Constructing and Analyzing Structures of Regular

Fullerenesl

2.1 Introduction

Since the discovery of buckminsterfullerene C60 by Kroto, Smalley and

Curl in1985,[6,9]_{and its subsequent synthesis in macroscopic amounts by}

Krätschmeret al.in1990,[10]_{the chemistry and physics of fullerenes have}

advanced remarkably in the past two decades.[233,234]There are, however, many open problems related to the structure and stability of fullerenes that remain to be solved. While there exists a number of heuristics that estimate the thermodynamic stability of a particular fullerene, for example by way of spectral analysis of its graph, it is not yet feasible to do so systematic- ally for all isomers of large fullerenes. The number of regular fullerene isomers Niso(CN) for a speciﬁc number of carbon atoms N increases as Niso(CN)∼ O

N9. As an illustration of the growth, C60 has1,812isomers,

and C180 has 79,538,751 isomers.[35] The order N9-growth follows from

l_Sections₂_.₁_to₂_.₁₃_{of this chapter have previously been published by Schwerdtfeger et}

al.[36]and are reproduced with kind permission from the authors and John Wiley & Sons, Ltd.. The candidate’s contribution to the article is quantiﬁed on pages299ff. The last section of the article (Examples) is omitted. Section2.14is added to describe in more detail the structure of the program, in particular functionality that has been added to the back end since this article was published. Furthermore it gives an outlook on possible future extensions of the program.

Thurston’s parameterization of triangulations on geodesic domes.[20]Even though the number of isomers is reduced considerably when pentagons are attached to hexagons only—the so-called isolated pentagon rule (IPR), which

increases the thermodynamic stability of fullerenes considerably[19,52]—the number ofIPRisomers still seem to increase asymptotically asN9, as shown on Figure1.1. This is likely due to the proportion ofIPRto non-IPRfullerenes growing with increasing N: Out of the1,812C60 isomers only one isIPR,

but out of the isomers for C180,4,071,832—one in twenty—areIPR.[35]As a

consequence, for large fullerenes it becomes difﬁcult to ﬁnd the thermody- namically most favored isomer. Moreover, graph theoretical indicators yield only rough estimates for stability. Once candidates for stable isomers have been narrowed down, the3D fullerene structure must be generated before they can be subjected to accurate quantum theoretical treatment.

To obtain a reasonably accurate structure for larger fullerenes is non- trivial. For this, the ring-spiral algorithm, developed by Fowler and Ma- nolopoulos, with a subsequent structure generation using the adjacency matrix eigenvector algorithm (AME) helped to generate and list approximate fullerene structures up to C50, and forIPRfullerenes up to C100, creating

“An Fowler-atlas-2006of Fullerenes” and the ﬁrst useful database.[32]_More

recently, Brinkmannet al. constructed fullerene isomers up to C400, using

theBuckygenprogram.[35,66,235]They also list special fullerenes without ring spirals starting at a pentagon and fullerenes without any ring spiral at all up to C400.[35,236]

The initial structure obtained through various embedding algorithms such as theAME [32,102]or the3D-TEmethod (described later) is usually far

from the physical3D structure and requires further optimization, for example by carefully adjusted force fields.[237–239]Standard force fields usually distinguish between single and double bonds. However, the assignment of double bonds (Kekulé structures) to fullerenes is equivalent in graph theory to finding perfect matchings,[240] _{of which fullerene graphs have}

exponentially many.[166]_{While the exponential theoretical lower bound for}

the number of perfect matchings in fullerenes only kicks in at C380, the

rapid growth in their numbers starts much sooner: TheIPRfullerene I_h-C60

(I_h-C70) has as many as12500(52168) Kekulé structures, of which158(2780)

are non-isomorphic.[163,241]As this number grows, searching through all Kekulé structures for the most stable one becomes impractical, and force

fields for fullerenes should therefore not be designed with double bonds in mind. While it is easy to findoneperfect matching inO(n2)time, if it exists, finding the optimal Clar structure with the maximum number of isolated sextets requires searching through the exponentially many perfect matchings.[242,243]

A similar situation is found if one searches for the lexicographically smallest of the longest carbon chains in a fullerene, required for example for theIUPACnomenclature of organic compounds.[199]This requires searching

through all Hamiltonian cyclesmin the fullerene graph.[245]Since determ- ining whether a Hamiltonian cycle in a planar cubic graph exists is an

NP-complete problem, ﬁnding all Hamiltonian cycles is at least #P-hard and likely cannot be solved in polynomial time. It is not known whether all fullerenes have Hamiltonian cycles, although most 3-regular graphs are Hamiltonian.[246]_{Brinkmann, Goedgebeur, and McKay demonstrated}

that Hamiltonian cycles exist for all fullerenes up to C316by generating all

isomers and testing for Hamiltonicity.[66]

There are many more open problems concerning fullerene structures related mostly to topological and graph theoretical aspects.[41]We only men- tion here the problem to ﬁnd all fullerenes that cannot be generated from ring spirals, the two smallest ones being C380and C384,[236] and the close

packing problem of fullerene polyhedra to estimate the solid-state3D structure, which is related to one of Hilbert’s famous problems in mathematics and to Kepler’s conjecture.[247]

While properties of smaller fullerenes have been explored extensively in the past by both experimental and computational methods,[248]_structural,

topological and electronic properties of larger fullerenes remain relatively unexplored. Although there are several other computer programs already available to deal with fullerene structures and graphs likeCaGe(plantri,[249]

fullgen,[250] Buckygen[66,235]),[251] Vega,[252], FuiGui,[253] or the routines introduced in the book by Fowler and Manolopoulos,[32] _{a comprehensive}

computer program addressing most of the problems mentioned using efﬁ-

m_{Or longest non-repeating paths if no Hamiltonian cycle exists: it is still an unsolved problem}

whether all fullerenes have Hamiltonian cycles or not. If Barnette’s conjecture[244]_{is true, it}

may be possible to construct fast algorithms for counting Hamiltonian cycles in fullerenes. However, it is still an open problem, and consequently the only known algorithms take exponential time.

cient algorithms (of course within the limitations of #P-hard orNP-complete problems) is to our knowledge not available. We therefore felt that there is a need to develop a general purpose and open-source program whose feature set is a superset of what other programs can do. The program introduced in this paper aims to ﬁll this role.

2.2 General structure and history of the

program

Fullerene

Fullereneis written in standard Fortran andC++ and can easily be installed on a Linux orUNIXenvironment using theGNUcompiler collection (gfortran and g++). The ﬁrst version of the programFullerenewas written in Fortran only for calculating the surface area and volume of a fullerene to determine the corresponding changes due to endohedral incorporation of rare gas atoms in a fullerene cage.[254]A much improved version2allowed for the creation of fullerene structures using the ring-spiral algorithm introduced in the book by Fowler and Manolopoulos,[32] and version3was already dealing with some topological and graph theoretical properties analyzing basic ring patterns. It soon became clear that many open problems remain in fullerene structure generation and corresponding topological properties, which needed to be addressed from a computational point of view. It was therefore decided to extend version3to a general purpose program package for fullerenes and to make it available to the scientiﬁc community.

In the current version4.4, only regular fullerenes are considered (i.e., of genus0and consisting of pentagons and hexagons only) fulﬁlling Euler’s polyhedral theorem,

nv−ne+nf =2 (2.1)

where nv is the number of vertices (carbon atoms) (nv ≥ 20 and nv =

22),[53] _n

e =3nv/2 the number of edges (C–C bonds), and nf = nv/2+2

the number of faces (pentagons or hexagons). From Euler’s polyhedral theorem it follows that the number of pentagons in a regular fullerene is exactly12, no more and no less, known as the12pentagon theorem.[255] The main task of the program is to determine the structure of a speciﬁc fullerene and to calculate its topological properties. The results can be

used for plotting planar drawings of fullerene graphs (also called Schlegel diagrams) and3D structures, and serves as a good starting point for further quantum theoretical treatment. A number of approximate physical and chemical properties are calculated as well. A ﬂow diagram for the main steps performed in the program is depicted in Figure2.1.

if ‘istop’= 0 Date and time

Timer Datain CoordC20C60 CoordBuild Isomers MoveCM Diameter Distmatrix Connect Hueckel GoldbergCoxeter Hamilton HamiltonCyc Paths Ring Ring RingC StoneWalesTrans EndoKrotoTrans YoshidaFowler4 YoshidaFowler6 BrinkmannFowler SpiralSearch Chiral TopIndicators OptFF MoveCM Distmatrix Diameter Volume MinCovSphere2 MinDistSphere MaxInSphere Graph2D Timer Isomers if ‘ICart’ = 0 if ‘ICart’ ∈ {2,3,4,5, 6,7} if ‘leap’>0 or if ‘leapGC’>0 if ‘iHam’= 0 if. . . = 0 if ‘iOpt’= 0 if ‘iSchlegel’= 0 if ‘loop’≥2

Figure 2.1 Flow diagram for the main tasks of the program_Fullerene.

In document Graph theoretic and electronic properties of fullerenes, &, Biasing molecular modelling simulations with experimental residual dipolar couplings : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy at Masse (Page 70-74)