PCBM is a mixture of chiral (ab-[70]-PCBM) and achiral isomers (cc₁- and cc₂-[70]-PCBM) (Fig. 20).¹⁶² Their polarizabilities and anisotropies have been recently calculated by the B3LYP/6-311G(d,p) method in terms of finite-field approach.¹³⁷ Calculations show that cc₁-[70]-PCBM and ab-[70]-PCBM are the least and the most anisotropic isomers in this set.

Fig. 20 [70]-PCBM isomers, their mean polarizabilities (in ų, blue numbers) and anisotropies a² (in Å⁶, red numbers), calculated by B3LYP/6-311G(d,p). Adapted with permission from ref. 137© 2014 Elsevier.

In addition, we should mention the recent theoretical work that compares molecular characteristics of [70]-PCBM and its 5.6-open isomer.¹³⁶ The work reports the mean polarizabilities for this species 106.5 and 107.2 ų, respectively (calculated by CAM-B3LYP/6-31G(d,p)). The C₇₀ fullerene has unequivalent bonds in its structure, so several closed and open isomers of [70]-PCBM may exist. Unfortunately, the authors did not clearly indicate what isomers they theoretically investigated in the work.¹³⁶ Nevertheless, their computational data agrees with the general trend of higher polarizability for 5.6-open fullerene species.

5 Polarizability of fullerene dimers and oligomers

5.1. Polarizability of C₆₀ [2 + 2]-dimers and oligomers

Fullerene dimers and oligomers are an interesting class of fullerene derivatives that contain two or several cores in the molecules. Due to this structural peculiarity, we have reviewed these compounds separately. Nowadays, many types of such derivatives are known.¹⁶³ Among them, [2 + 2]- and [1 + 1]-dimers of C₆₀ and C₇₀ are the most studied. Fullerene [2 + 2]-dimers (and oligomers) also attract attention due to the closest placement of fullerene cores in a molecule. In addition, these are rigid compounds that can play role of all-carbon building blocks for nanoarchitecture (See ref. 164 and 165, and references therein). Molecular constructions with four C₆₀ fullerene moieties, capable to move like wheels, have been tested as the thermally-driven nanocars.¹⁶⁶ The less-stable [1 + 1]-dimers can produce fullerenyl radicals (reversibly or irreversibly), which are used for generation of free radicals in fullerene-containing systems (see, e.g., ref. 167).

Polarizability and hyperpolarizability of exotic endohedrals [Na@C₆₀][F@C₆₀] with different cage connections have been theoretically studied by Ma et al. (Fig. 21).¹¹ However, that work was focused mainly on the hyperpolarizabilities.

Fig. 21 Mean polarizabilities (in ų, blue numbers) and intercage distances (in Å, black numbers) of [Na@C₆₀][F@C₆₀] dimers, calculated at CAM-B3LYP level. Adapted with permission from ref. 11© 2010 American Chemical Society.

Polarizabilities of the following (C₆₀)_n [2 + 2]-oligomers with n up to 5 have been studied by PBE/3ζ method in our studies (Fig. 22 and 23).¹⁶⁵ Calculations show that α values of (C₆₀)_n are approximately n-fold higher than the polarizability of the pristine fullerene. The obtained values have been analyzed in terms of two additive schemes. Both of them deduce mean polarizabilities of (C₆₀)_n from the polarizability of the isolated C₆₀. According to the first additive scheme, mean polarizability of (C₆₀)_n equals to n-fold C₆₀ polarizability:

α^(I)_add((C₆₀)_n) = nα_C60 (41)

Fig. 22 Structures of the fullerene dimer and trimers identified experimentally. Their mean polarizabilities (in ų, blue numbers) and anisotropies (in 10⁴ Å⁶, red numbers) are shown. Adapted with permission from ref. 165© 2013 Royal Society of Chemistry.

Fig. 23 Structures of the hypothetical fullerene oligomers under study. Their mean polarizabilities (in ų, blue numbers) and anisotropies (in 10⁴ Å⁶, red numbers) are shown. Adapted with permission from ref. 165© 2013 Royal Society of Chemistry.

This estimation is rough as far as it does not consider the change of polarizability when several carbon atoms alter the initial hybridization at oligomer forming. The second scheme takes it into account. Accordingly, each of (C₆₀)_n molecules is divided into n subunits of two types: one central fullerene core to which the other (n − 1) C₆₀ fragments are attached. In this case, the additive polarizability equals to

α^(II)_add((C₆₀)_n) = α_C60 + (n − 1)α_[C60], (42)

where α_[C60] = α((C₆₀)₂) − α_C60 = 97.3 ų is the increment. It describes the change of polarizability at the addition of one fullerene subunit to the central C₆₀ core and the disappearance of π-components of 6.6 bonds in both fullerene cores. Deviations from the schemes

Δα^(I) = α_DFT − α^(I)_add, (43)

Δα^(II) = α_DFT − α^(II)_add, (44)

have been found positive¹⁶⁵ (Fig. 24), i.e. the exaltation of polarizability occurs when two or several cores present in the molecule. This computational fact was unexpected but reproducible with other DFT^61,165 and long-range corrected DFT techniques.⁶¹ Generally, double bonds in a molecule enhance its mean polarizability. Conversely, polarizability diminishes if π-bonds disappear at chemical transformation without addition of electron-rich chemical groups.^14,29 For example, when two C₆₀ molecules are linked resulting in [2 + 2]-(C₆₀)₂, two π-bonds cease to exist and, according to the conventional notions, the final polarizability should be lower than the twofold mean polarizability of C₆₀, that is, the deviation from the additive scheme should be negative. However, as follows from the calculated data, Δα^(I) > 0 and the effect increases with n.

Fig. 24 Dependence of mean polarizability on the number of fullerene cores in (C₆₀)_n molecules. Circles correspond to α_DFT values; lines present the mean polarizabilities according to the additive schemes. Reprinted with permission from ref. 165© 2013 Royal Society of Chemistry.

For unsaturated hydrocarbons, the analogous exaltation is observed when π-conjugation arises in a molecular system.^66–68 Formation of oligomers results in the increase of the total number of double bonds in a molecule. However, π-electrons do not form an overall system since the subsystems of the conjugated C=C bonds of fullerene units remain isolated at least by three C–C single bonds (two 5.6 bonds of fullerene moiety and one of the cyclobutane fragment). Nevertheless, the subsystems of π-electrons can communicate through space or through σ-bonds, according to the concept of orbital interaction through space.¹⁶⁸ Indeed, the analogous exaltations are observed for the calculated polarizabilities of oligomers (C₂₀)_n, dimers (C₂₄)₂, (C₃₀)₂, (C₃₆)₂, (C₅₀)₂, (C₇₀)₂, and C₆₀ adducts with bowl-shaped π-conjugated hydrocarbons (sumanene, hemifullerene, etc.).^61,165

In the case of (C₆₀)_n, Δα^(II) values linearly increase with the largest distances between the centers of the crosslinked C₆₀ cages (L_max) (Fig. 25). Only (C₆₀)₂ and cyclic trimer (C₆₀)₃, both characterized by the smallest intercage distances (L_max ≈ 9.1 Å), are out the correlation. To explain the physical meaning of the correlation, we have appealed to the basic definitions. Polarizability reflects the molecule's ability to acquire the induced dipole moment in the external electric field (eqn (2)). On the other hand, the magnitude of dipole is proportional to the distance between the centers of positive and negative charges of the molecule.^14,29 Presumably, such centers are induced on the maximally remote atoms in (C₆₀)_n, so the exaltation increases with the distance between the C₆₀ cages. This explains why Δα achieves its highest value for linear nanostructures, having the largest L_max values. Note that analogous dependence of Δα (and, consequently α) on the distance between the cages is observed for isomeric (C₇₀)₂⁶¹ but violated in the case of the endohedrally functionalized dimer [Na@C₆₀][F@C₆₀] (Fig. 21).¹¹

Fig. 25 Dependence of the exaltation of polarizability according to additive scheme II (eqn (44)) on the maximal distance between fullerene units in (C₆₀)_n molecules. White circles, red triangles, blue circles, and white squares correspond to oligomers with n = 2, 3, 4, and 5, respectively. The data for (C₆₀)₂ and cyclic (C₆₀)₃ have not been included in the correlation. Reprinted with permission from ref. 165© 2013 Royal Society of Chemistry.

5.2 Polarizability exaltation of C₆₀ dimers: evidence from experimental studies

Fullerene dimers and, especially, trimers are novel compounds, so the data on their polarizability are absent. In addition, experimental obstacles to its measurement may be due to propensity of (C₆₀)_n for decomposition¹⁶⁹ and very low solubility in ordinary solvents.¹⁷⁰ However, correlations between the calculated polarizabilities and observables can be found. As known, the deduction of C₆₀ polarizability from the solid-state measurements¹⁷¹ leads to 89.9 ų that is ∼15% larger than the values demonstrated by isolated C₆₀ molecules (76.5 ± 8.0 ų; molecular beam deflection³⁸) or small clusters (79.0 ± 4.0 ų; time-of-flight technique³⁷). That enhancement has been explained by excitations and charge-transfer between the neighboring C₆₀ molecules in a crystal.¹⁷¹ In the case of C₆₀ oligomers, somewhat similar enhancement takes place. Indeed, according to DFT-calculations, the mean polarizability per one cage for (C₆₀)₂ is ∼8.8% higher than the mean polarizability of the isolated C₆₀. Though it is lower than the solid-state effect, there is a significant difference in the experimental and computational situations: unlike to crystalline C₆₀, fullerene cores are chemically bonded in oligomers. Notwithstanding, the similar tendency in both cases allows proposing a common nature of these enhancements caused by the interactions between π-electronic systems of C₆₀ cores.

Recent advances in application of polarizability to chemical reactions^172–174 provide another way to find a correlation between the calculated exaltations and the observed chemical properties of (C₆₀)₂ and (C₆₀)₃. For example, the simplest decomposition reactions A_mB_n…C_p → mA + nB + … + pC are described by the following change in polarizability Δα_R:¹⁷⁴

Δα_R = mα_A + nα_B + … + pα_C − α_AmBn…Cp (45)

Moreover, linear correlations between the heat effects of the mentioned reactions and Δα_R values have been found, as well as the minimum polarizability principle which states that thermodynamically more stable isomers are characterized by lower Δα_R values^173,174 (however, there are compounds that violate this principle¹⁷⁵). As known, fullerene dimer (C₆₀)₂ and its derivatives are unstable under the thermal treatment.^163,169 The deviation of polarizability from the additive scheme I and Δα_R of the dimer decomposition (C₆₀)₂ → 2C₆₀ are in a simple relation:

Δα_R = −Δα^(I), (46)

that is, the calculated exaltations should reflect the stability of the studied compounds. Therefore, Δα^(I) values of (C₆₀)₂ and its two derivatives have been compared with available experimental data.¹⁶⁵ We have used the results of the experimental studies^176,177 on the formation/decomposition of two auxiliary compounds [C₆₀(CR₂)₅]₂ and C₆₀(C₆₀F₁₆) (Fig. 26). The exaltations of [C₆₀(CH₂)₅]₂ and C₆₀(C₆₀F₁₆) are approximately equal (12.9 and 12.5 ų, respectively) and lower than Δα^(I)((C₆₀)₂) = 14.6 ų. It correlates well with the higher stability, observed experimentally for C₆₀(C₆₀F₁₆)¹⁷⁷ and carboxy-derivative of [C₆₀(CH₂)₅]₂.¹⁷⁶

Fig. 26 Structures of dimers [C₆₀(CH₂)₅]₂ (top) and C₆₀(C₆₀F₁₆) (bottom). Reprinted with permission from ref. 165© 2013 Royal Society of Chemistry.

Comparison of Δα^(I) values of (C₆₀)₂ and trimers (14.6 against 28.3–37.0 ų) predicts that the more polarizable trimers should be less stable. Indeed, the total yield of isomeric trimers is almost ten times lower than the yield of (C₆₀)₂.¹⁷⁰ Moreover, the exaltation increases in the series e-(C₆₀)₃ < trans-4-(C₆₀)₃ < trans-3-(C₆₀)₃ < trans-2-(C₆₀)₃ that is inversely correlated with the measured content of isomeric (C₆₀)₃ in the experimentally obtained (C₆₀)₃ fraction (Fig. 27). Thus, the exaltation correlates with the stability of fullerene trimers. Extrapolating this approach, we can make a reasonable prognosis of the increasing instability of the highest oligomers, especially, having linear structure. It can explain, for example, why linear trans-1-(C₆₀)₃ with the maximally distant cages, which seems to have no steric hindrances at its formation, has not been detected in the mixture of synthesized (C₆₀)₃.¹⁷⁰ However, such structures may become synthetically achievable if fullerene cores are functionalized that results in the wastage of their π-electronic systems and, as consequence, in higher stability (similar to C₆₀(C₆₀F₁₆) and [C₆₀(CR₂)₅]₂ in comparison with (C₆₀)₂).

Fig. 27 The calculated exaltations of polarizability versus the measured yields of the isomeric fullerene trimers (C₆₀)₃. Experimental data are taken from ref. 170. Reprinted with permission from ref. 165© 2013 Royal Society of Chemistry.

The highest enhancement of mean polarizability is observed in the case of many-cage fullerene-based molecules with the maximally remote C₆₀.¹⁶⁵ Unfortunately, this also makes the molecules less stable. The linear correlation found may be a landmark for rational design of fullerene-based nanostructures with adjustable response to electric fields. Currently, it seems to be the only way to estimate mean polarizability of the C₆₀ fullerene linear polymers with analogous structure (such polymers have been synthesized and recommended as nanofuses; at the same time the fragility of (C₆₀)_n has been noted¹⁷⁸).

5.3 Polarizability exaltation of other fullerene dimers

More recently, the phenomenon of polarizability exaltation has been theoretically found for [2 + 2]-dimers of other fullerenes with different structure (C₂₀)₂, (C₂₄)₂, (C₃₀)₂, (C₃₆)₂, (C₅₀)₂, (C₇₀)₂ (Fig. 28)⁶¹ as well as for C₆₀ [1 + 1]-dimers (Table 8),¹⁷⁹ which are formed in radical reactions of C₆₀. Therefore, the exaltation of polarizability can be considered as a common property of fullerenes family, regardless of the type of fullerene cages in a molecules and type of their connection.

Fig. 28 Structures of hypothetical fullerene dimers. Their mean polarizabilities and polarizability exaltations (in parentheses) are shown in ų (PBE/3ζ calculations). Adapted with permission from ref. 61© 2013 Royal Society of Chemistry.

Table 8 Mean polarizability α, additive polarizability α_add, and its deviation from the additive scheme Δα for 1,4,1′,4′-XC₆₀–C₆₀X dimers (in ų; PBE/3ζ calculations). Reprinted with permission from ref. 179© 2013 American Chemical Society

Dimer	α	αadd^a	Δα
a Twofold polarizability of the respective fullerenyl radical.
^tBuC60–C60^tBu	197.57	2 × 93.63	10.31
^tBuOC60–C60O^tBu	199.56	2 × 95.37	8.82
^tBuOOC60–C60OO^tBu	202.36	2 × 96.96	8.44
Ph(CH3)2CC60–C60C(CH3)2Ph	216.92	2 × 102.77	11.38
Ph(CH3)2COC60–C60OC(CH3)2Ph	214.74	2 × 104.48	5.78
Ph(CH3)2COOC60–C60OOC(CH3)2Ph	218.79	2 × 105.33	8.13

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6 Insights into molecular switch from fullerene polarizability

Molecular switch is a widespread concept in molecular machinery. It consists in the ability of a molecular system to exist in two or more stable states, differing by any physical property.¹⁸⁰ Fullerene derivatives have become very attractive for switchable molecular devices due to their unique structural and electronic properties.^181,182 Recently, we have briefly overviewed the synthesized and tested fullerene-based switches to recognize what novel fullerene derivatives may be useful for this application.⁹ In this context, we have paid attention to their polarizabilities, calculated in our previous works.

Polarizability is very informative physical quantity and it is sensitive even to small changes in chemical structure of a molecular system.¹⁵ However, it is hard to use it directly to monitor switching in the bistable molecular systems because its measurements usually require a complicated methodology. Meanwhile, polarizability defines other quantities that could be (easily) measured such as refractivity indices, ion mobility, and dielectric constant.¹⁴ Therefore, it can be useful for screening potential compounds for molecular switch because the difference in polarizabilities of two states of molecular switch (split polarizability) may lead to the difference in their polarizability-dependent properties. We have selected three types of the fullerene-based molecular systems that may be considered for the application as molecular switches. These are (1) pairs of isomeric 6.6-closed and 5.6-open fullerene adducts; (2) fullerene dimers with flexible bridge between the cages; and (3) singly bonded fullerene dimers (the systems are listed in order of the increasing split of mean polarizability).⁹

Pairs of isomeric 6.6-closed and 5.6-open fullerene adducts (Fig. 4). It is well-known that [2 + 1]-addition to C₆₀ and C₇₀ results in at least two types of adducts, viz. 6.6-closed (addition to 6.6 bond) and 5.6-open (addition to 5.6 bond with its simultaneous cleavage).^89,140 The preliminary examination of 6.6-closed versus 5.6-open fullerene adducts allows pointing out the minimal difference in their nuclear frameworks. It means that these compounds should be almost isoenergetic though 5.6-open species are less stable and isomerized to their 6.6-closed counterparts.^89,140 Meanwhile, the electronic structures of the mentioned compounds differ significantly. The π-electronic system of the parent fullerene remains almost intact in 5.6-closed fullerene adducts whereas it loses one π-bond when the 6.6-closed adduct is formed. As we have shown in our previous theoretical works, mean polarizability reflects the last fact.^8,131 Indeed, 5.6-open adducts, due to the richer π-electronic systems, have mean polarizabilities, which slightly higher than the respective values of the 6.6-closed isomers (Table 4).^8,131

The difference in mean polarizabilities (Δα split) may be increased if several addends are attached to the fullerene core.⁹ Such a structure can be useful for molecular switch if it remains propensity for isomerization. Quantum-chemical calculations show that difference in 4–5 ų is achieved in the case of C₆₀X₆ multiadducts with simple X (Table 4). Unfortunately, these are also small values, so another opportunity should be considered to enhance Δα split, for example, varying X moiety to obtain such a pair of monoadducts that would demonstrate a desirably high split.

Another obstacle to creation of the molecular switch, based on the described possibilities, is caused by the irreversibility of the reaction in Fig. 4; i.e. in the most cases, the 6.6-closed fullerene adducts are much more thermodynamically stable than their 5.6-open counterparts. However, this problem seems to be solvable due to the recently demonstrated conversion of the 6.6-closed fullerene adducts into 5.6-open derivatives.¹⁸³ Thus, the systems under discussion can be recommended to test their switching, though with reservations.

Molecular switch based on the reversible transitions in bicage fullerene derivatives. Recently, we have performed theoretical studies of diverse fullerene [2 + 2]-dimers and oligomers and found that resulting mean polarizability of such multicage structures depends on the maximal remoteness of the cages.^61,165 Unfortunately, such oligomers are rigid compounds with no transitions between the forms with the closest and the remotest locations of the fullerene cages. However, we have also shown that the described split polarizability may be typical for bicage structures regardless of the type of bridging (for both [2 + 2]- and [1 + 1]-dimers¹⁷⁹).

Therefore, we assume a bicage fullerene system with the cores, connected by a flexible bridge (Fig. 29).¹⁶⁵ The initial state of the system A is characterized by a short distance between the fullerene cores. If the bridge undergoes a transformation that increases its linear size (e.g., the well-known photo-induced isomerization or conformational change), the system enters state B with a greater remoteness of the fullerene fragments (Fig. 29). In the second state, the molecular system obtains the higher polarizability and, consequently, the enhanced response to an external electric impact. In contrast to state A, it can be facilely manipulated by the external electric field.

Fig. 29 Hypothetical bicage fullerene-containing system existing in to states characterized with lower (A) and enhanced (B) response to external electric field. Reprinted with permission from ref. 165© 2013 Royal Society of Chemistry.

The dimer, very similar to the described above, has been previously tested (Fig. 30).¹⁸⁴ In the experiments, one of the fullerene cages hosts a nitrogen atom that is decisive for functioning of the dimer as a molecular switch. Its two forms are distinguishable by EPR measurements due to the unpaired electrons of the encapsulated atom. The authors¹⁸⁴ have studied EPR signals of syn- and anti-isomers and analyzed electron spin lattice and spin–spin relaxation times. Indeed, the molecular rotational correlation time τ_c, deduced from pulse EPR measurements in degassed CS₂, is slightly longer for anti-form of fullerene dimer compared to its syn-form. The authors explain different EPR behavior of the compound in its syn- and anti-forms with the changing distance from the nitrogen center of the unpaired electrons and the empty cage (it is more distant in anti-isomer, so the influence on its EPR characteristics is weaker).

Fig. 30 Molecular switch based on fullerene bicage derivative with photoactive azobenzene bridge and studied in ref. 184.

Additionally, we have found that such a split also depends on the number of double bonds of the subunits in the dimer molecules.¹⁶⁵ Theoretically, if the number of double bonds is larger than 30 (their number in the C₆₀ molecule), one should expect the strengthening of Δα. Thus, the replacement in a fullerene dimer of one of the C₆₀ cages by a moiety with a rich π-electronic system should also demonstrate the split. The list of potential species for such enhancement contains compounds with conjugated double bonds, e.g., porphyrins, polycyclic aromatic hydrocarbons, higher fullerenes and carbon nanotubes. Additional opportunities may arise when carbon nanotubes are used as a part of fullerene-containing molecular switch because the electron-transfer processes between C₆₀ and nanotubes strongly depend on the type of the last one.¹⁸⁵

Molecular switch, based on the reversible dissociation of bicage fullerene derivatives. The reaction of dimerization of fullerenyl radicals C₆₀˙ is well-studied.¹⁸⁶ It results in the singly bonded fullerene dimers XC₆₀–C₆₀X (so-called [1 + 1]-dimers):

XC₆₀–C₆₀X ↔ XC₆₀˙ (47)

Theoretical studies predict that C₆₀ singly bonded dimers should demonstrate outstandingly high values of the first¹⁷⁹ and the second¹¹ polarizabilities. It makes these chemicals prospective for optical applications and nanodevices.

Reversible reaction (47) can change its direction depending on the conditions. EPR spectrometry is the easiest way to monitor in what state, dimer or radical, the system is. Additionally, according to our DFT study,¹⁷⁹ the mean polarizabilities of XC₆₀–C₆₀X are more than twice higher compared to those of the respective fullerenyl radicals. It means that the switchable system should be more facilely manipulated by external electric fields when it is in the dimer state. The possible disadvantage of this molecular system is that one of its states has unpaired electron. This may involve XC₆₀˙ in side processes. However, as shown in experimental and theoretical works, fullerenyl radicals C₆₀˙ slowly react with other species, e.g., with molecular oxygen.^179,187,188

It is noteworthy that systems, such as described, attract attention in another way. Thus, in a solid-state study,¹⁸⁹ it has been shown that thermolysis of the solid fullerene dimers results in both reversible and irreversible generation of radicals XC₆₀˙ (X = –CH₂Si(CH₃)₂C₆H₄OCH₂CH(C₂H₅)(C₄H₉)). The free radicals, irreversibly generated after the first cooling of the solid, do not recombine because of the emergence of a second state of molecular packing, which produces a solid containing a long-lived radical. This free radical acts as a dopant for the fullerene solid and increases the electron mobility.¹⁸⁹

The situation above is based on the structural control of the dimers' response in the external electric field. There is another opportunity of the use of such highly-polarizable structures for molecular devices. It deals with the reversed influence when the structure of the compound with flexible moieties (capable for conformational transits) is affected by the electric field. This case has been theoretically studied on the example of the bowl-shaped hydrocarbons,¹⁹⁰ which are considered as fullerene precursors. Such hydrocarbons (e.g., coronene C₂₀H₁₀, the smallest one) are sufficiently anisotropic: when they interact with the electric field E, it is useful to divide their polarizabilities into two parts, parallel (α_‖) and orthogonal (α_⊥) to the field applied. The authors¹⁹⁰ have analyzed the energy gain ΔU_tot of bowl-shaped hydrocarbons in the presence of the field with the intensity E and fixed orientation. In general, it has the following view:

where, ϑ is the angle of the rotation of a molecule that should move in the field to obtain the equilibrium position. For field intensities smaller than a defined critical value E_crit:ϑ = 0 always corresponds to a stable stationary point (minimum) and ϑ = π to an unstable one (maximum). For fields stronger than E_crit, a new stationary point appear:

The stability of the configuration depends on the difference between α_⊥ and α_‖: when α_⊥ − α_‖ > 0 it is a minimum and a maximum otherwise. Thus, a new metastable state (existing only if the field applied) may arise (Fig. 31). The authors¹⁹⁰ propose the exploiting the response to electric stimulus to tune the molecular orientation. As an example, they have demonstrated it by the structural relaxation of C₂₀H₁₀ molecule linked to a (5,5) nanotube fragment (Fig. 32).

Fig. 31 Model prediction for the total energy of coronene C₂₀H₁₀ as a function of the molecule orientation with respect to external electric field. The curve corresponding to the critical field is shown in red. Red points designate the positions of energy minima. Reprinted with permission from ref. 190© 2013 American Chemical Society.

Fig. 32 Field-induced conformational transits, proposed for molecular junctions. Reprinted with permission from ref. 190© 2013 American Chemical Society.

7 Endofullerene polarizability

Fullerene molecules have empty space inside. This fact was immediately used to put atoms and molecules into their cages that result in endofullerenes (or endohedral complexes), a new class of topological compounds promising for wide-range applications such as radiopharmaceuticals, quantum bits, or photoswitchable devices.^191–193 Such encapsulation may change molecular properties of both the host fullerene and the guest atom/molecule. As consequence, this causes changes in exohedral reactivity of the fullerene moieties of such complexes^194–197 as well as the measurable physicochemical properties of the atoms trapped.^193,194,198

Currently, endofullerene polarizability has been studied only by theoretical methods. In the review, we discuss it starting with the simplest endofullerenes and moving to the more complex species. The chosen order does not reflect chronology of these studies. Indeed, analysis of periodicals shows that studies on endofullerenes started with theoretical treatment of Sc_n@C_m polarizabilities in Torrens's works.^199–201 In 2007, a monopole–dipole interaction approach was applied to study dielectric properties of giant multi-shell carbon nano-onions^202–204 and then double-shell C₆₀@C₂₄₀ nano-onion was studied by DFT in Zope's work.⁴⁸ Later, polarizabilities of endofullerenes with encapsulated noble gas atoms,^205–207 simple molecules (H₂O or CH₄),^55,208,209 and relatively large hydrocarbons (norbornadiene and quadricyclane)⁵⁷ were calculated.

Polarizability of the most known endofullerenes with noble gas atoms inside Ng@C₆₀ Ng = He–Kr has been exhaustively studied with several DFT methods.^55,205,206 Being intrinsically non-polar, these species obtain induced dipole moments in the presence of the external field. The induction of the dipole is clearly demonstrated by the electrostatic potential (Fig. 33). Though noble gases have the saturated electronic shells and C₆₀ has enough empty space inside to avoid interactions between the C₆₀ cage and the trapped atoms, mean polarizabilities of Ng@C₆₀ do not equal to the sum of the contributions α_Ng and α_C60 (Table 9). This was first found by Yan et al.²⁰⁵ and later in our work.²⁰⁶ Yan et al.²⁰⁵ analyzed such a decrease in polarizability in terms of decomposing the polarizability of a molecular system into site-specific contributions (the details of this approach are discussed in ref. 210). For this purpose, the polarizability of each Ng@C₆₀ has been partitioned as:

where P and Q denote local atomic and charge-transfer contributions, respectively. The values of have been found very close to those of the pristine C₆₀ whereas increases slightly from He to Kr. On the contrary, the local polarizabilities α^P_Ng increase from He to Kr and α^Q_Ng remains null. However, this scheme does not definitely explain why polarizabilities of the encapsulated noble gases are lower than polarizabilities of the respective free atoms.

Fig. 33 B3LYP/aug-cc-pVDZ calculated electrostatic potential surfaces of Ar@C₆₀ under external field of 0.001 a.u. The blue and red surfaces represent the positive and the negative parts the electrostatic potential, respectively. Reprinted with permission from ref. 205© 2008 Elsevier.

Table 9 Mean polarizabilities of noble gas endofullerenes (the respective depression of polarizability, calculated according to eqn (52), are shown in parentheses; mean polarizability of C₆₀, calculated by the same methods, is given for comparison). All values are in ų

Endofullerene	Mean polarizability α (depression of polarizability Δα)
	B3LYP/aug-cc-pVnZ^a	SVMN/aug-cc-pVnZ^a	PBE/aug-cc-pVnZ^a	PBE/3ζ^b	M06-2X^c
a n = D and T for C and noble gas atoms, respectively. The data are extracted from ref. 205.b The data taken from ref. 206.c Basis sets used: 6-31+G(d,p) was applied to C atoms; MG3 basis was applied to He and Ar. The data are extracted from ref. 55.
He@C60	82.0 (−0.2)	82.7 (−0.4)	83.1 (−0.2)	82.6 (−0.1)	78.5 (−0.1)
Ne@C60	82.0 (−0.3)	82.9 (−0.3)	83.1 (−0.4)	82.6 (−0.3)	—
Ar@C60	82.2 (−1.3)	83.1 (−1.5)	83.3 (−1.4)	82.9 (−1.0)	78.8 (−1.2)
Kr@C60	82.4 (−2.1)	83.2 (−2.2)	83.4 (−2.2)	83.0 (−1.4)	—
Xe@C60	—	—	—	83.0 (−2.6)	—
C60	81.9	82.8	83.1	82.7	78.4

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Delaney and Greer have studied screening effects taking into account interaction of the hollow C₆₀ with the electric field.²¹¹ The C₆₀ fullerene has rich π-electronic system that generates its own electric field. This intrinsic field makes an obstacle for penetration for the external one (Fig. 34). Indeed, according to the estimations,²¹¹ only ∼25% of the external field penetrate the interior of the C₆₀ fullerene, i.e. C₆₀ acts as a small Faraday cage. It is reflected on the polarizability of the encapsulated atoms: it becomes smaller compared to the free state.

Fig. 34 Equipotentials and field vectors of ΔE on the cut plane. The vertical direction is the C₅ axis, along which a field of 8.23 V nm⁻¹ is applied, and values of the axis are in Å. There are strong fields near the atoms; for clarity, field vectors longer than 10 V nm⁻¹ were reduced to this length. Reprinted with permission from ref. 211© 2004 American Institute of Physics.

Later,²⁰⁶ we have analyzed the screening effects for noble gas endofullerenes with C₂₀, C₂₄, C₂₈, C₃₆, C₅₀, and C₆₀ as the cage molecules in terms of the additive schemes considering the deviation:

Δα = α_X@Cn − (α_C60 + α_X) (52)

We have found that the deviation Δα can be both positive and negative. In the case of noble gas endofullerenes with C₃₆, C₅₀, and C₆₀ cages, Δα < 0 as in the previous theoretical work²⁰⁵ (Table 9) (note that Δα < 0 is also observed for endohedral complexes of C₃₀ and C₃₂ fullerenes with noble gases as we have shown in the unpublished data), i.e. the depression of polarizability takes place. In the case of the small cages C₂₀, C₂₄, and C₂₈, we have observed the exaltation (Δα > 0) (Fig. 35).

Fig. 35 Polarizability depression of endofullerenes versus atomic radii of the encapsulated noble gas atoms. Reprinted with permission from ref. 206© 2010 Springer.

Polarizability in classic theory is interpreted as the molecule's electronic cloud. Therefore, we have tried to explain the differences in signs of Δα by the compression of electronic clouds. In the general case, the relation between the sizes of the guest atom and the host cavity is decisive for polarizability. We assume that the endoatom induces high pressure on the carbon cage; it “extrudes” electronic density from the inner cavity of the cage and therefore increases the total polarizability of the endohedral complex (Δα > 0). Another situation likely occurs for larger fullerene. In this case, the pressure of the carbon cage prevails, so the endoatom is compressed and the total polarizability decreases (Δα < 0) (Fig. 36). This explanation is too mechanistic and ignores the charge transfer between the guest and the host molecules. Nevertheless, the known compression of metal atoms in endohedral complexes¹⁹⁸ supports our assumption.

Fig. 36 Polarizability exaltation (a) and depression (b). The first one corresponds to the pressing out the electronic cloud in endofullerenes with less than 30 atoms; the second represents the compression of the endoatom in endofullerenes with more than 30 atoms. Reprinted with permission from ref. 206© 2010 Springer.

The model of quenched polarizability, stating that the polarizability of an atom or functional group may significantly vary depending on the environment,⁵⁵ also suitable for explanation of the nonadditivities observed for endofullerene polarizability (though it seems less effective for understanding the positive deviations). The resulting polarizability depends on how many space is reserved for the location of the chemical group. Later, in ref. 55 and in our works^208,209 the depression of polarizability of endohedral complexes of C₆₀ with simple molecules has been stated (Table 10). Thus, this phenomenon is a general trend. As shown within the quenched polarizability model,⁵⁵ the fullerene cage does not change its polarizability when it traps atoms/molecules; i.e. depression Δα is related to the decrease in the polarizability of the guest. Therefore, we have recommended²¹² the value:

as a screening coefficient. It lies in the range 0.78–0.91 for most of the studied molecules (Table 10). However, we should mention the found exception when c > 1. This is observed for DFT-calculated mean polarizability and its depression for NHe@C₆₀ complex.²¹² Such a complex has been synthesized recently²¹³ and contains the N⋯He inside, which exist in a free state only in specific conditions. Thus, very high depression (|Δα| > α_X) may be due to the stabilization effect of the fullerene on this complex.

Table 10 Polarizability, its depression, and screening coefficients of the encapsulated molecules, calculated by eqn (52) and (53), respectively

X@C60	α, Å^3	αx, Å^3	–Δα, Å^3	c	Method & reference
H2O@C60	82.8	1.1	1.0	0.91	PBE/3ζ^209
H2O@C60	77.4	0.2	0.05	0.22	BPW91/6-311+G(d)//BPW91/D95V^214
NH3@C60	82.9	1.7	1.5	0.88	PBE/3ζ^208
NH3@C60	77.6	1.4	1.1	0.77	BPW91/6-311+G(d)//BPW91/D95V^214
CH4@C60	83.0	2.3	2.0	0.87	PBE/3ζ^208
CH4@C60	78.7	2.0	1.7	0.85	M06-2X/6-31+G(d,p)^55
CH4@C60	77.8	1.8	1.2	0.69	BPW91/6-311+G(d)//BPW91/D95V^214
SiH4@C60	83.7	4.6	3.6	0.78	PBE/3ζ^208
CF4@C60	78.7	2.4	2.1	0.88	M06-2X/6-31+G(d,p)^55
CO@C60	78.7	1.7	1.4	0.82	M06-2X/6-31+G(d,p)^55
HF@C60	77.4	0.1	0.01	0.09	BPW91/6-311+G(d)//BPW91/D95V^214
NHe@C60	82.9	0.8	1.2	1.43	PBE/3ζ^212
NHe@C60	67.5	0.6	0.8	1.33	M06-2X/6-31G(d)^212

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In context of the screening we mention the pnictogen endofullerenes N@C₆₀ and P@C₆₀, promising as quantum qubits.^215–217 Their mean polarizabilities and depressions (in parentheses) are 83.3 (−0.3) and 83.2 (−0.83) ų, respectively (PBE/3ζ calculations²³). It is necessary for the quantum processing to manipulate the spin of the endo-atom. At the same time, it would be better if the guest atom remain available for external electric fields that create a harmful interference in the qubit functioning. Polarizability allows numerical estimations how deeply the trapped atom is affected by the external impacts. For example, according to eqn (53), the screening of P atom in P@C₆₀ (c = 0.83) is more effective compared to N in N@C₆₀ (c = 0.31).²³ The larger depression in the case of P@C₆₀ reflects its higher stability compared to N@C₆₀ that has been shown in the theoretical study of pnictogen endofullerenes.²¹⁸ The mentioned study²¹⁸ also predicts that trapped nitrogen atom can be released by the complexation of N@C₆₀ with CS₂. This fact is indicated by the relatively low screening coefficient for N@C₆₀.

Exaltation of polarizability takes place in the functionalized fullerenes that has been demonstrated for the Ne@C₂₀H_n fullerene hydrides with n = 2, 6, 12 and 20 by PBE/3ζ and TD-HF/6-31G(d) methods.²⁰⁷ We have expected the decrease of exaltation but found the non-monotonic dependence of Δα on n number (Ne@C₂₀H₆ < Ne@C₂₀H₂₀ < Ne@C₂₀ ≈ Ne@C₂₀H₁₂ < Ne@C₂₀H₂). As it turned out, Δα values well inversely correlate with the covalency factors of the interaction between Ne atom and the most remote carbon atoms of the C₂₀H_n cages (Fig. 37).

Fig. 37 Dependences of polarizability exaltation of the series of Ne@C₂₀H_n endofullerenes on the reversed covalency factors χ⁻¹ that correspond to the maximal (L_max) and minimal (L_min) internuclear distances between the Ne atom and the cage (PBE/3ζ calculations taken from ref. 207).

The described nonadditivities are typical for the other fullerenes and more complex encapsulated molecules (e.g., in the case of the fullerenes C₇₀, C₈₀, C₉₀, C₁₀₀, and C₁₂₀ with the trapped quadricyclane and norbornadiene molecules⁵⁷). The most complex objects, for which depression of polarizability has been postulated, are C₆₀@C₂₄₀ endofullerene (studied by DFT⁴⁸) and multi-shell carbon nano-onions (e.g., 2(C₅₄₀@C₉₆₀)@C₂₉₁₀, studied in terms of monopole-dipole interaction approach^202–204). For C₆₀@C₂₄₀, depression of polarizability and screening coefficient equal to −74.1 ų and 0.90, respectively (PBE/NRLMOL calculations⁴⁸).

However, the list is not limited by the mentioned nanostructures. Depression of polarizability have been observed in computations of metal–silicon clusters,²¹⁹ endohedral complexes of silsesquioxanes,²²⁰ boron-nitride fullerene B₃₆N₃₆,⁵⁷ hollow silicate Si₁₆O₂₄(OH)₁₆ (ref. 57) and even in supramolecular ensembles (e.g., CH₄@nH₂O (ref. 221)). In addition, polarizabilities of encapsulated fullerene dimers,¹¹ endometallofullerenes (e.g., Li@C₆₀ (ref. 222) and Ti@C₂₈ (ref. 223)), metal-trinitride endofullerenes,²²⁴ and the structures with both-side functionalization (e.g., Ln@C₂₀–glycine conjugates with Ln = Ce and Dy²²⁵) have been also theoretically studied. Though these works did not describe the phenomenon of polarizability depression, it obviously takes place. Thus, the influence of confinement on the polarizability is a general property for diverse chemical objects.

In the lists above, we do not include the endohedral complexes of C₇₂ and C₇₄ fullerenes because their calculations^226,227 have been incorrectly performed (ref. 227 provides the unreliably high values for mean polarizability for La@C₇₂ and its derivative; ref. 226 states zero values for α_xx and α_yy of the polarizability tensor of Ba@C₇₄ and unreliably high α_zz).

Mean polarizabilities correlate with chromatographic retention times.²²⁸ Moreover, such a correlation has been found for endometallofullerenes³⁵ but with the reservation. To estimate retention times, the authors³⁵ have used polarizabilities of the empty fullerenes instead of the values of the filled ones. We consider that this is possible due to the depression of polarizability.

Unfortunately, there are no theoretical works on the inadditivities of endometallofullerene polarizability though this is rather widespread class of the endohedral species.¹⁹⁴ Based on the reviewed studies, we propose two cases for mean polarizability of endometallofullerenes based on C₆₀ and its higher analogues. The absence of covalent bonds or charge transfer between the trapped atom/cluster and the fullerene skeleton (the first case) must lead to the depression of polarizability due to the compression of the guest's electronic cloud. When the covalent bonds or charge transfer arise (the second case), the electronic clouds of the fullerene cage and the guest atoms are united and mean electronic polarizability of endohedral structure is contributed from the polarizabilities of the guest and the host. In addition, the endometallofullerenes in the second case have permanent dipole moments. It means that orientational effects should be taken into account according to eqn (10) when calculating the total polarizability.

Indeed, polarizability of X@C₆₀ and accompanied values can provide useful information about in what state the encapsulated species are. For example, in our study of the pressure-induced transformation of H₂O@C₆₀,²⁰⁹ we have theoretically considered the reaction (Fig. 38), in which water dissociates and O and H atoms react with the inner surface of C₆₀. This fact can be monitored by the evolution of polarizability (though the changes are small): depression of polarizability takes place in H₂O@C₆₀ but it vanishes in the product of the transformation (Fig. 38).

Fig. 38 Evolution of mean polarizability (in ų) upon the chemical transformation of water endofullerene (PBE/3ζ calculations taken from ref. 209).

The question about the placement of the trapped atoms inside the fullerene cages is not simple. Anisotropy of polarizability may be useful to assist in uncovering this issue. It reflects symmetry (or its absence) in the molecular structure. For example, it has been computationally demonstrated that after encapsulation H₂₀@C₈₀H₆₀ and H₂₀@C₈₀F₆₀ remain isotropically polarizable (a² = 0).²²⁹ Thus, despite the significant structural changes, these molecular systems retain the pristine symmetry of the hollow cage.

In the end of this section, we mention theoretical works in the field of chemical physics of endofullerenes.^230,231 These works approximate the fullerene cages as confining potentials and try to reveal relationships between the electronic structure of the encapsulated hydrogen atom and its polarizability. As shown by this simple case, polarizability presents aspects which are essentially related to the behavior of the wavefunctions.²³⁰ The interesting result has been obtained for the series of multishell endohedral complexes H@C₆₀, H@C₆₀@C₂₄₀, and H@C₆₀@C₂₄₀@C₅₄₀. It is shown that the addition of new walls does not modify the polarizability of the ground state of H atom but changes significantly those of the excited states.²³¹ However, such a detailed analysis seems to be hardly transferrable to the cases of the more complex endo-atoms.

8 Polarizability of fullerene ions and derivatives with ionic bonds

Fullerenes can generate negative and positive ions (e.g., C₆₀^z−, z = 1–6 and C₆₀^z+, z = 1–3).²³² The volumes of C₂₀, C₃₆, C₆₀, and C₇₀ have been previously studied in comparison with the uncharged states.²³³ In all cases, the cage enlarges when positively or negatively charged. However, this trend (expected from common notions) is not typical for mean polarizability, which has the dimension of volume. When negatively charged, mean polarizability of the C₆₀ cage increase according to DFT-calculations (Fig. 39). However, in the positive part of the plot, α(C₆₀^z+) remain almost unchanged up to z = 6 (PBE/3ζ) or grows up very slowly (B3LYP/Λ1) with increasing charge. If electrons leave the cage upon formation of C₆₀^z+, one can expect the decrease of polarizability. Possibly, it does not occur due to the cage expansion that makes C–C weaker and, consequently, more polarizable. This compensates the decrease in polarizability, caused by electron loss. The analogous situation has been observed in high-level CCSD/UCCSD computations of unsaturated hydrocarbons and their cations.²³⁴ Except for the smallest members in each series (benzene and butadiene), mean polarizability increases upon ionization (electron removal), and this increase becomes more pronounced with larger molecular size.²³⁴

Fig. 39 Dependence of mean polarizability of C₆₀ ions on the charge, obtained by PBE/3ζ (red diamonds) and B3LYP/Λ1 methods (white circles).

This compensation seems impossible when endohedral atoms has the bonds with the internals of the cage. For example, the charged endofullerene with the coordinated Li atom [Li@C₆₀]⁺ demonstrates the lower polarizability than the neutral Li@C₆₀, according to the accurate MP2 and DFT calculations.²²²

Fullerene may also form ionic compounds and clusters with metals, which contain ionic bonds in their structure. These derivatives are mainly clusters in their nature. Some of polarizability and electric susceptibility measurements for mixed clusters M_nC₆₀ (M is an alkali metal) have been previously reviewed.^15,235 In the present paper, we mention those and focus on a few newer compounds.

High dipole moments are typical for most of the C₆₀–alkali metal clusters.^15,236 In the context of the dielectric properties, it means that in measurements of dielectric susceptibility χ one should take into account the orientational polarizability, defined by eqn (10):

The measured susceptibilities are of 10³ ų order of magnitude^15,236–238 The second (orientational) term of eqn (54) is decisive for χ values. It was clearly demonstrated with the reverse temperature dependence for KC₆₀ (Fig. 40).²³⁷ The authors²³⁷ observed no permanent dipole moment for KC₆₀. This fact and the giant susceptibility allows proposing that KC₆₀ has no fixed structure and potassium atom skates on the C₆₀ surface.

Fig. 40 Temperature dependence of the polarizability of KC₆₀. Line represents the simulation by eqn (54). Reprinted with permission from ref. 237© 2000 The American Physical Society.

The molecular analogues of the clusters above K⋯C₂₀F₂₀, K⋯C₆₀F₆₀ (C_3v) and K⋯C₆₀F₆₀ (C_6v) have been theoretically studied (Fig. 41).²³⁹ After the DFT-optimization, their mean polarizabilities have been estimated by the HF/6-31+G(d) method: these are 30.0, 100.0, and 94.5 ų, respectively. The most interesting in this results is that the oblong carbon nanostructure K⋯C₆₀F₆₀ (C_6v) has higher mean polarizability than the compact K⋯C₆₀F₆₀ (C_3v) (potassium atom is more remote from the center of the C₆₀F₆₀ (C_6v) cage). This is reminiscent to the structural dependence of fullerene oligomer polarizability and differs from the case of the exohedral fullerene derivatives with non-fullerene addends.

Fig. 41 Optimized geometries of K⋯perfluorofullerene species (B3LYP/6-31G(d) calculations). Reprinted with permission from ref. 239© 2000 Springer.

Another perspective class of ionic fullerene derivatives can arise from fullerenols that are able to form salts with metals.²⁴⁰ Dielectric properties sodium fullerenol salt have been measured recently.²⁴¹ The measured values of conductivity and dielectric response of C₆₀(ONa)₂₄ were interpreted in terms of polarizability. Thus, polarizability of this material is mainly due to the distortion of the ionic bonds. The last ones are tight enough that no ionic contribution in the conductivity is observed up to 550 K. In this work, the reduction of the orientational effects upon increasing temperature has been also observed.²⁴¹ This means that C₆₀(ONa)₂₄ has a non-zero dipole moment. Based on this statement, we conclude that C₆₀(ONa)₂₄ has the non-uniform distribution of C–O⁻⋯Na⁺ bonds on the C₆₀ surface. This information is valuable, regarding the lack of the structural information about fullerenols.

9 Conclusion and perspectives

In the review, we have analyzed the results of theoretical and experimental studies of polarizability of different fullerene-containing compounds, trying to cover all the possible classes. Thus, nonadditivity accompanies their polarizability. The negative deviation from the additivity (depression of polarizability) has been found in the case of fullerene exohedral derivatives with simple addends and endofullerenes with more than 30 carbon atoms in the molecule. The positive one (exaltation of polarizability) is typical for fullerene bi- and multicage derivatives and endohedral derivatives of small fullerenes (C₂₀, C₂₄, and C₂₈). These additivity violations correspond to the conventional additive schemes, so one can find such an additive scheme that will be able to take into account all the effects described. However, the rough schemes, shown in the review, allowed deducing general formula for calculations of mean polarizability of fullerene adducts (eqn (37)) and screening coefficient for encapsulated atoms/molecules for endofullerenes (eqn (53)) as well as correlations between the polarizability of fullerene dimers and the remoteness of fullerene cores in their structure. These should be helpful for the relevant experiments on fullerene derivatives polarizability.

Despite the fact that the discussed data are mainly computational, these well correlate with the known chemical and physical properties of fullerenes and their derivatives. Moreover, the reviewed data demonstrates that this material is not tacit numbers but very effective tool for understanding of unusual physicochemical processes in fullerene-containing systems and the design of the novel fullerene derivatives with improved molecular and macroscopic properties. Among them, we point insights into fullerene-based molecular switch and design of fullerene bisadducts for organic solar cells.

Based on the cited papers, we consider that the application of polarizability to fullerene science is in the starting phase. Indeed, its applications are striking but non-numerous. However, we may expect the novel cases, which may arise from the remarkable achievements in the studies of correlations between the polarizability of organic compounds and their interaction with positrons,²⁴² its use for understanding physicochemical processes in photovoltaic devices,^243,244 fullerene-containing polymers,²⁴⁵ and biological systems.²⁴⁶ Polarizability should be taken into account by mechanistic studies, when classic reactions are performed in electric fields.²⁴⁷ Thus, theoretical studies of polarizability in terms of transition state theory are currently performed.²⁴⁸

Endofullerene polarizability provides a universal model for description of screening effects and the confined molecules' behavior under electric fields. This model is easily extrapolated to the more complex encapsulated objects (see, e.g., ref. 249) as well as more complex encaging nanostructures.

Another opportunity for this field arise from the applications of carbon nanomaterials. As shown recently, polarizability is sensitive to their mechanical deformations²⁵⁰ and may be important for understanding of the encapsulated molecules' behavior.²⁵¹ In addition, it may provide new insights into the methane²⁵² and hydrogen^253,254 storage by carbon materials under the external electric fields. Another perspective way to use polarizability deals with moving and trapping nano-objects by laser pulses and electric impacts.^255–257 Obviously, this list can be extended.

The main challenges of application of polarizability originate from the difficulties of its measurements or (in certain cases) accurate calculation. Thus, the polarizability has been used mainly in qualitative aspect for understanding the processes in fullerene-based nanosystems. We hope that the performed review on polarizability of fullerenes and their derivatives will be useful for their further theoretical and especially experimental studies dealing with applications. The title of the review might seem audacious. However, we believe that in coming years it will be justified.

Acknowledgements

The author is grateful to Markus Arndt (University of Vienna, Austria), Ramil G. Bulgakov (Institute of Petrochemistry and Catalysis of RAS, Russia), Eugene Katz (Ben-Gurion University of the Negev, Israel), Sergey L. Khursan (Institute of Organic Chemistry, Ufa Scientific Centre of RAS, Russia), Eiji Ōsawa (Nano Carbon Institute, Japan), Roberto Rivelino (Universidade Federal da Bahia, Brazil), and Ajit J. Thakkar (University of New Brunswick, Canada) for their assistance at the diverse steps of the theoretical study on fullerene polarizability. In 2009–2014, the Presidium of Russian Academy of Sciences financially supported the author's works on polarizability (Program no. 24 “Foundations of Basic Research of Nanotechnologies and Nanomaterials).

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