Skeletal Rearrangements of the C₂₄₀ Fullerene: Efficient Topological Descriptors for Monitoring Stone–Wales Transformations

Abstract

Stone–Wales rearrangements of the fullerene surface are an uncharted field in theoretical chemistry. Here, we study them on the example of the giant icosahedral fullerene C₂₄₀ to demonstrate the complex chemical mechanisms emerging on its carbon skeleton. The Stone–Wales transformations of C₂₄₀ can produce the defected isomers containing heptagons, extra pentagons and other unordinary rings. Their formations have been described in terms of (i) quantum-chemically calculated energetic, molecular, and geometric parameters; and (ii) topological indices. We have found the correlations between the quantities from the two sets that point out the role of long-range topological defects in governing the formation and the chemical reactivity of fullerene molecules.

1. Introduction

Most chemical studies on fullerenes are focused on the reactions and compounds of C₆₀ and C₇₀, the two most abundant fullerenes. Nevertheless, the diversity of the sizes and shapes within the fullerenes’ family is much wider. Indeed, there are many fullerene structures, smaller [1] and larger [2], which are produced in smaller amounts. The chemistry of such small and large fullerenes has growing experimental and theoretical interests.

We pay attention to the giant C₂₄₀ fullerene with icosahedral symmetry. Its structure is the next after C₆₀ in the series of icosahedral Goldberg polyhedra [3,4,5] having 60a² vertices (where a is the integer). In the fundamental aspect, such chemical structures are very close to their mathematical models, so direct bridges between the chemistry and mathematics could be built, which means the possible correlations between the structural and topological (mathematical) indices and molecular (chemical) properties of the molecules [4,5,6]. It is noteworthy that the C₂₄₀ structure obtained with quantum chemical calculations corresponds to the idealized one, i.e., the highest symmetry (I_h) allowed by the topology of the molecule (there is a possibility of the Jahn–Teller symmetry reduction in the case of other icosahedral fullerenes, e.g., C₂₀ and C₈₀) [6]. In practical aspects, such giant cages as C₂₄₀ attract as they have a large size and may be classified as nano-objects, but they are still molecules. Due to their size, giant fullerenes are considered as nano-vessels for gas storage [7] and the closed structure makes C₂₄₀ a molecular Faraday cage screening the guest atoms/molecules from the external electric fields [8]. Note that C₂₄₀ (I_h) has not been yet synthesized nor isolated. Under arc-discharge fullerene synthesis, it should be produced in extremely minute quantities (it is also potentially insoluble compound due to its size; this makes an obstacle of its extraction) [9,10]. However, its complexes like C₆₀@C₂₄₀, C₂₄₀@C₅₆₀, and C₆₀@C₂₄₀@C₅₆₀ have been obtained and visualized with atom-force microscopy [11,12,13,14]. As follows from these studies, the C₂₄₀ molecule is most likely icosahedral. The thermodynamic [5,15,16] and dielectric properties [8,17,18,19] of C₂₄₀ and other giant fullerenes have been studied with relevant computational techniques.

Giant fullerenes relate to the processes of classical fullerene formation [20]. They are proposedly formed from the C₂ gas and convert to C₆₀ and C₇₀ [21]. The presence of a large cavity (much larger than in C₆₀) makes giant fullerenes chemically unstable [21] and sensitive to various transformations. The Stone–Wales isomerization is one of such topological processes that may occur on the fullerene surface [22,23]. Since 1986 [24], it is well-known that Stone–Wales (SW) rotations have a theoretical role in connecting the isomers of a given C_n fullerene.

Under the action of the SW_5|6 operator (Figure 1a), the 1812 isomers of C₆₀ are grouped in 13 sets. The larger one includes 1709 cages connected to C₆₀ (I_h) through one or more SW_5|6 transformations whereas 31 isomers remain separated from the rest of the isomers [24]. Later, the introduction of the so-called generalized Stone-Wales transformations (Figure 1b) overcame this limitation offering a method of generating the complete C₆₀ isomeric space starting from just one C₆₀ isomers [25]. The instrumental role of chemical graphs in studying fullerene rearrangements was immediately evident [25]. The description of fullerene molecule C_n as a planar graph $G$ made by n nodes (carbon atoms), $B=3n/2$ edges (carbon–carbon bonds) forming f₅ = 12 pentagons and f₆ = n/2 – 10 hexagons is a convenient and powerful tool that is also suitable for computerized algorithms. Chemical graph $G$ represents the topological skeleton of the molecule and allows a clear picture of the Stone–Wales isomerization mechanism. In Figure 1a, the SW_5|6 transformation (the so-called pyracylene rearrangement) that is capable of bridging C_n isomers with different symmetries is shown in the direct (bottom) and dual graph (top). In $G$ the rotation of the central bond is obtained by cutting the two graph edges indicated by the red arrows and then by connecting, pairwise, the green and the orange nodes. This topological transformation is perfectly isomeric since the numbers of nodes, edges, and faces are conserved. The generation of the SW_5|6 rotation became even simpler when occurring on the dual fullerene graph $\overline{G}$ composed by $\overline{n}=\raisebox{1ex}{$n$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+2$ dual nodes, which correspond to the faces of the direct C_n fullerene and $B=3n/2$ edges. Both fullerene graphs, direct $G$ and dual $\overline{G}$, have the same number of edges $B$. The top part of Figure 1a illustrates the pyracylene transformation in the dual graph $\overline{G}$. In this case the SW_5|6 rearrangement is obtained just by rotating the central edge (red arrow) evidencing a general feature of the SW transformations the dual space $\overline{G}$:

• In the dual space, a SW rotation consists in the reversible rotation of just one dual bond.

This straightforward mechanism allows the generation of extended topological defects, so-called generalized Stone-Wales transformations (gSW) that modify the fullerene surfaces in a peculiar fashion.

Figure 1b shows the typical gSW involving two 5|6 pairs (shown in Figure 1a) and η pairs of the internal faces (white circles) included between those 5|6. The white circles represent indifferently hexagons or pentagons. The surface transformation of the dual fullerene $\overline{G}$ (Figure 1b) corresponds therefore to the gSW_η=3 of size η = 3 whereas the generalized rotation gSW_{η = 0} of order η = 0 represents instead to the pyracylene rearrangement SW_5|6. Operators like gSW depend on integer parameter η (Figure 1b) and constitute only one set of the infinite class S of reversible non-local topological rearrangements. The extension (or the size) of the gSW defect is expressed by integer η.

• The gSW size η corresponds to the number of pairs of internal faces (white circles) included between the two 5|6 pairs.

All SW operators represent full-isomeric rearrangements of the fullerene surface, which therefore preserve the numbers of direct nodes n, dual nodes $\overline{n}$, bonds B and faces. They also preserve the type of internal faces included in a dislocation dipole. Figure 1 gives nice examples of these topological modifications.

In S another interesting group of operators is represented by the specific transformations that generate and then propagate 5|7 pairs in a hexagonal network [26,27]. In a regular hexagonal mesh, this original mechanism produces a peculiar linear extended defect, the so-called SW wave (SWw) with variable length η. Figure 2 describes in the dual graph the sequence of specific SW operators that create (SW_6|6) and move (SW_6|7) that wave in a graphenic lattice. The first rotation SW_6|6 (Figure 2a) of the arrowed dual edge creates the 5|7 double pairs (Figure 2b). The second rotation SW_6|7 separates them by the insertion of a couple of hexagons (Figure 2c). At this point, the propagation of the 5|7 pair has begun and successive SW_6|7 transformations propagate it in the lattice producing the topological SW wave. This topological defect SWw is known in the literature also under the name of dislocation dipole, whose length η corresponds to the minimal number of hexagon pairs between the 5|7 pairs. Figure 2 shows the dislocation dipoles (SWw) with η = 0, 1, and 2.

Theoretically, the existence of extended topological defects such as SWw with η ≥ 3 has been documented by molecular dynamics and ab initio studies [28,29] reporting about the presence of 5|7 dislocations in pristine graphene and at the grain boundaries of polycrystalline graphenic lattices. In the latter case, the authors state that these linear defects cannot be annealed by local reorganization of the honeycomb mesh. Specifically, work [28] affirmed that “neither an isolated pentagon, or a heptagon, or their pair 5|7” cannot be corrected by lattice rearrangements – an erroneous conclusion not considering that the SWw mechanism allows the 5|7 dislocations to be annealed by moving the 5|7 pair backward by applying the SW transformations (Figure 2) in the reverse direction.

Electronic properties of honeycomb planar systems indicate that single heptagon–pentagon dislocations are stable defects whereas 5|7 adjacent pairs are dynamically unstable [29]. The energy E_f of the formation of the 5|7 double pairs (Figure 2b) determined by the extended Hückel [30] and molecular mechanical simulations [31] depends on multiple structural factors:

(i) the size of the honeycomb flat lattice around the defect, with E_f that assumes the minimum value E_f ≅ 6 eV for “hexagonal layers” made by 9,000 carbon atoms [31];

(ii) the presence a cylindrical curvature in the honeycomb mesh that facilitates furthermore the creation of the dislocation dipole by lowering the energy barrier to E_f ≅ 3 eV for carbon nanotubes. Formation energy of the 5|7 double pair drops even more, down to E_f ≅ 2 eV, when interstitial defects or ad-atoms are present in the networks [32,33].

Pentagon–heptagon pairs have been moreover detected by high-resolution TEM studies [34,35,36,37] whereby energetic particles, such as electrons and ions, generate 5|7 pairs in graphite layers or single-walled CNTs because of atom displacements.

We consider that both appearing elements favor the creation of dislocation dipoles, i.e., curvature and presence of hexagonal regions of variable size, are specific characteristics of fullerenes’ morphology. Therefore, the aim of this work is addressed, mainly from a topological corner, to the following basal theoretical questions:

• In which ways is the fullerene surface modified by SWw topological defects?
• Is the creation of SWw defects energetically favored?

In the following, the first attempt for answering the above points is made by taking into account the C₂₄₀ (I_h) molecule as a case study. The reasons for this choice are the high symmetry of the molecule and the previous of well-documented ab initio [16] and topological studies [27] about its stability. The next paragraph is devoted to the description of the mechanisms for the formation of dislocation dipoles on the C₂₄₀ (I_h) surface; then the energetic and topological considerations complete this work.

2. SW Waves on the C₂₄₀ (I_h) Fullerene

The theoretical evidences of the stability of 5|7 defects in various hexagonal systems [28,29,30,31,32,33,34,35,36] suggest the possibility to create extended defects even on the surface of a fullerene large enough to have graphene-like zones tiled only with hexagons. This section provides a concise introduction to the topological wave-like mechanism for the creation and the diffusion (or annihilation) of the pentagon–heptagon linear defect on the fullerene surface. The present investigation, combining extended topological rearrangements of the carbon networks and specific modifications of fullerene topology, sheds new light on possible mechanism that could be a base of fullerene formation and chemical properties.

Figure 3 shows the icosahedral C₂₄₀ cage with f₆ = 110 hexagonal rings and the symmetry-independent rearrangements of type SW_6|6, which may take place on its surface (images sourced from the web [38]). Due to the icosahedral structure of the C₂₄₀ molecule, one has in fact just three symmetry-independent ways A, B, and C of selecting quartets of hexagons admitting the SW_6|6 rotation. The three regions A, B, and C are shown in Figure 3a prior to applying the SW operator according to the mechanisms illustrated above (Figure 2). The SW_6|6 operator rotates the dual bonds pictured in black in each diamond in Figure 3a.

Our density functional theory (DFT) calculations show that, among the three isomers of the C₂₄₀ (I_h) fullerene generated by the SW_6|6 operator applied to the quartets A, B, and C (Figure 3a), the energy effect favors the rotation of the quartet A. Indeed, the calculated ΔE values for modes A, B, and C equal 147.9, 323.5, and 347.4 kJ/mol (see Appendix A for the details of the calculations). Note that the structures obtained via the rotations of quartets A and B have the C_S symmetry whereas the structure derived by the quartet C rearrangement has no symmetry. The configuration of the molecule after the SW_6|7 rotation of quartet A is given in Figure 3b, which evidences the new 5|7 double pair. At this point, the SW wave with η = 0 has been created on the surface of the new isomer ${\mathrm{C}}_{240}^{\eta =0}$ with f₅ = 14, f₆ = 106, and f₇ = 7. The two extra pentagons and the two heptagons are labeled with the number of respective edges in Figure 3b. According to the propagation mechanism (Figure 2c,d), the second SW_6|7 operator rotates the dual bond between the heptagon and the nearby hexagon (the arrowed dual bonds in Figure 2b,c), moving the 5|7 pair along the fullerene network. At each step η, the 5|7 pair swaps its place with a 6|6 one (see also Figure 3b); after η iterations of the SW_6|7 rearrangement, the ${\mathrm{C}}_{240}^{\eta }$ isomer is formed. Figure 4 shows the isomer with η = $2$. The described propagation process is applied to the isolated 5|7 dislocation monopole as well as to the 5|7 double pair arising from the SW_6|6 rearrangement. It is worth mentioning that similar topological tools are used in other disciplines such as biology whereby wave-like diffusion processes and icosahedral patterns of viruses are normal mechanisms to model biochemical processes [39,40,41,42].

From the pure topological point of view, it is noteworthy that each fullerene network in the current study could be considered as the result of an instantaneous transformation generated by a single non-local generalized SW rotation. This kind of transformations, previously proposed in [25], constitutes a potentially infinite class of global rearrangements aiming to generate the entire isomeric space of a given C_n fullerene starting from a limited number (even only one) of inequivalent cages.

The final part of this section briefly introduces the topological invariants and how they measure the relative stability of fullerene isomers ${\mathrm{C}}_{240}^{\eta }$ modified by SW waves. Graph-theoretical results are presented and compared with energy values of the systems coming from quantum chemical simulations in the next sections. According to the topological modeling (TM) methods [43], chemical structures with n atoms and B bonds are schematically described as simple graphs G with n nodes and B edges. Geometrical, structural, and other physicochemical properties (maximal molecular symmetry being in the opinion of the authors the prominent, and somehow surprising, one) are embedded in the graph topology as the study of distance-based invariants evidenced some decades ago in pioneering studies [44,45]. In this context, distances d_ij are integer numbers computed by counting the number of edges connecting two atoms i and j with the shortest path in G. The distances are graph invariants (i.e., they do not depend on the labeling of the nodes) and generate a vast parade of distance-based graph descriptors. Graph diameter is defined as M = max{d_ij}.

TM techniques mainly make use of two invariants W (1) and ${\rho }^E$ (3) to measure graph compactness and roundness, respectively. Assuming d_ii = 0, the Wiener index W is derived as the semi-sum of all distances in G:

$$W\left(G\right)=\frac{1}{2}{\sum }_{i,j=1}^n{d}_{ij}={\sum }_{i=1}^n{w}_i ,$$

(1)

counting the contributions $w_i$ arising from each node in the graph:

$$w_i=\frac{1}{2}{\sum }_{j=1}^n{d}_{ij},$$

(2)

Intuitively, Equation (2) admits the minimum ($\underset{\_}{w}$) and maximum ($\overline{w}$) values for central (peripheral) nodes in G. The quotient ${\rho }^E$ between the two $w_i$ extremals:

$${\rho }^E=\overline{w}/\underset{\_}{w},$$

(3)

called extreme topological roundness denotes the topological symmetry of the graph: the lower (3) is the more symmetric G is (with the general constraint ${\rho }^E\ge 1$). We assign the role of the topological potential of the system described by G to the Wiener index and extreme topological roundness. Indeed, both quantities obey the minimum principle. Therefore, the TM model assumes that similar carbon systems arrange their structures to minimize W and ${\rho }^E$. Heuristically, this approach is confirmed by the fact that icosahedral fullerenes C₆₀ and C₂₄₀ minimize both indices in the respective isomeric sets. Recent works [46,47,48] on C₆₆ and C₈₄ molecules confirm that stable fullerenes combine high topological symmetry (low ${\rho }^E$) with high compactness (low W).

This fast and elegant computational approach is applicable to the evolution of the SWw-defected fullerene molecules ${\mathrm{C}}_{240}^{\eta }$. The electronic configurations of the atoms constituting the C₂₄₀ isomers with defective structures in Figure 3; Figure 4 keep the sp²-hybridization state. On the other hand, surface reconstructions induced by SWw cause complex modifications of “the electronic charge density and this varies the bond lengths within and around” [49], with effects also on the local curvature of the molecular cage.

We have performed the preliminary density functional theory (DFT) computations to monitor the topology effect on the geometrical parameters and molecular properties of the defected C₂₄₀ cages. The DFT computations have been performed with standard procedures; the details are collected in Appendix A. We have computationally studied SWw on the C₂₄₀ surface up to η = 6 (Figure 5). The SW_6|6 operator implies replacing six original hexagons with the emergence of four “defected” polygons (two 5|7 pairs). From the point of structural chemistry, we approximate SWw as a movement of one “defected” pair from the other due to similar operations. Thus, each ${\mathrm{C}}_{240}^{\eta }$ with η = 0–3 contains 12 original pentagons, 2 heptagons and 2 pentagons, static and migrating, generated by the wave (the remaining faces being hexagons). In Figure 5, static and dynamic defects are colored differently for clarity. The ${\mathrm{C}}_{240}^{\eta =4}$ structure contains one migrating tetragon resulting from involving an original pentagon by SWw. In further structures, this tetragon converts to a new static pentagon. In other words, the wave shifts the position of the original pentagon in two operations. Note that the structures with η = 3 and η = 5 contain pentagon–pentagon fusions. These structural features are presented in

Table 1. Structural parameters of original and defected C₂₄₀ fullerenes (DFT computations).

Fullerene Cage	Number of Polygons 1	Volume (Å 3)	Sphericity, Ψ	Curvatures of Original Pentagons (Å–1)	Curvatures of Defected Pentagons (Å–1)
					Static	Migrating
C240 (Ih)	12 110 0 0	1433.73	0.9829	0.2365	none	none
${\mathrm{C}}_{240}^{\eta =0}$	14 106 2 0	1436.10	0.9828	0.1850–0.2475	0.1641	0.1257
${\mathrm{C}}_{240}^{\eta =1}$	14 106 2 0	1430.11	0.9797	0.1763–0.2586	0.2112	0.1537
${\mathrm{C}}_{240}^{\eta =2}$	14 106 2 0	1425.73	0.9774	0.1763–0.2502	0.2171	0.1915
${\mathrm{C}}_{240}^{\eta =3}$	14 106 2 0	1420.88	0.9751	0.1763–0.2680	0.2151	0.2722
${\mathrm{C}}_{240}^{\eta =4}$	12 107 2 1	1417.49	0.9762	0.1754–0.3277	0.2135	0.3932 2 (tetragon)
${\mathrm{C}}_{240}^{\eta =5}$	14 106 2 0	1420.49	0.9750	0.1765–0.2589	0.2140 (0.2679)	0.2754
${\mathrm{C}}_{240}^{\eta =6}$	14 106 2 0	1421.94	0.9759	0.1578–0.2588	0.2140 (0.2383)	0.1939

¹ In the order: pentagons, hexagons, heptagons, tetragons. ² For tetragon. ³ Curvatures of static pentagons, which resulted due to SWw, are in parentheses.

.

Local curvature (k) is a key parameter we pay attention as it strongly correlates with the stability [1,2,50] and reactivity [51,52,53] of the fullerene species. The SWw transformations occur locally on the C₂₄₀ surface. However, the calculated curvatures indicate the change in the shape throughout the whole surface (Equation (A2), Appendix A). To demonstrate it, we have chosen the curvature in the pentagon regions, both original and emerged upon SWw. As follows from Table 1, all pentagons have the same curvature in original C₂₄₀ but SWw makes the curvatures more diversified. Accordingly, some of pentagons become more flattened as the others become more curved. Herewith, the structures with a tetragon (${\mathrm{C}}_{240}^{\eta =4}$) and fused pentagons (${\mathrm{C}}_{240}^{\eta =3}$ and ${\mathrm{C}}_{240}^{\eta =5}$) manifest the highest curvatures in the set.

Sphericity (Ψ) is another parameter indicating the changes in the shape (Equation (A1), Appendix A). The calculated sphericities decrease from the starting value 0.9829 as the C₂₄₀ system loses its initial icosahedral symmetry. Note that original C₂₄₀ (I_h) and ${\mathrm{C}}_{240}^{\eta =0}$ have very close values because the structure with η = 0 still has a low symmetry (C_S). The further changes are more pronounced. Nevertheless, Ψ does not become lower than 0.97 though SWw perturbs the shape. For comparison, Ψ may be significantly reduced (Ψ < 0.9) in highly functionalized fullerene compounds (e.g., C₆₀ halogenides [54]). Sphericity (with associated quantity of volume, V) does not change monotonically with SWw propagation, i.e., there are no (and should not be) correlations between Ψ (or V) and η. However, we demonstrate below the correlations between these geometric parameters with topological ones.

Pristine and almost spherical C₂₄₀ (I_h) has n = 240 direct nodes and B = 360, three symmetry-distinct sets of atoms (orbits) having multiplicities 60, 60, and 120, and 240 atoms with the same $\overline{w}=\underset{\_}{w}=2312$. This last condition implies W = 277,400 and ${\rho }^E=1$, indicating that the C₂₄₀ (I_h) fullerene is a maximal roundness molecule even in presence of non-symmetry equivalent atoms (a special condition featured by tetrahedral C₄₀ too [49]). We would like to remember here that the complete individuation of all families of non-transitive fullerenes with ${\rho }^E=1$ is still an open task.

3. Topological Simulations and Electronic Structure

3.1. Topological Modeling

The first step of the SW wave on the C₂₄₀ (I_h) fullerene graph consists of a SW_6|6 bond flip that produces the 5|7 double pair (Figure 3b). This surface transformation decreases the lattice compactness to W = 277,122 derivable from the direct computation of the graph chemical distances d_ij and Equation (1). In our approximated model, this –0.11% variation of the Wiener index represents the gain in topological compactness allowed by the creation of the SW defect in the molecule ${\mathrm{C}}_{240}^{\eta =0}$.

The subsequent SW_6|7 rotation (η = 1) splits the 5|7 pairs by inserting in between one pair 6|6 according to the topological mechanism described in (Figure 2a,b). This decreases by another –0.14% the value of the topological descriptor W = 276,730 for newly formed ${\mathrm{C}}_{240}^{\eta =1}$. This “gaining” behavior holds for each propagation steps in the interval η = [0, 3] of SWw over the fullerene surface, increasing the topological compactness of the system.

The reduction of the topological index W follows a quasi-parabolic trend (Figure 6). The curve evidences that the creation of an extended topological defect up to η = 3 is favored; the topology of the fullerene network allows the topological diffusion of the 5|7 pair over a quite extended region. The SWw propagation is then impeded by a small +0.02% increment of the topological index W having a local peak for η = 4. The Wiener index curve in Figure 6 evidences, and this result is reported here for the first time for the fullerene molecules, that the 5|7 defects are able to migrate on the fullerene surface to form extended linear defects. This is a specific behavior that matches similar findings previously reported for graphene planar layers [26] and curved carbon nanotubes and nanotori [33]. The present TM simulations show that creation of SW waves is allowed also on quasi-spherical surfaces tiled by hexagons. Based on this outcome, we may therefore propose the following important conjecture: the formation and propagation of the SW waves are allowed on surfaces tiled with hexagons independently of the surface genus.

This effect has a pure topological root and strongly correlates with the connectivity properties of the pentagon–heptagon pairs embedded in the hexagonal mesh and the “edge” effect induced by the presence of pristine pentagons on the surface. We note in fact here that the small barrier effect encountered by the 5|7 pair at η = 4 (Figure 6) is an effect generated by SWw colliding with one of the original pentagons of the C₂₄₀ (I_h) fullerene. A different situation is encountered in the variations of the topological roundness (Equation (3)) of the system upon SWw diffusion, see Figure 6.

The generation of the 5|7 double pairs reduces the topological symmetry and moves the ${\rho }^E$ curve upward after the first SW_6|6 rotation, with a topological increase ${\rho }^E=1.0241$ for ${\mathrm{C}}_{240}^{\eta =0}$ compared with the unitary values of the original C₂₄₀ (I_h). The successive SW_6|7 flip starts the dipole propagation still increasing the topological invariant ${\rho }^E$. The diffusion of the wave is opposed by the action of the invariant ${\rho }^E$, which shows a barrier effect confirmed at the successive steps in Figure 5. At η = 2 and 3, the ${\rho }^E$ quantity show an almost flat curve in the same region where W shows a local minimum (Figure 6). This result is quite interesting since it confirms that system, when the dislocation dipole has size η = 2 and 3, stops the obstacle to the creation of defective molecules (${\mathrm{C}}_{240}^{\eta =2}$ represented in Figure 4).

Topological invariants are listed in

Table 2. Topological descriptors of the original and defected C₂₄₀ fullerenes.

Fullerene Cage	W	M 1	$\underset{\_}{\mathit{w}}$	$\overline{\mathit{w}}$	${\mathit{\rho }}^{\mathit{E}}$
C240 (Ih)	277,440	19	1156	1156	1
${\mathrm{C}}_{240}^{\eta =0}$	277,122	19	1140	1167.5	1.0241
${\mathrm{C}}_{240}^{\eta =1}$	276,730	19	1125.5	1171.5	1.0409
${\mathrm{C}}_{240}^{\eta =2}$	276,607	19	1118.5	1178.5	1.0536
${\mathrm{C}}_{240}^{\eta =3}$	276,557	19	1124	1185.5	1.0547
${\mathrm{C}}_{240}^{\eta =4}$	276,607	19	1118.5	1192.5	1.0662
${\mathrm{C}}_{240}^{\eta =5}$	276,560	19	1125	1199.5	1.0662
${\mathrm{C}}_{240}^{\eta =6}$	276,551	19	1120	1196.5	1.0683

¹ M is the graph diameter M = max{d_ij}.

and represented in Figure 6. We have found the inverse correlations between the extremal roundness (Table 2), volumes and sphericities of the C₂₄₀ cages (Figure 7a). The found correlations are the same as in the case of the C₈₄ isomeric series (made up with the isomers obeying the isolated-pentagon rule) [55] and similar to the findings of comprehensive work [56] on all possible isomers of C₆₀. The geometry and topology of the molecules obtain direct linkage in the relations of Ψ and ${\rho }^E$. Indeed, Ψ → 1– for sphere-like species whereas ${\rho }^E$ → 1+ in the case of structures with “equilibrated connectivity”. These terms, in the case of the fullerenes, coincide with the notion that is manifested with the found correlations. As for the correlation between the volume and extremal roundness of the C₂₄₀ cages, it follows from the interdependence of the Ψ and V values (Figure 7b).

3.2. DFT Simulations on Energetic and Molecular Parameters of the Defected C₂₄₀ Cages

We have assessed the possibility of SWw transformations on the C₂₄₀ cage from the thermodynamic point of view. For this purpose, we have operated with the DFT-computed total energies of the original and defected cages (see Appendix A for the details of the DFT study). As expected, the energies of all SWw-generated C₂₄₀ cages are much higher as compared with the pristine one (

Table 3. Energetic and molecular parameters of original and defected C₂₄₀ fullerenes (DFT computations).

Fullerene Cage	Relative Energy, Erel (kJ/mol) 1	Energy Effect, ΔE (kJ/mol) 2	Dipole Moment, μ (D)	εHOMO (eV)	εLUMO (eV)	HOMO–LUMO Gap (eV)
C240 (Ih)	0	n/a	0	–5.53	–4.30	1.23
${\mathrm{C}}_{240}^{\eta =0}$	147.8	147.8	0.86	–5.26	–4.46	0.80
${\mathrm{C}}_{240}^{\eta =1}$	315.5	167.7	0.51	–5.18	–4.41	0.76
${\mathrm{C}}_{240}^{\eta =2}$	430.4	114.9	1.57	–5.30	–4.57	0.72
${\mathrm{C}}_{240}^{\eta =3}$	510.5	80.1	2.53	–4.98	–4.70	0.28
${\mathrm{C}}_{240}^{\eta =4}$	642.6	132.1	0.83	–4.89	–4.61	0.27
${\mathrm{C}}_{240}^{\eta =5}$	489.2	–153.5	4.62	–4.94	–4.83	0.10
${\mathrm{C}}_{240}^{\eta =6}$	384.0	–105.2	3.57	–5.03	–4.79	0.24

¹ Calculated as $E_{rel}=E\left({\mathrm{C}}_{240}^{\eta }\right)-E\left({\mathrm{C}}_{240}^{original}\right)$. ² Calculated as $\mathsf{\Delta }E=E\left({\mathrm{C}}_{240}^{\eta }\right)-E\left({\mathrm{C}}_{240}^{\eta -1}\right)$.

). However, the energy effects of the stepwise transitions between ${\mathrm{C}}_{240}^{\eta =0}$ shows that some of the transitions may be favorable. Indeed, the transitions are endothermic up to ${\mathrm{C}}_{240}^{\eta =3}\to {\mathrm{C}}_{240}^{\eta =4}$ and exothermic for the two next steps (Table 3). The exothermicity is explainable in terms of the structural chemistry of fullerenes. The structure ${\mathrm{C}}_{240}^{\eta =5}$, a participant of the two exothermic steps, has two adjacent pentagons. This is the instability factor making the conversion ${\mathrm{C}}_{240}^{\eta =5}\to {\mathrm{C}}_{240}^{\eta =6}$ favorable. As for η = 4, the precursor of ${\mathrm{C}}_{240}^{\eta =5}$, it has the tetragon–heptagon junction, and exothermicity ${\mathrm{C}}_{240}^{\eta =4}\to {\mathrm{C}}_{240}^{\eta =5}$ indicates that such an introduction of tetragon to the C₂₄₀ cage makes the structure less favorable as compared with one containing the fused pentagons.

The HOMO–LUMO gap is another indicator of the fullerene stability [55] (HOMO and LUMO are the highest occupied and lowest unoccupied molecular orbitals, respectively). It is calculated as the difference in the HOMO and LUMO energies. In our isomeric C₂₄₀ series, only the original structure manifests the largest gap value that indicates its stability. Other (defected) cages, as follows from the orbital energies, should be kinetically unstable. We have tried to collate the obtained HOMO–LUMO gaps with the results of the TM modeling and found a correlation between the gap and extreme topological roundness values (Figure 8). Indeed, both values, being different in their origin, should reflect the stability of the C₂₄₀ cages. However, the correlation is weak (R² = 0.861) and requires further investigation, which includes a larger number of the defected fullerene structures.

4. Discussion

The results of the TM simulations are summarized in Table 2 and Figure 6. The analysis of the evolution of the topological compactness W of the isomers ${\mathrm{C}}_{240}^{\eta }$ (Figure 6) allows observing how the generation η = 1 of the SW dipole causes an immediate decrease in the network invariant W = 277,122 (Equation (1)). In our topological model, this negative variation in the Wiener index represents the topological gain in the graph compactness in respect to the pristine C₂₄₀ (I_h) fullerene. The creation of the SW defect is therefore able to make the fullerene cage more compact. The successive SW_6|7 rotations further augment the topological compactness of the structure, the reduction in W showing an almost parabolic trend for the first propagation steps η = 1, 2, 3. This characteristic behavior of the W index, which favors the expansion of the SW wave, has a pure topological root and it relates to the connectivity properties of the hexagonal regions partially covering the fullerene surface. This result, in fact, matches with previously reported behaviors of SW waves when they propagate in pure honeycomb lattices [26]. The topological diffusion of the 5|7 pair with creating extended dislocation dipoles is stopped at η = 4 and W reaching a local maximum (W = 276,607). Here, the SW wave “collapses” with one of the pentagons of the pristine icosahedral fullerene with the creation of a 4-ring of carbon atoms.

Figure 6 shows instead that the topological order gets diminished by the creation and the propagation of the SW wave. The increase in the extreme topological roundness ${\rho }^E$ (Equation (3)) represents, in fact, an evident “topological response” of the cage to the diffusion of the pentagon-heptagon pairs over the fullerene surface. The evolution of ${\rho }^E$ demonstrates a characteristic plateau (η = 3, 4 with ${\rho }^E$ ≅ 1.05 and η = 4, 5, 6 with ${\rho }^E$ ≅ 1.06) where the topological symmetry of the $C_{240}^{\eta }$ cages remains pretty uncanged, opening the door for the presence of extended dislocation dipoles embedded in the fullerene mesh.

In summary, the main outcomes of the TM simulations state that isomers ${\mathrm{C}}_{240}^{\eta }$ have greater compactness (lower W) than the parent C₂₄₀ (I_h), favoring the creation and the propagation of the SWw defects over the fullerene surface. In these molecules, however, at each step of the SWw propagation, the ${\rho }^E$ index increases (lowering their extreme topological roundness) and obstructs the diffusion of the SWw itself; this barrier effect is reduced in the case of the large η values.

The DFT-estimated energy parameters, in general, indicate endothermicity of the transformations towards the defected fullerene isomers. Thus, such SW processes may take place under high-energy (nonequilibrium) conditions (plasma, interstellar environments etc. [20,21]). The computed HOMO–LUMO gaps also reveal the kinetic instability of the defected cages (in contrast to the parent one having the largest gap). We do not deeply discuss the HOMO–LUMO vs. ${\rho }^E$ plot (Figure 8). There values seem correlated, but the justification of the trend requires extending the correlation field.

The title defect introduced to the honeycomb carbon-containing system is called a dislocation dipole. In the case of the fullerenes, we associate it with nonzero dipole moments (Table 3) of the defected fullerene cages (indeed, it is impossible to make such attribution in the case of introducing a dislocation dipole to the infinite systems, like graphene [57], as their dipole moment is not defined). The influence of the defects on the polarizability of fullerenes has been studied in a few works concluding the increase with the topological defects (see in review [58]). In further studies, we will investigate this important property in the context of SW waves.

As usually considered, the title defect also generates negative-curvature regions on the fullerene surface. In our estimates deduced from the pyramidality angles (Equation (A2)), we have found the atoms with decreased curvatures of the surroundings but which are still positive. Thus, the presence of the pair 5|7 is not the only condition of the negatively curved surface. Additionally, upon SWw, more curved regions also emerge, especially with tetragon inclusion or pentagon–pentagon fusions. Curvature is an important parameter for assessing the reactivity of fullerenes and their derivatives [1,50,51,52,53]. We think that our estimates may be useful for assessing chemical properties of the giant fullerenes, hollow [4,5,6,50] or filled (as a part of carbon nano-onions [59,60]).

The calculated curvature values correspond to the local changes in the C₂₄₀ structures upon SWw. Sphericity is another parameter reflecting the general change in the shape of the molecule. As expected, it goes down with introducing the defects. More important here, sphericity, a geometric structural parameter, is correlated with extreme roundness, a topological structural parameter. This indicated the deducibility of the structural and energetic parameters from the topology of the molecules.

5. Conclusions and Prospective

In the present work, topological, energetic and structural modifications induced on the C₂₄₀ fullerene by the presence of a propagating expanding SWw have been investigated for the first time. Although cage compactness is enhanced for the dislocation-dipole isomers, their energies increase with the size η of the SW dislocation. This means a relatively low probability of the formation of these defective fullerenes. However, the existence of these defective isomers, especially those molecules with large η, which combine low W with non-increasing ${\rho }^E$, may relate to special thermodynamic conditions like during fullerene growth or in interstellar media.

As found, the extreme topological roundness correlate with volumes and sphericities of the C₂₄₀ cages. Such correlations are general for fullerene structures (as they have also been found for other isomeric series). Considering these regularities, we consider the efficiency of the presented topological approach for structural studies on fullerenes and deducing their molecular properties.

These results are perfectly in line with the topological determination of the relative stability of smaller fullerenes. The ability of W in selecting the most compact cage as the physically stable one is documented by the non-trivial case of the C₆₀ fullerene [44] that, among its 1812 non-isomorphic isomers, minimizes the Wiener index in correspondence to the C₆₀ (I_h) with W = 8340 and ${\rho }^E=1$. In more complex C₆₆ case [46], the stability of the whole set of 4478 isomers has been ranked with maximizing two graph properties, i.e., topological compactness (low W) and topological roundness (low ${\rho }^E$). As concluded, the most stable C₆₆ isomers minimize both graph invariants and, hence, may be found in the local minima of the (W,${\rho }^E$) plane. We recently found that this purely topological approach works also for the C₈₄ fullerene. Its most stable forms have minimal graph invariants W and ${\rho }^E$ [47,48]. The latter result also evidences the computational convenience of the TM simulations that rank just in a matter of minutes the topological stability of 51,592 distinct C₈₄ isomers. Thus, we highlight that the TM approximated method is generally applicable to the formation and stabilization mechanisms of complex sp² carbon structures, like large fullerenes. As an additional advance, topological studies allow quick sieving the possible structures before detailed but time-consuming ab initio computations.