Topology-Based Approach to Predict Relative Stabilities of Charged

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Topology-Based Approach to Predict Relative Stabilities of Charged and Functionalized Fullerenes Yang Wang, Sergio Díaz-Tendero, Manuel Alcami, and Fernando Martin J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b01048 • Publication Date (Web): 27 Jan 2018 Downloaded from http://pubs.acs.org on January 28, 2018

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Journal of Chemical Theory and Computation

Topology-Based Approach to Predict Relative Stabilities of Charged and Functionalized Fullerenes Yang Wang,∗,†,‡ Sergio Díaz-Tendero,†,¶,‡ Manuel Alcamí,†,§,‡ and Fernando Martín†,§,¶ †Departamento de Química, Módulo 13, Universidad Autónoma de Madrid, 28049 Madrid, Spain ‡Institute for Advanced Research in Chemical Sciences (IAdChem), Universidad Autónoma de Madrid, 28049 Madrid, Spain ¶Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, 28049 Madrid, Spain §Instituto Madrileño de Estudios Avanzados en Nanociencia (IMDEA-Nanociencia), 28049 Madrid, Spain E-mail: [email protected]

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Abstract Understanding the relationship between structure and stability is one of the fundamental aspects of fullerene chemistry, as the number of possible cage isomers is very large and complexity increases by orders of magnitude when chemical groups are attached to the fullerene cage. The well-established stability rules valid for neutral fullerenes do not apply to many charged or functionalized fullerenes. Here we present the theory, implementation and applications of two simple topology-based models that allow one to predict the relative stability of charged and functionalized fullerenes without the need for quantum chemistry calculations: (i) the charge stabilization index (CSI) model, based on the concepts of cage connectivity and frontier π orbitals, which offers a general framework for the relative stability of both positively and negatively charged fullerenes, as well as endohedral metallofullerenes, and (ii) the exohedral fullerene stabilization index (XSI) model, which incorporates all key factors governing the stability of exohedral fullerenes, namely, π delocalization, σ strain and steric hindrance between addends. Based exclusively on topological information, both models are powerful prescreening tools for predicting the most stable structures of a large number of charged and functionalized fullerenes. For easy use by fullerene chemists, both models have been implemented in the FullFun (for Fullerene Functionalization) software package, whose effectiveness and efficiency are demonstrated by some illustrative examples.

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INTRODUCTION

Fullerenes, a family of closed-cage carbon molecules, have received widespread attention in physics and chemistry 1–4 since their discovery in 1985 by Curl, Kroto and Smalley. 5,6 Although fullerene chemistry usually involves neutral species, it should not be ignored that fullerenes were actually detected in the form of positive ions by mass spectrometry. Produced by photoionization of neutral fullerenes, fullerene cations are naturally present in outer space, 7–10 and play a relevant role in the growth of carbonaceous species. 11,12 They have also been intensively studied in collision experiments, 13,14 where neutral fullerenes are multiply ionized by high energetic ions, electrons or XUV and x-ray light. 13–17 Experiments 17 and theory 18 have demonstrated that the maximum charge that a C60 cage can carry without being torn apart can be as high as +14. Positively charged fullerenes are also found in solution, 19–21 fullerenium salts 22–24 and polymeric films, 24 where they are formed and stabilized by superacids or chemical oxidation. Fullerenes can also be found in the form of anions in, e.g., cyclotron resonance traps, 25 storage rings 26 and electrospray mass spectrometry experiments. 27–30 They can also be prepared in solution by chemical 30–32 or electrochemical reduction. 33–39 Moreover, due to their high electron affinity, fullerene cages are negatively charged in the form of fullerides 40 formed by intercalation of metals 41–43 or organic reducing agents. 44 Another common way to obtain an anionic fullerene cage is by endohedral functionalization with metals, as will be discussed in more detail below. Chemical functionalization endows fullerenes with large structural diversity and versatile properties. 45–47 Owing to their cage-like cavity, fullerenes can encapsulate small chemical species, such as atoms, molecules or atomic clusters, forming a new family of complexes named endohedral fullerenes. 48–50 In particular, when the encaged species is a metal or a metal-containing cluster, the carbon cage is usually negatively charged and its electronic and/or magnetic properties are modulated because of the charge transfer. These complexes, the so-called endohedral metallofullerenes (EMFs), 51–53 have vastly enriched the chemistry 3



of fullerenes, since many otherwise unstable cage structures have been successfully produced and isolated, 54 even in macroscopic amounts. 51,55 EMFs have unique properties 56 and prospective applications in biomedicine 57–59 and photovoltaics. 60,61 Alternatively, the functionalization of fullerenes can be achieved by attaching chemical groups to the cage outer surface, leading to the so-called exohedral fullerenes. 4,62,63 The exohedral derivatization of fullerenes can enhance their solubility and proccessibility, and endow them novel properties. 63 Generally speaking, the functional groups can be any chemical species, from simple atoms, such as hydrogen 64–66 or halogens, 67–71 to very complex moieties, such as sugars 72,73 and macrocycles. 74,75 Therefore, exohedral fullerenes have numerous applications in medicine 76–78 and materials science. 79–82 Here, we put special emphasis on fullerenes functionalized by atoms or simple groups (hereafter referred to as prototype exohedral fullerenes 83 ), which are the starting point for much more complex derivatives and the building blocks of supramolecular fullerene polymers and networks. 63,84 Hydrogenated fullerenes or fulleranes have been detected in samples of the Allende meteorite, 85 and are probably carriers of diffuse interstellar bands and other interstellar and circumstellar features. 86,87 The hydrogenation of fullerenes may also have important implications for catalysts, hydrogen storage and electronic devices. 64,88–90 Halogenated fullerenes can be conveniently used in organic synthesis of more complex fullerene derivatives, 91 for they are precursors for subsequent nucleophilic substitution and/or addition at free sites elsewhere in the molecule. 92–98 The enhanced electron affinity makes halogenated fullerenes excellent electron acceptors for novel charge-transfer complexes, 99 donor-acceptor dyads, 100,101 and photonic/photovoltaic devices. 102,103 Perfluoroalkylated fullerenes have also been intensively studied, 104 with practical applications as powerful and tunable acceptors in the field of organic electronics. 104–106 One of the fundamental aspects of the chemistry of neutral, charged and functionalized fullerenes is the relationship between stability and structure, due to the huge number of possible isomers. For a pristine empty fullerene, the number of possible cage forms is,

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e.g., almost ten thousand for C90 , and grows dramatically with cage size. 107 This complexity becomes even more striking when adding functional groups to the fullerene cage. Taking the highly symmetric buckminsterfullerene as an example, there are over 20 million possible ways of binding eight addends of the same kind to the cage, 108,109 and this number scales factorially with the number of addends. When we consider both the cage isomers and the regioisomers (or say, addition patterns) for each cage, the total number of possible structures would be unmanageable, e.g., over 50 billion for C60 X8 . It is also important to point out that, depending on the synthetic conditions, the formation of exohedral fullerenes can be either thermodynamically or kinetically controlled. In very hot environments during arc-discharge, 110 combustion 111,112 or radio-frequency furnace synthesis, 113,114 fullerene cages are formed through carbon-clustering process 115 at temperatures typically higher than 4 000 K 116 and thus the final derivatized products usually correspond to global minima of the potential energy surface. Due to the effect of the addends, the topology of the derivatized cage may differ from that of the most stable pristine cage. 54 Conversely, in moderate-temperature conditions (typically from room temperature to 500 ◦ C), as in solution chlorination, 70 solid/gas hydrogenation, 117,118 solid-phase fluorination, 119 solid/liquid bromination, 120 or gas phase ion-molecule reactions, 121 fullerene cages are already formed before the addition and their topology usually remains unaltered during the reaction. Occasionally exceptions are found, where the fullerene cage may undergo Stone-Wales 122 skeletal rearrangement 123 or C2 loss 123–126 or insertion. 127,128 The final products may be kinetically controlled and correspond to local minima of the potential energy surface. Over the last decades, many theoretical efforts have been made to understand and predict stable structures of neutral, charged fullerenes and their derivatives. For neutral pristine fullerenes, it has been revealed that the major factor determining the relative isomer stability is σ strain caused by the adjacency of pentagonal rings (pentagons for short) in the cage framework. 129–132 Consequently, in stable neutral cages, pentagons are inclined to locate as far as possible from each other, which is the basis of the well-known isolated pentagon rule

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(IPR) 129 and the pentagon adjacency penalty rule (PAPR). 130,131 By applying these rules, the number of possible isomer candidates is substantially reduced when looking for the stable structures. However, these stability rules do not apply to many charged fullerenes, 14,133,134 EMFs 51,54 and exohedral fullerenes. 54 Many non-IPR cages or cages containing a larger number of adjacent pentagon pairs (APPs) have been observed in structures synthesized experimentally or predicted computationally. In a recent work, 135 we have introduced a simple and unified model, called the charge stabilization index (CSI) model, that allows one to predict the relative stability of both positively and negatively charged fullerene isomers, based on the concepts of cage connectivity and π frontier orbitals. 135 Requiring solely the knowledge of the cage topology, with no need for either geometry optimizations or iterative electronic structure calculations, the CSI model has successfully predicted the cage structures of many charged fullerenes and EMFs observed in experiments or determined from elaborate quantum chemistry calculations. It is worth mentioning that earlier theoretical models based on different physical arguments, such as electrostatic interaction, 133,136 energy gaps between LUMOs 137 and aromaticity, 138 are difficult to apply to general families of fullerene compounds as they require computationally demanding geometry and/or electronic structure calculations. As far as exohedral fullerenes are concerned, as mentioned above, we need to take into account two types of isomerism, cage forms and addition patterns. Regarding the former aspect, Kroto and Walton 139 suggested that small hydrogenated fullerenes (from C24 H12 to C50 H10 ) can be stabilized by maximizing conventional aromatic domains concomitant with alleviating cage strain. Consequently, in derivatized non-IPR fullerenes, adjacent-pentagon sites are preferentially occupied in order to release the strain. However, the rule does not always apply, as in C54 Cl8 , 140 C64 Cl8 141 and C50 X12 (X = H, F, Cl), as a result of the delicate interplay between π delocalization, σ strain and steric hindrance. 83 With respect to the addition patterns, several basic rules have been proposed. Fowler et al 142,143 pointed out that addition patterns yielding an odd number of bare carbon atoms isolated by addends have open-shell

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configuration and can thus be ruled out. This rationalization reduces the number of possible regioisomers, but usually to a relatively small extent. They also proposed that bulky addends should never bind to adjacent sites on the fullerene cage. 143 Using this assumption combined with the above-mentioned non-radical condition, Fowler et al predicted the maximum stoichiometries for bromofullerenes and prescreened regioisomers of those compounds solely based on topological information. 143 However, exceptions have been found in some experimentally identified exohedral fullerenes with bulky addends, including C60 Br6 , 144–146 C60 (CF3 )14 , 147 C60 (CF3 )16 , 148 C70 (CF3 )18 149 and C70 (CF3 )20 . 150 It has also been observed that exohedral additions rarely take place at triple-hexagon junction (THJ) sites, 104,151,152 for the carbon atom at such a site has a planar sp2 configuration precluding it from converting to a pyramidal sp3 one. 151 This empirical rule fails in many synthesized fullerene derivatives, especially with a higher degree of addition, as in C70 Cl28 , 153 C70 F38 , 154 C76 (CF3 )16,18 , 155,156 C78 Cl18,30 , 157–159 C88 Cl12 , 160 C90 Cl24 , 161 C94 (CF3 )20 , 162 C100 Cl30 , 123 C102 Cl18,20 , 163 etc. Additionally, some addition motifs were proposed to understand the experimental addition patterns of X groups (X = H, 164–172 F, 165,167,168 Br 165,169–171,173–175 and CF3 104 ), which, nevertheless, are rather empirical observations a posteriori and unable to provide quantitative predictions. To solve these problems, we have recently extended the CSI model and introduced the exohedral fullerene stabilization index (XSI) model, which allows one to predict the relative stability of prototype exohedral fullerene isomers. 83 The model incorporates the three key stability factors, namely, the change in π stability, the release of σ strain and the introduction of steric effects among addends upon the binding of atomic or molecular species to the fullerene cage. The XSI model, in combination with graph theory 176,177 and the stepwise addition algorithm, 164,175,178–184 has allowed us to rapidly predict a good number of experimentally observed structures of exohedral fullerenes, without resorting to expensive quantum chemistry calculations. In this work, we present a detailed description of the theory behind these models, as well as their computer implementation and applications to systems not considered in previous

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works, such as nonclassical fullerenes. To facilitate their use by other chemists, we have developed the FullFun (for Fullerene Functionalization) software package, and provide some illustrative examples to demonstrate its performance. In particular, by applying the CSI model to C4− 42 , we correctly predict the low-energy isomers of Ti@C42 detected in mass spectra 185 and characterized by elaborate DFT computations. 186 By combining the CSI model with a recently proposed structural-motif model, 132 we also correctly predict the relative stability of large fullerene cations, such as C6+ 104 . We have also used the FullFun package to determine the global minimum-energy structure of C36 H6 187 and shown that it is different from the one proposed in earlier work. 188 Finally, the FullFun package has been used to search for the low-energy regioisomers of C104 Cl24 containing a nonclassical cage. 189 The paper is organized as follows. In Section 2 we describe in the detail all the theoretical aspects that are the basis of the CSI and XSI models and in Section 3 its implementation in the FullFun package (more technical details are presented in an Appendix). Applications to the different cases mentioned above are presented in Section 4. The most important conclusions of this work are summarized in Section 5. For convenience, we provide in Appendix A a list of abbreviations used in this article.

2

THEORY

It is well known that the frontier orbitals of fullerenes and related systems are π orbitals delocalized over the cage surface, which play an important role in their stability, reactivity, aromaticity, electronic and spectroscopic properties. 1–4 As one of the simplest quantum models, the simple Hückel molecular orbital (HMO) theory 190–193 has achieved a great success for describing the delocalized conjugated π system of organic molecules. 194 In a nutshell, the simple HMO method derives from a linear combination of atomic orbitals (LCAO) treatment of a delocalized π system, with some simplifying approximations. The wave function of the whole system is a linear combination of pz atomic orbitals centered at each atom and

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perpendicular to the molecular plane for a perfectly planar π system. In the case of curved molecules like fullerenes, each pz atomic orbital is locally perpendicular to the mean plane of all σ bonds formed by the corresponding atom. By neglecting all interatomic overlap integrals, these pz atomic orbitals form a minimum orthonormal basis. The corresponding Hamiltonian matrix depends only on the connectivity between atoms: all diagonal elements are equal to a parameter called Coulomb integral, denoted as α, which is the Hamiltonian integral between pz orbitals of the same atom; the off-diagonal elements are zero if the corresponding pair of atoms are not bonded; for a bonded atom pair, they are equal to a parameter called resonance integral, denoted as β, being the Hamiltonian integral between pz orbitals of the bonded atoms. As a result, we obtain the energies and coefficients of molecular orbitals simply by diagonalizing the adjacency matrix of the molecular graph. The construction of adjacency matrix for bare and derivatized fullerene cages will be detailed in subsections 2.1.2 and 2.2.2. The HMO method has several advantages. Based solely on the connectivity between atoms, a simple HMO calculation is almost computationally costless, compared to other quantum chemistry methods involving iterative electronic structure computations and geometry optimizations. This benefit is especially powerful for dealing with a huge number of isomers of fullerenes and their derivatives. Moreover, unlike more sophisticated methods, the simple HMO model describes exclusively the π effect, which can thus be clearly separated from other effects like σ strain and steric hindrance. In addition, it also provides a conceptual parameter-free framework 194 for many fundamental properties of a molecular system, such as atomic charges, bond orders, free valence, band gaps, local aromaticity, resonance energy, etc. In the present work, we have applied the simple HMO method to charged and derivatized fullerenes, which leads to two simple topology-based models, as detailed in the following subsections.

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2.1 2.1.1

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Charged Fullerenes and EMFs The Hamiltonians for charged fullerene systems

The Hamiltonian of the π electrons of a fullerene Cq2n with charge q consists of three parts: kinetic energy, electron-nucleus potential energy and electron-electron repulsion energy, as given below: 2n−q

ˆq = H

X k=1

2n−q 2n X −∇2k ZA∗ 1X 1 − + 2 |rk − RA | 2 l6=k |rk − rl | A=1

! (1)

where rk and rl are the positions of electron k, and l, respectively; RA is the position of nucleus A and ZA∗ is its effective nuclear charge approximating the shielding effect of inner electrons. In the following, we demonstrate that for a fullerene system the π electron Hamiltonian can have a much more simplified form, thanks to the nearly spherical shape of the molecule. Firstly, let’s look at the electron-nucleus interaction energy. For a highly symmetric fullerene cage (such as icosahedral Ih (1)-C60 and Ih (7)-C6− 80 ), all nuclei are approximately located on a sphere of radius Rn . If we assume that the total electric field exerted by the nuclei is also spherical, the electron-nucleus potential energy for a single electron depends only on its radial position r (with respect to the cage center) and can be evaluated as an angular average: π

Z

Z

Ven (r) = − 0

0

2n 2π X

ZA∗ sin θdθdϕ. |r(r, θ, ϕ) − RA | A=1

(2)

On the other hand, being spherical, the electric potential of nuclei can be calculated by using Gauss’ law. Assuming that all nuclei have the same effective charge ZA∗ , we thus obtain the electron-nucleus interaction energy for a single electron at radial position r simply as

Ven (r) = −2nZA∗ U (r; Rn ),

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(3)

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where the function U (r; R) is defined as

U (r; R) =

 1      R

if r < R

     1 r

if r > R

equation

(4)

We have calculated Ven (r) using equations 2 and 3 for Ih (1)-C60 and D5h (1)-C70 . In the latter case, the cage is not spherical (instead, an elongated spheroid) and we take the nuclei averaged value of Rn in equation 3 by neglecting the small corrugations of nuclear positions on the sphere. As can be seen in Figure 1, the results obtained by using both equations 2 and 3 are in perfect agreement with each other, even for the non-spherical C70 cage.

Rn(C70)

−4

Ven/ZA* (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Rn(C60) −6

C60 (exact) C60 (Gauss) C70 (exact) C70 (Gauss)

−8

−10 0

2

4

6

8

10

r (Å) Figure 1: Electron-nucleus interaction energy per nuclear charge, Ven /ZA∗ , for a single electron as a function of radial position r, for fullerenes Ih (1)-C60 (red color) and D5h (1)-C70 (blue color). Empty points present the exact values computed numerically by using equation 2, while solid curves denote the analytic results from Gauss’ Law by using equation 3. The average cage radius, Rn , is indicated by a vertical line. Secondly, for the electron-electron repulsion energy, we can apply similar approximations. We assume that the electron density distribution is spherically homogeneous. This is generally a reasonable approximation, as it has been proposed that the π electron density can be represented to zero order by a uniform electron density distribution (or a conducting 11



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sphere). Indeed, our DFT/B3LYP calculations demonstrate that the charge inhomogeneities in charged fullerenes are typically of the order of 0.05 e− in the vicinity of the atomic centers, representing a rather small fraction of the total number of electrons occupying the π orbitals. This approximation has been employed in earlier work to study the ionization of fullerenes, 195 the fragmentation dynamics of charged fullerenes, 196 and to reproduce the experimental/DFT ionization potentials and electron affinities for various fullerenes, such as C60 , 197 C76 , C78 and C84 . 198 Consequently, the potential that an individual π electron feels due to the other (2n − q − 1) π electrons is spherical. According to Gauss’ law, the electron-electron interaction energy per electron can be approximated as

Vee (r) =

2n − q − 1 U (r; Rπ ), 2

(5)

where q is net charge of the fullerene molecule and Rπ the average radius of the sphere of the electron cloud. Thus, the π electron Hamiltonian of Cq2n is approximately expressed as (cf. equation 1) 2n−q

X −∇2 2n − q − 1 k ∗ H ≈ − 2nZA U (rk ; Rn ) + U (rk ; Rπ ) . 2 2 k=1 ˆq

(6)

ˆ q is separable and can be written as the sum of individual As can be seen, the Hamiltonian H ˆ q }: single-electron Hamiltonians {h k 2n−q

ˆq = H

X

ˆq h k

(7)

k=1

where 2 ˆ q = −∇k − 2nZ ∗ U (rk ; Rn ) + 2n − q − 1 U (rk ; Rπ ). h A k 2 2

12


(8)

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Then, the Coulomb integral is

ˆ q |φA i = α0 − αq = hφA | h k

q hφA | U (r; Rπ ) |φA i 2

(9)

where {φA } are atomic orbitals (more specifically, 2pz orbitals of carbon) comprising the molecular orbitals, and α0 is the Coulomb integral for neutral systems, written as 2 ˆ 0 |φA i = hφA | −∇ − 2nZ ∗ U (r; Rn ) + 2n − 1 U (r; Rπ ) |φA i . α0 = hφA | h k A 2 2

(10)

We define the effective π electron cloud radius as the inverse of the last integral in equation 9:

Rπ∗ =

1 . hφA | U (r; Rπ ) |φA i

(11)

It can be shown that Rπ∗ defined in this way is in practice almost identical to Rπ (see Appendix B). By simply denoting α0 as α, we now have a simpler form of the Coulomb integral, as follows:

αq = α −

q . 2Rπ∗

(12)

The resonance integral is given by ˆ q |φB i = β 0 − q hφA | U (r; Rπ ) |φB i , β q = hφA | h k 2

(13)

where β 0 is the resonance integral for neutral systems, written as −∇2 2n − 1 0 ˆ β = hφA | hk |φB i = hφA | − 2nZA∗ U (r; Rn ) + U (r; Rπ ) |φB i . 2 2 0

(14)

It follows from the simple HMO theory that the last integral in equation 13 is zero, as

13



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demonstrated in the following. From equation 4, it is obvious that

0 < U (r; R) 6

1 . R

(15)

Noticing that pz orbitals φA and φB are of adjacent atoms, and thus are more or less parallel to each other, indicating that they have the same sign in the integrand. Using the fact that the overlap between φA and φB is neglected in the simple HMO theory, we have

0 6 hφA | U (r; Rπ ) |φB i 6 hφA |

1 1 |φB i = hφA | φB i = 0. Rπ Rπ

(16)

Hence,

hφA | U (r; Rπ ) |φB i = 0.

(17)

This means that the resonance integral is independent of the fullerene charge q:

β q = β 0 ≡ β.

(18)

We note that the Coulomb and resonance integrals are always negative and are assumed to be the same for all isomers of the same charge q.

2.1.2

The charge stabilization index (CSI)

Following the same approximations and procedures as in the simple HMO theory, the wave function is obtained variationally by diagonalizing the adjacency matrix of the i-th cage isomer of fullerene Cq2n , {aiAB }, whose elements are equal to 1 if atoms A and B are bonded, and 0 otherwise. The π orbital energies {qk,i } can be expressed in terms of the eigenvalues

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{χik } of the adjacency matrix: i i,q k = α + βχk −

q 2Rπ∗

(19)

Hence, the total π energy of the i-th cage isomer of charged fullerene Cq2n (for simplicity, we consider a closed-shell system where q is an even number) is

Eπi,q

n−q/2 X q = (2n − q) α − + 2β χik . 2Rπ∗ k=1

(20)

Note that the sum is over the highest (n − q/2) eigenvalues of the connectivity matrix {aiAB }. That is to say, the eigenvalues {χik } should be arranged in descending order. From equation 20, we can calculate the Hückel charge stabilization energy (HCSE) of charged fullerene isomers. 135 The HCSE is defined as the π energy of isomer i of charged fullerene Cq2n , referred to that of the neutral cage with the same isomeric form, is calculated as

∆Eπi,q = Eπi,q − Eπi,0 = −2βXiq − qα +

(2n − q)q 2Rπ∗

(21)

where

Xiq

≡

         

n X

χk,i if q > 0

k=n−q/2+1

(22)

   n−q/2  X    χk,i  

if q < 0

k=n+1

As can be seen, for positively charged fullerenes, Xiq sums over the eigenvalues of the HOMOs from which |q| electrons have been removed, whereas for negatively charged fullerenes, the sum runs over the LUMOs that have accepted the extra |q| electrons. Assuming that the last two terms in equation 21 are constant for all cage isomers with the same charge, Xiq 15



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represents the variation of π energy, in units of −2β, due to addition (removal) of |q| electrons to (from) the frontier π orbitals of the corresponding neutral isomer i. In order to establish a simple model to evaluate the relative energy of charged fullerene isomers, we start by estimating the relative energy of neutral isomers, which can be simply approximated, following the PAPR, 130,131 as

i,0 Erel = 0.2 NAPPi (−2β)

(23)

where NAPPi is the number of APPs of cage isomer i. The coefficient 0.2 is a characteristic value of the energy penalty per APP (in units of −2β), which falls in the accepted range, 19–24 kcal/mol, established in previous work. 131,199 Although strictly speaking the energy penalty per APP may vary with cage size within the above range, we use 0.2 as a universal value for all sizes and charge states of fullerenes for the sake of simplicity. We have shown that the predictions of our models remain unchanged when the actual values of the energy penalty per APP are used. 135 Now, we can evaluate the relative energy of a given fullerene isomer i with charge q as

i,q i,0 Erel = Erel + ∆Eπi,q

= (Xiq + 0.2 NAPPi )(−2β) − qα +

(2n − q)q . 2Rπ∗

(24)

We can see that the first term depends on the isomeric form of cage i and the charge state q, while the last two terms are constant for all isomers of the same cage size and at the same charge state. We thus define the charge stabilization index (CSI) 135 of a given fullerene isomer i with charge q as

CSIqi ≡ Xiq + 0.2 NAPPi .

(25)

In comparison with equation 24, the CSI can be regarded as the relative isomer energy, in

16


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units of −2β, with reference to the constant −qα + (2n − q)q/(2Rπ∗ ). We would like to stress that the CSI is a purely topology-based quantity that depends only on interatomic connectivity of the cage framework. This remarkable advantage of CSI allows us to efficiently determine the most stable cage forms of charged fullerenes, even among a vast number of possible isomers. 135 For a better visualization of the results, it is useful to define relative values of CSI as ∆CSIqi = CSIqi − CSIq0 , where CSIq0 is the charge stabilization index for the isomer of charge q corresponding to the lowest-energy structure in neutral state. Thereby, a negative value of ∆CSIqi implies that an isomer with charge q is more stable than the isomer with the same charge produced from the lowest-energy neutral isomer. In such a case, π charge stabilization surpasses the possible destabilizing strain effect. Consequently, the lowest-energy isomer in a given charge state can usually be found in the set of isomers with negative ∆CSIqi . Thus, for a given cage size 2n, ∆CSIqi serves as an indicator of the relative stability of different fullerene isomers of the same charge.

2.1.3

Improvement of CSI using the structural motif model

As discussed previously, we have used the PAPR to estimate the relative energy of neutral isomers, as given by equation 23. This approximation works fairly well for small or mediumsized fullerenes or anionic fullerenes. However, this is not the case for large fullerene cations. On the one hand, the charge effect on the relative stability of cationic fullerenes is less strong 135 and thus an accurate estimation of the relative energy of neutral isomers becomes more important. On the other hand, large fullerenes have many isomers with the same number of APPs. For instance, there are over 80 IPR isomers for fullerenes larger than C90 . According to the PAPR, however, all neutral isomers with the same number of APPs have exactly the same relative energy. Consequently, the prediction performance of CSI formulated by equation 25 would be worse for large fullerene cations. Very recently, we have developed a generalized structural motif model to reproduce the

17



Page 18 of 80

excess energy of IPR fullerenes. 132 This topological model is valid for a wide range of cage sizes up to graphene limit, with typical deviations of 3.4 kcal mol−1 from the DFT energies. For large fullerene cations, since the lowest-energy isomers usually correspond to the IPR ones, 135 we can apply the motif model to improve the performance of the CSI model for these cases. For IPR isomers, the CSI can be computed using an improved formula, as follows:

i,0 CSIqIPR,i = Xiq + Emot .

(26)

i,0 Here, Emot is the excess energy of isomer i estimated using the motif model: 132

i,0 Emot (C2n ) =

30 X

i E m Nm + J ln

m=1

2n + L , 60

(27)

i is the number of motifs m appearing in cage isomer i; Em is the corresponding where Nm

excess energy contribution of motif m; J and L are parameters related to the global curvature effect. The optimized values of all these parameters can be found in Ref 132. Note that a typical value of −2.49 eV 200,201 can be chosen for the resonance integral β when converting the units of energy from kcal/mol to −2β.

2.2 2.2.1

Exohedral Fullerenes The Hamiltonians for exohedral fullerene systems

We can also apply the simple HMO method to prototype exohedral fullerenes C2n X2m , where 2m atoms or chemical groups X are bonded to the outer surface of the carbon cage C2n . We refer to isomer [i, j] of C2n X2m as the isomer with a particular cage form i and a particular distribution pattern j of the 2m addends on that cage. In other words, we use i and j to denote, respectively, the cage isomer and the regioisomer. Compared to the pristine cage C2n , the HMO model for C2n X2m excludes the 2m carbon atoms that are bonded to addends X, for these atoms no longer participate in the π system of the derivatized cage.

18


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Following procedures similar to those described in the previous subsections, we can deduce the Coulomb and resonance integrals for exohedral fullerenes. The electron-nucleus potential energy for a single electron in C2n X2m can be approximately written as Ven (r) = −2(n − m)ZA∗ U (r; Rn ).

(28)

To validate this formula, we have compared the values of the Ven (r) potential obtained from equations 28 and 2, for the exohedral fullerenes Ih (1)-C60 X12 (X = H, Cl and CF3 ). 83 The experimental addition patterns have been taken for X = Cl 202 and CF3 . 203 As to X = H, the lowest-energy regioisomer has been chosen based on self-consistent charge (SCC) density functional tight-binding (DFTB) calculations. 83 As shown in Figure 2, the electron-nucleus interaction energies calculated by equations 2 and 28 are in excellent agreement. The results for X = H, Cl and CF3 are almost identical, regardless of the very different addition patterns. We can also see that the Ven (r)/ZA∗ curves for C60 X12 lie above that for pristine C60 , and are shifted by 2mU (r; Rn ) (see equation 28) due to the removal of 2m carbon atoms in the HMO model. The electron-electron interaction energy in C2n X2m can be approximately written as

Vee (r) =

2(n − m) − 1 U (r; Rπ ) 2

(29)

and thus the single-electron Hamiltonians are given by 2 ˆ 2m = −∇k − 2(n − m)Z ∗ U (rk ; Rn ) + 2(n − m) − 1 U (rk ; Rπ ). h A k 2 2

(30)

So, we obtain the Coulomb integral for C2n X2m : ˆ 2m |φA i = α + 2mZ ∗ hφA | U (rk ; Rn ) |φA i − m hφA | U (r; Rπ ) |φA i α2m = hφA | h k A

19


(31)


−2

(a)

−3

Rn(C60X12) Rn(C60)

−4

Ven/ZA* (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 80

−5

C60H12 (exact) C60H12 (Gauss) C60Cl12 (exact) C60Cl12 (Gauss) C60(CF3)12 (exact) C60(CF3)12 (Gauss) C60 (exact) C60 (Gauss)

−6 −7 −8 −9 0

1

2

3

4

5

6

7

8

9

(b)

10

r (Å)

(c) C60H12

(d)

C60Cl12

C60(CF3)12

Figure 2: (a) Electron-nucleus interaction energy per nuclear charge, Ven /ZA∗ , for a single electron as a function of radial position r, for exohedral fullerenes Ih (1)-C60 X12 (X = H, 83 Cl 202 and CF3 , 203 indicated by blue, red and green color, respectively) and pristine Ih (1)-C60 (black color). Empty points present the exact values computed numerically by using equation 2, while solid curves denote the analytic results from Gauss’ Law by using equation 28. The average cage radius, Rn , is indicated by a vertical line. (b–d) Schlegel diagrams showing the distribution pattern of addends X on the cage for X = H, Cl and CF3 , 104 respectively. By defining the effective cage radius, Rn∗ , as follows (cf. equation 11), Rn∗ =

1 hφA | U (r; Rn ) |φA i

(32)

we have a simpler form of the Coulomb integral:

α2m = α +

m ∗ RX

(33)

where 2ZA∗ 1 1 − . ≡ ∗ RX Rn∗ Rπ∗

(34)

Similar to the case of charged fullerenes, the resonance integral for C2n X2m is the same as

20


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that for the neutral pristine cage:

β 2m = β.

2.2.2

(35)

The exohedral fullerene stabilization index (XSI)

In view of the above, the π energy of isomer [i, j] of C2n X2m , with respect to that of the pristine C2n with the same cage form i, can be written as ∆Eπ[i,j]

=

Eπ[i,j]

−

Eπi,0

=2

n−m X

(α

2m

+

β 2m χjk,i )

k=1

−2

n X

(α + βχk,i )

(36)

k=1

[i,j]

where {χjk,i } are the highest (n − m) eigenvalues of the adjacency matrix {aAB } associated with isomer [i, j] of C2n X2m . To build the adjacency matrix of C2n X2m , we start with that of the pristine C2n with the same cage form. Then, all rows and columns corresponding to the 2m carbon atoms that are bonded to addends X are deleted. Consequently, the first sum in equation 36 runs over (n − m) orbitals instead of n. Using equations 33 and 35, we have

∆Eπ[i,j]

By choosing −2m α −

n−m = −2m α − − 2β ∗ RX n−m ∗ RX

n X

χk,i −

k=1

n−m X

! χjk,i

(37)

k=1

as a reference energy, which is assumed to be the same for all

isomers of C2n X2m , the relative π energy of isomer [i, j] can be expressed as (in units of −2β)

Xij

≡

∆Eπ[i,j]

n n−m X X j n − m + 2m α − /(−2β) = χ − χk,i . k,i ∗ RX k=1 k=1

(38)

It is worth mentioning that the idea of removing atoms at addition positions in the HMO framework can be dated back to 1940s. 204,205 This method was devised by Wheland 204 and Brown 205 for studying the substitution reactions of aromatic molecules. Recently, it has also been applied to understanding site selectivity of Diels–Alder reactions of polyaromatic hydrocarbons, where the HMO reaction energies are shown to correlate well with DFT 21



Page 22 of 80

activation free energies. 206 Besides π delocalization effect, there are two other factors determining the relative stability of exohedral fullerenes, namely, cage strain induced by fused pentagons, and steric repulsion between adjacent addends. 83 The cage strain can be approximated, by the PAPR, as 0.2 NAPPji (in units of −2β). Here, NAPPji is the effective number of APPs for isomer [i, j] of C2n X2m . It is the same as NAPPi of the pristine cage i unless any addend X is attached to the carbon atom of a pentagon adjacency, converting its hybridization from sp2 to sp3 (and thus releasing the strain energy). Therefore, NAPPji is determined as follows: (i) pentagon adjacencies holding no addends X count as one each; (ii) those holding one addend count as half each; (iii) those holding two addends do not count. Regarding the steric hindrance effect, we have shown in earlier work 83 that the repulsion energy between addends is approximately proportional to the number of adjacent addends, NAXj , in isomer [i, j] of C2n X2m . Obviously, NAXj depends only on the distribution pattern j of addends X, not on the cage form i. Combining all the above three stability factors, we define the exohedral fullerene stabilization index (XSI) 83 of isomer [i, j] of C2n X2m , as follows XSIji ≡ Xij + 0.2NAPPji + γX NAXj

(39)

where the coefficient γX accounts for the steric repulsion energy between every two adjacent addends X. The steric coefficient γX depends only on the chemical nature of addends X. We have determined its value for some typical chemical groups, X = H, F, Cl, Br, and CF3 , based on systematic SCC-DFTB calculations for the Ih (1)-C60 X4 regioisomers. 83 The obtained γX values are listed in Table 1. As can be seen, the XSI quantifies the relative isomer energy (in units of −2β) of exohedral fullerenes, by analogy with the CSI for charged fullerenes. As a purely topology-based tool, it enables rapid determination of the most stable isomers of exohedral fullerenes, among millions or even billions of possible structures.

22


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Table 1: Optimized steric coefficients γX for chemical groups X = H, F, Cl, Br, and CF3 , obtained by fitting the SCC-DFTB energies of the Ih (1)-C60 X4 regioisomers.

2.3

X

γX (−2β)

H

0.2230

F

0.2305

Cl

0.2827

Br

0.3139

CF3

0.3338

Canonical labeling of fullerene atoms and addends

Considering the large number of possible fullerene structures of a given cage size, it is necessary to uniquely identify different cage isomers based on topological information. The most convenient and generally accepted way of numbering fullerene isomers is the scheme established by Fowler and Manolopoulos, based on ring spiral codes. 107 In this scheme, all rings (including 12 pentagons and (n − 10) hexagons) of a given fullerene isomer of C2n are ordered in a unique way that follows a ring spiral algorithm. Then, the order of appearance of the 12 pentagons defines the so-called ring spiral code that is uniquely associated to a cage isomer. By lexicographically sorting the ring spiral codes of all isomers (or alternatively, all IPR isomers) of a given fullerene size, distinct cage isomers can be numbered. When it comes to exohedral fullerenes C2n X2m , the nomenclature of regioisomers is much more complicated, as the number of possible addition patterns is many orders of magnitude larger than that of cage forms. To identify symmetrically distinct regioisomers, one suggested method is to compare the HMO energies. 207 However, this method is not only very inefficient (involving matrix diagonalization) and less accurate (due to numerical errors), but also unreliable, as we have found that different regioisomers may have exactly the same HMO energies (see Appendix C for the simplest example). Better methods 208–212 for nomenclature of regioisomers of C2n X2m rely on a unique and systematic labeling of the cage atoms,

23



which is called the canonical labeling. Once all cage atoms are canonically labeled, each symmetrically nonequivalent regioisomer can be uniquely identified, based on the canonical labels of its addition positions. Although several nomenclature schemes 208–212 have been proposed for canonical labeling of the atoms of a fullerene cage, including the recommendations by IUPAC 211,212 and by Chemical Abstracts Service (CAS), 210 their nomenclature rules are less general (inapplicable to fullerenes without a vertex spiral, 176 or to nonclassical fullerenes), and less straightforward to implement. In this work, we have employed the breadth-first-search (BFS) 213 numbering scheme proposed by Fowler et al 176 to canonically label the cage atoms. Among the computer-codable and mathematically well-founded methods, 214 the BFS labeling algorithm has been shown to be more efficient. 176 In the following, we give a brief description of this labeling scheme. We take each atom of a given fullerene cage C2n as a starting atom, and label this atom as number 1 (see the first column in Figure 3). The rules for subsequent labeling of atoms are as follows: (i) The next atom to be labeled must be adjacent to any of the already labeled atoms. (ii) Among all unlabeled atoms, we label firstly the one adjacent to the atom having the smallest label. (iii) If there are more than one unlabeled candidate fulfilling the above rules, the labeling order for these candidates can follow either a clockwise or counterclockwise manner, as specified in the following. We take the already labeled atom commonly adjacent to the unlabeled candidates as a reference atom. Among all labeled neighbors of the reference atom, we choose the one with the smallest label and refer to it as the smallest neighbor of the reference atom (in the case of a cubic graph like fullerenes, this choice is unique since there are two unlabeled and one labeled neighbors). Then, the labeling of the unlabeled candidates follows either a clockwise or counterclockwise order of the neighbors starting from the smallest neighbor. Once this labeling clockwiseness is decided, it must be followed for all subsequent labeling (when necessary). By applying these rules, there are three possible ways to label the second atom, which can be one of the neighboring atoms of the starting

24


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atom 1 (as a fullerene structure is a cubic graph), as shown in the second column in Figure 3. Next, we have two choices to label the third and the fourth atoms, following either a clockwise or a counterclockwise direction among the neighbors of the reference atom (i.e. atom 1): 2 → 3 → 4, as illustrated in the third column in Figure 3. Once the first four atoms have been labeled, the labeling of the remaining atoms is uniquely determined by the above rules (see the fourth column in Figure 3). Consequently, for each of the 2n starting atoms, there are 6 possible ways to label all atoms, each of which is called a labeling path. Hence, for a given fullerene cage C2n , we obtain 12n possible labeling paths. In order to identify distinct labeling paths, we define a connectivity code as a sequence of numbers consisting of the connectivity information bewteen atoms. For a given labeling path, we write down the labels of the three neighbors of each atom, in the order from atom 1 to 2n. For the sake of generality, the label of the last neighbor of each atom is followed by a terminal zero so that the whole graph of any degree can in turn be decoded from such a connectivity code. We thus obtain a sequence of 8n numbers as a unique code for each symmetrically nonequivalent labeling path. Examples are provided in the fifth column in Figure 3 showing the connectivity codes of the corresponding labeling paths. Now, we can choose the labeling path with the lexicographically smallest connectivity code as the minimum labeling path. Note that fullerene cages with symmetry have more than one minimum labeling path. We denote the number of minimum labeling paths as its degeneracy. Naturally, the canonical labeling is defined as the labeling of atoms along the minimum labeling path. Now, we are ready to label the addition positions of a given isomer [i, j] of C2n X2m . Suppose there are gi minimum labeling paths of cage i. On each minimum labeling path, we have a sequence of numbers, being the canonical labels of the 2m addition positions. We then choose the path with the lexicographically smallest numbers, among all gi sequences, as the minimum addition path. 177 This smallest sequence consisting of 2m numbers is defined as the canonical labels of addition positions. Symmetric addition patterns may have more than one minimum addition path with a degeneracy of gij . Based on the canonical labels

25



14 3

1

4 13

Page 26 of 80

8 15 9 4 3 1

10 7

2

6

2

12

17

2, 3, 4, 0, 1, 5, 6, 0, 1, 7, 8, 0, 1, 9, 10, 0, 2, 10, 11, 0, 2, 12, 7, 0, 3, 6, 13, 0, 3, 14, 15, 0, 4, 15, 16, 0, 4, 17, 5, 0, ...

1

13

2, 3, 4, 0, 1, 5, 6, 0, 1, 7, 8, 0, 1, 9, 10, 0, 2, 10, 11, 0, 2, 12, 7, 0, 3, 6, 13, 0, 3, 14, 15, 0, 4, 15, 16, 0, 4, 17, 5, 0, ...

1

15

2, 3, 4, 0, 1, 5, 6, 0, 1, 7, 8, 0, 1, 9, 10, 0, 2, 10, 11, 0, 2, 12, 13, 0, 3, 13, 14, 0, 3, 15, 9, 0, 4, 8, 16, 0, 4, 17, 5, 0, ...

2

16

2, 3, 4, 0, 1, 5, 6, 0, 1, 7, 8, 0, 1, 9, 10, 0, 2, 11, 12, 0, 2, 13, 7, 0, 3, 6, 14, 0, 3, 15, 9, 0, 4, 8, 16, 0, 4, 17, 11, 0, ...

3

13

2, 3, 4, 0, 1, 5, 6, 0, 1, 7, 8, 0, 1, 9, 10, 0, 2, 11, 12, 0, 2, 13, 7, 0, 3, 6, 14, 0, 3, 15, 9, 0, 4, 8, 16, 0, 4, 17, 11, 0, ...

3

11

2, 3, 4, 0, 1, 5, 6, 0, 1, 7, 8, 0, 1, 9, 10, 0, 2, 10, 11, 0, 2, 12, 13, 0, 3, 13, 14, 0, 3, 15, 9, 0, 4, 8, 16, 0, 4, 17, 5, 0, ...

2

16

5 11

1 16

9 15

2 4

1

4

3 17

7 10 5

2

11

12

1

3 8

10

4

17 2

14

7

1

5

4

1

6 12

6 13 2

3 11

3

1

2

2

14

8

1

9 16

18 12 2

1

5 11 10 2

4 13

3

17

4

1

9 6 7

3

14

8 15

18 17 4

1

2

10 4

16 3

11

5 2

1

6 9 8

3

15 1

12

7 14

2 14

3

1

3

2 15

4

12

7 13 6 2

1

5 8 9 16

4

10 17

Figure 3: Schematic diagram demonstrating the BFS procedure of labeling carbon atoms of a fullerene graph, staring from a given atom. The first (leftmost) column shows the labeling of the starting atom. The second column illustrates the three possible ways of labeling the second atom. The third column shows that for each case in the second column there are two possibilities of labeling the third and the fourth atoms: following either a clockwise or counterclockwise order for the three neighbors of atom 1, i.e., 2 → 3 → 4. The fourth column gives the subsequent labeling of all the remaining atoms. The fifth column lists the connectivity codes of the corresponding labeling paths. The last column indicates the ranking order of these labeling paths from the smallest to the largest connectivity codes.

26


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of addition positions, nonequivalent regioisomers are uniquely identified and can be coded using numbers and letters in the following manner. Given a sequence of 2m canonical labels of addition positions, {λk } = λ1 , λ2 , ..., λ2m , we firstly obtain a new sequence of numbers being λ1 followed by the differences between two neighboring numbers, namely, {σk } = λ1 , (λ2 − λ1 ), ..., (λ2m − λ2m−1 ). Evidently, 1 6 σk 6 n and thus we propose the rules of encoding each number σk , as: (i) For σk from 1 to 35, they are encoded by a single character, being 1–9 and A–Z, respectively. (ii) If σk > 35, it is encoded using 3 characters. The first one is always a zero, indicating that the input number is greater than 35. Then, we decompose σk as σk = ξ1 + 35ξ2 with 1 6 ξ1 , ξ2 6 35, and both ξ1 and ξ2 can be encoded using 1–9 and A–Z, as in rule (i). Hence, using only numbers and letters, we can encode labels of addition positions for fullerene cage sizes up to C2520 (= 2 × (35 + 352 )).

2.4

Other computational details

B3LYP/6-31G(d) computations were performed to compare with the results of the FullFun package by using the Gaussian 09 suite of programs. 215 SCC-DFTB calculations were also carried out by using the DFTB+ (version 1.2) code. 216 Finally, semiempirical PM3 calculations were also performed by using the MOPAC2016 program. 217

3

IMPLEMENTATION

The FullFun software package can be freely downloaded from our web page on SourceForge, 218 and consists of four programs: CSIOpt, XSI, XSIOpt and bkcage (as detailed below). They are implemented using the C and C++ languages and distributed under an open source licence. The code can be compiled and executed on Linux, Mac OS X and Windows (via DOS) operating systems. These programs are briefly outlined below (see Appendix D for detailed description). The CSIOpt program reads in a number of cage isomers in a given charge state and outputs

27



the values of CSI sorted in ascending order, so that the most stable candidate structures are identified. For IPR isomers, the program also prints out the CSI values evaluated by the improved model (using equation 26). The XSI program performs a “single-point” calculation of the XSI value of a given regioisomer of C2n X2m . It reads in a given cage topology from an external file and a series of addition patterns from stdin (standard input). The final output contains the computed XSIji values and their components, Xij , NAPPji and NAXj , as well as the corresponding canonicalized addition positions. The purpose of the XSIOpt program is to find the best addition patterns with the lowest XSI values for a given cage isomer of prototype exohedral fullerene C2n X2m . By employing the stepwise addition algorithm, 164,175,178–184 which requires a user-defined value of the cutoff energy, the program generates all possible regioisomer candidates of C2n X2m , and outputs their XSIji values in ascending order, so that the lowest-XSI isomers can be easily selected for subsequent refinement calculations using semiempirical or DFTB methods. The bkcage program is for general purposes of generation, identification and nomenclature of fullerene structures. 107,219,220 This is an extremely useful utility for scientists who work with fullerenes. The bkcage program can interconvert different formats of fullerene structures, including isomer numbering after Fowler and Manolopoulos, 107 ring spiral code and cage connectivity. One can also obtain atomic Cartesian coordinates by piping the topology output of bkcage to the embed program in the CaGe package. 221,222 It also provides all relevant topological information of a fullerene cage, such as symmetry, rings, number of APPs, adjacency matrix, etc. Additionally, simple HMO calculations can be performed to obtain π orbital energies and coefficients, atomic charges, bond orders, as well as distribution of wave functions.

28


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4

APPLICATIONS

In previous work, 83,135 the CSI model has successfully predicted the relative stability of many fullerene isomers from C28 to C104 with charges between +6 and −6, and of experimentally identified EMFs. It has also explained why the IPR is often violated for fullerene anions, whereas the opposite is found for fullerene cations. The XSI model has successfully predicted the structures of many experimentally synthesized exohedral fullerenes, 83 providing at the same time a simple quantitative description of the subtle interplay between π delocalization, cage strain and steric hindrance between addends. It has also allowed us to understand the variation of the addition patterns with the chemical nature of the addends, and the change in fullerene cage stability with the progressive addition of chemical groups. Here we apply the FullFun implementation of these models to find the most stable structure of some charged, endohedral and exohedral fullerenes recently produced in the laboratory. More specifically, we will first use the CSI model to find the most stable structures of one of the smallest EMFs synthesized so far, Ti@C42 , and for which the structure has not yet been determined experimentally. 185 Then, we will apply the improved CSI model that incorporates the structural-motif method described in Section 2.1.3 to investigate the relative stability of giant C6+ 104 fullerene cations. In the third example, we will demonstrate that the XSI model allows one to find the global minimum-energy isomer of C36 H6 , 187 whose structure has so far remained elusive to theoretical calculations. As we will show, the structure found in the present work is considerably more stable than that proposed in earlier theoretical investigations. 188 Finally, we will apply the XSI model to determine, for the first time, the relative stability of nonclassical exohedral fullerene isomers, namely, those of the chlorofullerene C104 Cl24 , in order to understand the origin of the structure found in recent experimental work. 189

29



4.1

Cage structures of the most stable EMFs Ti@C42

Small EMFs M@C2n (M = Ti, Zr, U; 2n = 28–50) have been synthesized in pulsed laser vaporization processes and observed by using high-resolution FT-ICR mass spectrometry. 185 Unlike EMFs with larger cages (2n > 66), whose structures have been unambiguously identified by X-ray crystallography, the structures of those small EMFs still remain unclear. 185 With the aid of DFT calculations, the cage structures of the lowest energy isomers of Ti@C2n (2n = 26–50) have been predicted. 185,186 Here, as an example, we apply the CSI model to predict the most stable cage isomers of Ti@C42 , and then compare the results with those obtained from AM1 and DFT calculations. 186 We have used the fullgen program to generate all 45 cage isomers of fullerene C42 . The output in the “code 6” format was directly piped into the CSIOpt program with a specified charge state of −4, which is the appropriate oxidation state of the encaged titanium atom. 185,186 The CSIOpt program printed out all CSI values computed using equation 25 as well as the corresponding isomer labels according to the generation order by the fullgen program. By convention, we converted the fullgen’s isomer numberings to those following the scheme of Fowler and Manolopoulos 107 by using the bkcage utility. 120

ΔEAM1 (kcal/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

C4− 42

100 80

9 APPs 10 APPs 11 APPs 12 APPs 13 APPs 14 APPs 15 APPs 16 APPs

60 40 20 0 3.0

3.5

4.0

4.5

5.0

5.5

6.0

CSI (−2β)

Figure 4: Correlation between CSI values and relative AM1 energies for all 45 isomers of C4− 42 . Isomers with different number of APPs are indicated by different colors and symbols. The top five cage isomers highlighted by filled symbols, namely, D3 (45), C1 (32), C1 (33), Cs (35) and C1 (39), correspond to the lowest DFT energies of EMFs Ti@C42 . 186

30


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Figure 4 shows that the CSI values correlate well with the AM1 relative energies for all 45 isomers of C4− 42 . The latter energies have been shown to be in reasonable agreement with the relative energies resulting from more elaborate DFT (BP86/TZP) calculations. 186 The filled symbols in Figure 4 indicate the five cage isomers with the lowest energy according to the DFT calculations: 186 D3 (45), C1 (32), C1 (33), Cs (35) and C1 (39). As can be seen, these isomers also have the lowest CSI values, thus showing the high predictive power of the CSI model at a negligible computational cost.

4.2

Relative isomer energies of C6+ 104 cations

In previous work, the predictive power of the CSI model has been systematically checked by considering fullerene cations from C28 to C104 with charges up to +6. 135 For the smaller species, the performance of the CSI model is excellent, since with very few exceptions it always provides the most stable isomer structures. The situation becomes slightly worse when the model is applied to the largest members of the series. Here, we investigate in depth the performance of the CSI model for the most difficult case: fullerene C6+ 104 . After a prescreening process by using the CSI model, we have selected 741 IPR isomers and 889 isomers with one APP, and performed SCC-DFTB calculations with full geometry optimization. Our calculations show that for C6+ 104 the isomers with one APP are at least 28 kcal/mol higher in energy than the lowest-energy IPR isomer. This is consistent with our previous conclusion that the most stable isomers of large fullerene cations are the IPR ones. 135 Therefore, in the following discussion, we will only take into account the IPR isomers. Figure 5a shows the correlation between the CSI values obtained from equation 25 (the original CSI model) and the SCC-DFTB relative energies for the 741 prescreened IPR isomers of C6+ 104 . The five lowest-energy isomers determined at the SCC-DFTB level, namely, C2 (810), D3 (814), C2 (635), C2 (450) and C2 (443), are highlighted by red filled symbols. As can be seen, although the latter structures have rather low CSI values, the correlation between these values and relative energies is not very satisfactory, since we find quite a few number 31



140

(a)

(b)

120

ΔEDFTB (kcal/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 80

100

C6+ 104

C6+ 104

80 60 40 20 0

2.2

2.4

2.6

2.8

3.0

8.4

CSI (−2β)

8.6

8.8

9.0

9.2

9.4

9.6

9.8

CSI (−2β)

Figure 5: Correlation between CSI values and SCC-DFTB relative energies for 741 prescreened IPR isomers of C6+ 104 . The CSI values in (a) and (b) have been calculated using equations 25 and 26, respectively. The five cage isomers with the lowest SCC-DFTB energies are highlighted by red filled symbols. of structures with similar CSI values whose relative energies are however substantially higher. This is mainly due to the fact that the PAPR, which is used in equation 25 to estimate the contribution of strain to the relative energy, does not make any difference between IPR isomers, the only ones relevant when considering large fullerene cations (see Section 2.1.3). To improve the performance of the CSI model when many IPR isomers are accessible, as in the case of large fullerenes, we have also evaluated the CSI values by using equation 26, which includes information about the structural motifs that appear in large fullerenes. The structural-motif model has been shown to do a good job in estimating the relative energies of neutral isomers of large fullerenes. 132 The correlation of the improved CSI values with the SCC-DFTB relative energies is shown in Figure 5b. As can be seen, the performance of the improved CSI model (eq 26) is significantly better than that of the original CSI model. Indeed, the structure with the lowest energy is associated with the lowest CSI value, and the ranking number of the highest-energy member of the five most stable isomers goes down from 143 (eq 25) to 47 (eq 26). Nevertheless, both are relatively small numbers compared to the total number (419 013) of possible isomers of C104 , indicating that either CSI model is a good prescreening tool to identify the lowest-energy isomers.

32


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4.3

Global minimum-energy structure of C36 H6

Hydrogenated fullerenes C36 H6 have been produced in experiments by means of the hightemperature laser-vaporization of graphite rods, catalyzed by Ni/Co and Ni/Y. 187 In such high-temperature conditions, the produced species should correspond to the global minimumenergy structures. Although the composition of C36 H6 was clearly identified by HPLC analysis, no further information was provided neither on the cage structures nor on the distributions of hydrogen atoms on the surface of C36 . Later on, Fowler et al. 188 determined the lowest-energy isomers of C36 H6 at the DFTB level. They have systematically explored all possible regioisomers of C36 H6 based on the two cage forms, D2d (14)-C36 and D6h (15)-C36 , the two lowest-energy isomers of neutral pristine C36 . The predicted minimum-energy of C36 H6 corresponds to a D6h -symmetry addition pattern on the D6h (15) cage (as shown in Figure 6a). 35

31

36

(a)

(b) 36

30

33

32

33

35

32 24

26 34

27

10

5

12

10

16

34

24

26

2

6

22

8

9 21

29

1 8 3

6

14

18

9

7

3

14

4

12

4 1

7

5

2

13 21

15

11 11

20

22

17

17 19

29

23 16

25

25

18

28

30

31

19

15

27

23

D6h(15)-C36H6 (1E3293)

13

20

28

C2v(9)-C36H6 (1112F4)

Figure 6: Schlegel diagram of the lowest-energy isomer of C36 H6 predicted by (a) Ref. 188 and (b) this work. Canonical labels of carbon atoms are indicated for cages (a) D6h (15)-C36 and (b) C2v (9)-C36 . The corresponding encoded labels of addition positions are given in parentheses. Addition positions are highlighted in yellow. Pentagonal rings are indicated in blue. Here, we revisit the search for minimum-energy structures of C36 H6 , by considering three more cage forms, namely, C2v (9), C2 (12) and C2 (11). All cage forms considered here cor33



respond to the five low-lying isomers, within 30 kcal/mol with respect to the lowest-energy isomer of neutral pristine C36 , according to our SCC-DFTB calculations. The total number of possible addition patterns (excluding open-shell cases) for these five cages is as large as 2 481 007. By using the XSIOpt program, this number is significantly reduced to 11 321 when applying the stepwise addition algorithm with a cutoff energy of 0.8 |β| (ca. 40–50 kcal/mol). Then, we selected the 100 lowest-XSI regioisomers for each cage form of C36 H6 , and performed full geometry optimizations at the SCC-DFTB level. Finally, we refined the results at the B3YLP/6-31G(d) level (with full geometry optimization as well) for the 50 lowest-energy structures at the SCC-DFTB level among all 500 candidates of different cage forms and addition patterns. The five lowest-energy structures of C36 H6 determined at the DFT level are summarized in Table 2. As we can see, the lowest-energy isomer has a cage form of C2v (9) with an addition pattern of C2 symmetry (see Figure 6b). Four hydrogens are added to the pentagon-pentagon bonds of the quadruply fused pentagonal rings. This addition manner should greatly release the σ strain of the cage framework and thus stabilize the molecule. In comparison, the lowest-energy isomer previously found in Ref. 188, which corresponds to a D6h (15) cage, lies about 9 kcal/mol higher in energy than the lowest-energy isomer predicted by this work. The other three of the five lowest-energy isomers of C36 H6 also have a cage form of C2v (9). As can be seen in Table 2, all these low-lying isomers have very low values of the XSI, in particular the lowest two ones and that with the D6h (15) cage. This result shows that evaluation of the XSI values prior to performing elaborate quantum chemistry calculations provides very useful information about possible isomer candidates for the global energy minimum, thus minimizing the risk of missing some of them.

4.4

Addition patterns of C104 Cl24 with a nonclassical fullerene cage

Very recently, several chlorinated giant fullerenes with cage sizes from C104 to C108 have been synthesized by chlorination with VCl4 and SbCl5 at 350 ◦ C. 189 The pristine cages were 34


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Table 2: The predicted five lowest-energy isomers of C36 H6 . Relevant information is given about the cage forms, canonical labels of addition positions and their corresponding encoded labels, relative values of XSI (in −2β) with respect to the lowest-XSI isomer, relative energies (in kcal/mol) with respect to the lowest-energy isomer obtained at the SCC-DFTB and the B3LYP/6-31G(d) levels. Cage form

Addition positions

Encoded labels

∆XSI

∆EDFTB

∆EDFT

C2v (9)

1, 2, 3, 5, 20, 24

1112F4

0.009

0.00

0.00

C2v (9)

1, 2, 3, 4, 20, 23

1111G3

0.002

1.66

5.07

C2v (9)

1, 2, 3, 5, 27, 34

1112M7

0.175

8.22

7.65

D6h (15) 188

1, 15, 18, 20, 29, 32

1E3293

0.005

5.80

8.92

C2v (9)

1, 2, 3, 5, 21, 22

1112G1

0.151

9.81

10.33

pre-obtained by HPLC extraction from arc-discharge produced soot. Among these chlorofullerenes, the C104 Cl24 is of particular interest, for it has a nonclassical fullerene cage containing one heptagon and 13 pentagons (see Figure 7). Here we show that the experimental addition pattern of C104 Cl24 , determined by single-crystal synchrotron X-ray diffraction, 189 corresponds to one of the low-XSI regioisomers, and can thus be efficiently predicted by the prescreening of the XSI model followed by semiempirical or DFTB calculations. In the experimental work, it is believed that the nonclassical cage of C104 is formed in arc-discharge soot before chlorination. 189 Therefore, we will fix the cage form as the one determined in the experiment (see Figure 7), and only look for the optimal addition patterns. In this nonclassical cage, there are no pentagon-pentagon adjacencies and only three pentagon-heptagon edges. To keep the model as simple as possible, we assume that there is no energy penalty for pentagon-heptagon adjacencies, which, compared to pentagonpentagon contacts, would not induce significant strain due to the compensation between pentagon’s positive and heptagon’s negative curvatures. With a cutoff energy of 0.4 |β| for the stepwise addition algorithm, we obtain a total of 13 545 872 distinct closed-shell regioisomers of C104 Cl24 by using the XSIOpt program. The experimental structure given in Figure 7 was identified as the 49th lowest-XSI regioisomer, with a relative XSI value of 0.076 35



57

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46 34

58 47

35

59 24 71

48

72 83

36

60

73

37

50

95 85

38

100 92

104

81

75 62

101

98

97 88

64 77

90

65 78

80

40

66

79

3 4

7

1 33

18 10

51

2

5

28

52

54

22

19 11 29

53

20 30

42

45

13 6

41

89

9

17

39 27

63 96 86 103 102 87 76

99 91

8

16

61

23 14

26

82

70

15

25

49

84 93 94 74

12

21

32

31

67 43

55

44

68 69

56

C104Cl24 (25352215621AEA4612436235)

Figure 7: Schlegel diagram of the lowest-energy isomer of C104 Cl24 predicted by this work. Canonical labels of carbon atoms are shown. The encoded labels of addition positions are given in parentheses. Addition positions are highlighted in green. Heptagonal and pentagonal rings are indicated in magenta and blue, respectively. (in units of −2β) with respect to the lowest-XSI isomer. It is the fourth lowest-XSI isomer among all those without adjacent chlorines. These ranking numbers are very low compared to the total number of generated isomers exceeding ten million. It thus allows us to perform subsequent semiempirical calculations for a much smaller number of candidates. Here, we chose the 2 000 lowest-XSI regioisomers to carry out full geometry optimizations at the PM3 level. According to these semiempirical calculations, the structure shown in Figure 7 is the lowest-energy isomer of C104 Cl24 , which is in agreement with the experimental identification of this compound. It is worth mentioning that we have tried several values of the cutoff energy, namely, 0.2, 0.3 and 0.4 |β|. The results are compared in Table 3. As we can see, the total number 36


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Table 3: Three different cutoff energies chosen for optimizing the addition patterns of C104 Cl24 with the experimentally determined nonclassical cage. We present the total number Ntotal of generated distinct regioisomers, the number of open-shell (Nopen ) and closed-shell (Nclose ) ones, as well as the XSI ranking number of the experimental structure among all generated closed-shell isomers. The execution times on a single processor are also compared. Cutoff

Ntotal

Nopen

Nclose

Rank

Time (h)

0.2 |β|

262 084

10 339

251 745

47

2.4

0.3 |β|

4 025 741

2 402 846

1 622 895

49

20.2

0.4 |β|

31 743 079

18 197 207

13 545 872

49

127.9

of generated regioisomers increases almost exponentially with the increasing cutoff energy. And so is the execution time, but the increase is relatively less dramatic. For higher cutoff energies, over a half of the generated patterns have open-shell configurations, because of the considerable number of chlorine addends. In all the cases, the ranking number of the experimental regioisomer, in ascending order of XSI values, is quite small and converges to constant at cutoff energy of 0.3 |β|. This indicates that we can reliably generate the experimental structure using a lower cutoff energy while the computation cost can be orders of magnitude lower.

5

CONCLUSIONS

We have presented in detail a simple topology-based theory to describe the relative stability of charged fullerenes and fullerene derivatives. Thanks to the closed three-dimensional shape of most fullerene cages, their π delocalized system can be effectively approximated by the simple HMO theory. As a result, the π stability is conveniently given by the eigenvalues of the cage connectivity matrix. By further including the σ strain effect as described by the PAPR, we have introduced the CSI index, which measures the relative stability of different charged fullerene isomers. This model has been refined to account for large charged IPR fullerenes, by replacing the PAPR by a more accurate motif model. As an extension of the 37



CSI model, the XSI model for exohedral fullerenes takes into account an additional stability factor, the steric repulsion between addends, which is proportional to the number of adjacent addends and depends on the nature of the chemical groups. Despite their simplicity, both models are shown to be very efficient tools to find, among a huge number of possibilities, the lowest-energy structures of charged fullerenes or prototype exohedral fullerenes. Both methods have been implemented in the FullFun computational package, which is easy to use and is available at the SourceForge page. 218 We hope that this computational tool will help chemists to easily predict and understand fullerene structures that will be produced in future experiments.

APPENDIX A

List of Abbreviations

APP = adjacent pentagon pair BFS = breadth-first search CSI = charge stabilization index DFT = density functional theory DFTB = density functional tight binding EMF = endohedral metallofullerene HMO = Hückel molecular orbital IPR = isolated pentagon rule LCAO = linear combination of atomic orbitals NAPP = number of adjacent pentagon pairs PAPR = pentagon adjacency penalty rule SCC = self-consistent charge THJ = triple-hexagon junction XSI = exohedral fullerene stabilization index 38


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B

Evaluation of the Effective π Electron Cloud Radius 1.4

Ih-C60 (Rn = 3.55 Å) Ih-C80 (Rn = 4.12 Å) I-C140 (Rn = 5.42 Å)

1.3

Rπ* / Rn

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.2

Rπ* = Rπ 1.1

1.0 0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

Rπ / Rn

Figure 8: Effective π electron cloud radius Rπ∗ as a function of average π electron distance to the cage center, Rπ , in units of cage radius Rn . The results for three icosahedral fullerenes Ih (1)-C60 , Ih (7)-C80 and I(121354)-C140 have been presented. We have evaluated the effective π electron cloud radius Rπ∗ , according to equation 11. We have chosen carbon’s 2pz atomic orbital centered at distance Rn with respect to the cage center, as follows: Z∗ p 2 1 ∗ 52 2 r + Rn − 2Rn r cos θ φ(r, θ; Rn ) = √ Z (r cos θ − Rn ) cos θ exp − 2 4 2π

(40)

where we take the value of 3.25 for the effective nuclear charge of carbon Z ∗ , following Slater’s rules. 223 Hence, Rπ∗ is calculated by performing numerically the following integration: 1 = 2π Rπ∗

Z 0

∞

Z

π

φ(r, θ; Rn )2 U (r; Rπ )r2 sin θdrdθ.

(41)

0

Obviously, Rπ∗ depends on the choices of the average π electron distance to the cage center, Rπ , and the cage radius, Rn . Figure 8 shows the calculated Rπ∗ as a function of Rπ for three icosahedral fullerenes Ih (1)-C60 , Ih (7)-C80 and I(121354)-C140 with different cage radii Rn . Usually, Rπ and Rπ∗ should be slightly larger than Rn . For instance, the effective radius of the π electron cloud of C60 has been estimated to be 4.43 Å, 224 about 1.25 times larger than 39



its cage radius Rn . For this Rπ /Rn ratio, as we can see in Figure 8, Rπ∗ is practically the same as Rπ .

C

An example showing different C20 X4 isomers having the same HMO energy

As an example, we consider the simplest classical fullerene C20 . For this cage, there are two distinct regioisomers of C20 X4 that have exactly the same HMO energy. The addition patterns of both isomers are shown in Figure 9. It can easily be seen that the corresponding adjacency matrices have the same eigenvalues.

(a)

(b)

Figure 9: Schlegel diagrams of two distinct regioisomers of C20 X4 , which have exactly the same HMO energy. Addition positions are indicated by cyan circles. The canonical labels of addition positions in (a) and (b) are, respectively, 1, 2, 8, 11 and 1, 2, 8, 19.

D

Detailed Description of the Suite of Programs Contained in the FullFun Package

To run the CSIOpt program, the charge state q and the energy penalty per APP (0.2 by default, in units of −2β) are specified as command line arguments. The input is read from stdin containing a series of cage isomeric forms of the same cage size 2n. The specification of each cage structure can be in three ways, as follows. (i) The widely used XYZ file format, providing Cartesian coordinates of atoms. (ii) The ASCII “code 6” output format generated by the fullgen program in the CaGe package. 221,222 (iii) The binary “planar code”, the default 40


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output format for the buckygen program. 225,226 The latter two formats provide the topological information, i.e., the connectivity between atoms. They are especially useful if one wants to enumerate all cage isomers of a given size by using the fullgen or the buckygen program. It is worth mentioning that the latter is considerably faster than the former and other fullerene generators. 225,226 The CSIOpt program then determines NAPPi and constructs the adjacency matrix of each cage isomer i, based on the input geometry or topology. The adjacency matrix is diagonalized using the Eigen 3 template library. 227 The CSI calculations are performed using equation 25 for all isomers and the results are ordered from the lowest value to the highest. The final results containing the values CSI and the corresponding X q and NAPP components are printed out to stdout (standard output), accompanied by the original input order of isomers. In the case of IPR isomers, the CSI value is also evaluated by the improved model (using equation 26) and printed out as an additional output. Optionally, the corresponding ring spiral codes can also be printed, which is useful for the identification and labeling of cage isomers by using the bkcage program. For each run, the XSI program treats only one isomeric cage form of exohedral fullerenes C2n X2m , which is from an ASCII file in the “code 6” format. The path of this file, as well as the energy penalty per APP and the steric coefficient γX (both in units of −2β) are specified as command line arguments. Analogous to the CSIOpt program, the XSI program can read from stdin a series of regioisomers (namely, addition patterns). Each addition pattern can either consist of 2m numbers indicating the labels of addition positions that are not necessarily canonical, or be encoded canonical labels of addition positions consisting of characters among 0–9 and A–Z (see Section 2.3). It firstly checks whether the addition pattern corresponds to an open-shell configuration, in which there exists an odd number of bare carbon atoms isolated by addends X. 142,143 If it is the case, this regioisomer will be skipped and the program moves onto the next one. Otherwise, the XSI program then determines NAPPji and NAXj based on the cage form i and the addition pattern j. It also builds the adjacency matrices of

41



the derivatized cage and the corresponding pristine cage. Eigenvalues are thus obtained by diagonalizing these matrices. With all this information, XSI is computed for regioisomer [i, j] using equation 39. After processing all input regioisomers, the final results are outputed, including the XSI values and the corresponding addition patterns in the form of encoded canonical labels. The addition patterns can also be printed as explicit numbers indicating the canonical labeling of addition positions. The XSIOpt program reads only one input file, named “XSIOPT.INP”, where all parameters such as the path of cage topology file, the number of addends, the energy penalty per APP, the steric coefficient and other settings (see below for details) are specified. Based on the input cage isomeric form i, the XSIOpt program canonicalizes the labeling of carbon atoms of the cage, which is the basis of canonical coding of addition patterns (see Section 2.3 for the algorithm). Meanwhile, the program builds and diagonalizes the adjacency matrix of the pristine cage to obtain the corresponding eigenvlaues. Next, the program enumerates all possible regioisomers of C2n X2 and computes the corresponding XSI values. If the desired number of addends 2m is greater than 2, then the subsequent addition patterns of C2n X2m are generated by using a stepwise addition model, 164,175,178–184 which has been evidenced by recent experiments on the chlorination of C74 . 228 The algorithm can be briefly described as follows. For generating the regioisomers of C2n X2l , we add two X addends to all best candidate regioisomers of C2n X2l−2 obtained in the previous step. All possible addition positions are explored and symmetrically equivalent regioisomers are discarded by comparing the canonical labels of addition positions. Among the generated regioisomers of C2n X2l , those corresponding to an open-shell configuration or with an XSI value higher than a given cutoff energy are ruled out. The remaining regioisomers are thus the best candidates of C2n X2l . We repeat the same procedure while increasing the number of addends until l reaches the given value of m. The bkcage program has included the database in the Program Fullerene version 4.4. 229 By using Elias omega coding, 230 we have compressed and reformatted the database. The

42


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input of the bkcage program can be specified as the cage size and the conventional isomer number according to the ring spiral algorithm, 107 or according to the generation order by the fullgen program. 221,222 In case that the nomenclature of a given fullerene isomer is unknown, the program can recognizes it from its atomic Cartesian coordinates or its ring spiral code (consisting of 12 integers). After reading an input fullerene structure, the bkcage program can print out all relevant topological information, such as symmetry, ring spiral code, rings, number of APPs, adjacency matrix, cage connectivity in the “code 6” format, etc. It can also carry out simple HMO calculations, providing energies and wave functions. The latter can be exported to cube files 215 for visualization of π orbitals.

Acknowledgement The authors thank allocation of computer time at the Centro de Computación Científica of the Universidad Autónoma de Madrid (CCC-UAM) and the Red Española de Supercomputación. Work supported by the MINECO projects FIS2013-42002-R, FIS2016-77889-R, CTQ2013-43698-P and CTQ2016- 76061-P, the CAM project NANOFRONTMAG-CM ref. S2013/MIT-2850, and the European COST Action CM1204 XLIC. Financial support from the Spanish Ministry of Economy and Competitiveness - MINECO - through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0377) is acknowledged. S.D.-T. gratefully acknowledges the “Ramón y Cajal” program of the Spanish MINECO (RYC-2010-07019).

References (1) Stephens, P. W., Ed. Advanced Series in Fullerenes: Physics and Chemistry of Fullerenes; World Scientific: Singapore, 1993; Vol. 1.

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(2) Dresselhaus, M. S.; Dresselhaus, G.; Eklund, P. C. Science of Fullerenes and Carbon Nanotubes; Academic Press: San Diego, 1996. (3) Omacrsawa, E. Perspectives of Fullerene Nanotechnology; Kluwer Academic Publishers: Dordrecht, the Netherlands, 2002. (4) Hirsch, A.; Brettreich, M. Fullerenes: Chemistry and Reactions; Wiley-VCH Verlag GmbH & Co. KGaA: Weinheim, 2004; pp 185–211. (5) Kroto, H. W.; Heath, J. R.; O’Brien, S. C.; Curl, R. F.; Smalley, R. E. C60 : Buckminsterfullerene. Nature 1985, 318, 162–163. (6) Smalley, R. E. Discovering the Fullerenes (Nobel Lecture). Angew. Chem. Int. Ed. 1997, 36, 1594–1601. (7) Cami, J.; Bernard-Salas, J.; Peeters, E.; Malek, S. E. Detection of C60 and C70 in a Young Planetary Nebula. Science 2010, 329, 1180–1182. (8) Berné, O.,; Mulas, G.,; Joblin, C., Interstellar C+ 60 . Astron. Astrophys. 2013, 550, L4. (9) Iglesias-Groth, S.; Esposito, M. A Search for near Infrared Bands of the Fullerene Cation C in the Protoplanetary Nebula IRAS 01005+7910. Astrophys. J. Lett. 2013, 776, L2. (10) Campbell, E. K.; Holz, M.; Gerlich, D.; Maier, J. P. Laboratory confirmation of C+ 60 as the carrier of two diffuse interstellar bands. Nature 2015, 523, 322–323. (11) Bohme, D. K. Multiply-charged ions and interstellar chemistry. Phys. Chem. Chem. Phys. 2011, 13, 18253–18263. (12) Zettergren, H.; Rousseau, P.; Wang, Y.; Seitz, F.; Chen, T.; Gatchell, M.; Alexander, J. D.; Stockett, M. H.; Rangama, J.; Chesnel, J. Y.; Capron, M.; Poully, J. C.; Domaracka, A.; Méry, A.; Maclot, S.; Schmidt, H. T.; Adoui, L.; Alcamí, M.; Tielens, A. G. G. M.; Martín, F.; Huber, B. A.; Cederquist, H. Formations of Dumbbell 44


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Tomita, S.; Jensen, J.; Schmidt, H. T.; Zettergren, H.; Cederquist, H. Fullerene collisions and clusters of fullerenes. Int. J. Mass Spectrom. 2006, 252, 117–125. (225) Brinkmann, G.; Goedgebeur, J.; McKay, B. D. The Generation of Fullerenes. J. Chem. Inf. Model. 2012, 52, 2910–2918. (226) Goedgebeur, J.; McKay, B. D. Recursive generation of IPR fullerenes. J. Math. Chem. 2015, 53, 1702–1724. (227) Guennebaud, G.; Jacob, B.; et al, Eigen v3. http://eigen.tuxfamily.org, 2010. (228) Gao, C.-l.; Abella, L.; Tan, Y.-Z.; Wu, X.-Z.; Rodríguez-Fortea, A.; Poblet, J. M.; Xie, S.-Y.; Huang, R.-B.; Zheng, L.-S. Capturing the Fused-Pentagon C74 by Stepwise Chlorination. Inorg. Chem. 2016, 55, 6861–6865. (229) Schwerdtfeger, P.; Wirz, L.; Avery, J. Program Fullerene - A Software Package for Constructing and Analyzing Structures of Regular Fullerenes. J. Comput. Chem. 2013, 34, 1508–1526. (230) Elias, P. Universal codeword sets and representations of the integers. IEEE Trans. Inf. Theory 1975, 21, 194–203.

69



Graphical TOC Entry

70


Page 70 of 80

Page 71 of 80

Ven/ZA* (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21


Rn(C70)

−4

Rn(C60) −6

C60 (exact) C60 (Gauss) C70 (exact) C70 (Gauss)

−8

−10 0

2ACS Paragon Plus 4 Environment 6

r (Å)

8

10

−2

Ven/ZA* (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24


(a)

−3

Page 72 of 80

Rn(C60X12) Rn(C60)

−4 −5

C60H12 (exact) C60H12 (Gauss) C60Cl12 (exact) C60Cl12 (Gauss) C60(CF3)12 (exact) C60(CF3)12 (Gauss) C60 (exact) C60 (Gauss)

−6 −7 −8 −9 0

1

2

3

4

5

r (Å)

6

(b)

7ACS Paragon 8 Plus 9Environment 10

(c) C60H12

(d)

C60Cl12

C60(CF3)12

Page 73 of 80

Journal of Chemical Theory and Computation 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

1

14

4 13

8 15 9 4 3 1

10 7

2

6

2

12

17

2, 3, 4, 0, 1, 5, 6, 0, 1, 7, 8, 0, 1, 9, 10, 0, 2, 10, 11, 0, 2, 12, 7, 0, 3, 6, 13, 0, 3, 14, 15, 0, 4, 15, 16, 0, 4, 17, 5, 0, ...

1

13

2, 3, 4, 0, 1, 5, 6, 0, 1, 7, 8, 0, 1, 9, 10, 0, 2, 10, 11, 0, 2, 12, 7, 0, 3, 6, 13, 0, 3, 14, 15, 0, 4, 15, 16, 0, 4, 17, 5, 0, ...

1

15

2, 3, 4, 0, 1, 5, 6, 0, 1, 7, 8, 0, 1, 9, 10, 0, 2, 10, 11, 0, 2, 12, 13, 0, 3, 13, 14, 0, 3, 15, 9, 0, 4, 8, 16, 0, 4, 17, 5, 0, ...

2

16

2, 3, 4, 0, 1, 5, 6, 0, 1, 7, 8, 0, 1, 9, 10, 0, 2, 11, 12, 0, 2, 13, 7, 0, 3, 6, 14, 0, 3, 15, 9, 0, 4, 8, 16, 0, 4, 17, 11, 0, ...

3

13

2, 3, 4, 0, 1, 5, 6, 0, 1, 7, 8, 0, 1, 9, 10, 0, 2, 11, 12, 0, 2, 13, 7, 0, 3, 6, 14, 0, 3, 15, 9, 0, 4, 8, 16, 0, 4, 17, 11, 0, ...

3

11

2, 3, 4, 0, 1, 5, 6, 0, 1, 7, 8, 0, 1, 9, 10, 0, 2, 10, 11, 0, 2, 12, 13, 0, 3, 13, 14, 0, 3, 15, 9, 0, 4, 8, 16, 0, 4, 17, 5, 0, ...

2

16

5 11

1 16

9 15

2 4

1

4

3 17

5

12

2

3 8

10

4

17 2

14

7

1

5

4

1

6 12

6 13 2

3 11

3 7

10

11

1

8

1

2

2

14

1

9 16

18 12 2

1

5 11 10 2

4 13

3

17

4

1

9 6 7

3

14

8 15

18 17 4

1

2

10 4

16 3

11

5 2

1

6 9 8

3

15 1

12

7 14

2 14

3

1

15 4 ACS Paragon Plus Environment

7 13 6 3

2

12 2

1

5 8 9 16

4

10 17

120

ΔEAM1 (kcal/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Page 74 of 80

C4− 42

100 80

9 APPs 10 APPs 11 APPs 12 APPs 13 APPs 14 APPs 15 APPs 16 APPs

60 40 20 0 3.0

3.5 4.0Plus Environment 4.5 5.0 ACS Paragon

CSI (−2β)

5.5

6.0

140

Page 75 of 80

ΔEDFTB (kcal/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


(a)

(b)

120 100

C6+ 104

C6+ 104

80 60 40 20 0

2.2

2.4

2.6

2.8

CSI (−2β)

3.0

8.4

8.6


8.8

9.0

9.2

9.4

CSI (−2β)

9.6

9.8

35

31

(a)

(b)

36

1 2 3 4 5 6 7 8 30 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

36

35


30

33

32

33

32

27

10

5

12

10

16

24

34

26

21

8 3 14

18

9

7

3

22

9

1

6

1

8

4

12

4 7

5

2

2

6 13

14

15

11 11

21

22

17

17 19

20

23 16

25

25

18

28

29

31

24

26 34

Page 76 of 80

19

15

27

23

D6h(15)-C36H6 (1E3293)


13

20

28

C2v(9)-C36H6 (1112F4)

29

57

Page 77 of 80 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 70 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

46


34

58 47

35

59 24 71

48

72 83

93

60

73

94

36

84 74

37

14

8

16

61

23

26

82

50

95 85

38

100 92

104

75 62

101

81

96 86 103 102

99 91

15

25

49

98

64

65

66

79

4

7

1 33

5

28

2

13 6

52 41

89 78

80

40

3

18 10

51

87 76 77

90

39 27

63

97 88

9

17

29

53 54

22

19 11 20 30 42

12

21

32

31

67 55

43 44

68 69

56 ACS Paragon Plus Environment

C104Cl24 (25352215621AEA4612436235)

45

1.4

Rπ* / Rn

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21


Page 78 of 80

Ih-C60 (Rn = 3.55 Å) Ih-C80 (Rn = 4.12 Å) I-C140 (Rn = 5.42 Å)

1.3

1.2

Rπ* = Rπ 1.1

1.0 0.5

0.6

0.7 Paragon 0.8 Plus 0.9Environment 1.0 1.1 ACS

Rπ / Rn

1.2

1.3

1.4

Page 79 of 80 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23


(a)


(b)

Journal of Chemical Theory and Computation Page 80 of 80 1 2 3 4


Topology-Based Approach to Predict Relative Stabilities of Charged

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