Spectral classification of fullerenes

Joint work with Artur Bille (Skoltech/UUlm) andVictor Buchstaber (Steklov Math. Inst./Skoltech)

Evgeny Spodarev | Ulm University | 3.12.2018

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Spectral classification of fullerenes

Outline of the talk

I What is a fullerene?I Why do we care?I Isomers of fullerenesI Spectral clustering of isomersI Example of C60

I Dimension reductionI 5 interesting isomers

I OutlookI References

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What is a fullerene?

Chemical point of view:I Molecule Cn with n carbon atomsI Closed cage-like hollow sphereI Any atom belongs to exactly three carbon ringsI Carbon rings can be pentagons or hexagons only

C60 C70

Nobel Prize in Chemistry 1996 (R. Curl, H. Kroto, R. Smalley)

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What is a fullerene?Mathematical point of view:A fullerene Cn is a simple 3–convex polytope with allfacets being regular pentagons or hexagons.Numer of vertices: n ≥ 20, even.

Schlegel diagram:

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Why do we study fullerenes?I Wide variety of applications: artificial photosynthesis,

nonlinear optics, nanostructures etc.I Young research area:

What fullerenes can be synthesized?Are there any particularly stable fullerenes?→ IPR–Isomers(IPR- isolated penagon rule. All pentagon facets are isolated)

I How can fullerenes be simulated?I How can one cluster fullerene isomers?

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What are isomers and how many do exist?

Two combinatorially nonequivalent fullerenes with the samenumber of hexagons are called combinatorial isomers.

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Number of Cn–Isomers

n # Cn–Isomers # IPR–Isomers20 1 022 0 024 1 060 1812 168 6332 070 8149 180 31 92 7100 285 914 450

⇒ #Cn–Isomers = O(n9)

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How to cluster these Isomers?Our approach:

1. Construct an adjacency matrix A of all hexagons .2. Calculate the k–th power of A and trace tk := tr

(Ak),

2 ≤ k ≤ N.3. For every k check how many different values one gets.4. Cluster all Cn–isomers based on these values.

Remark:

It holds tr (A) = 0 for all Cn–Isomers by definition of A,tk ∈ N for all k .

Question: What is minimal N needed?

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Example: Hierarchical clustering of C60

Vector # Cluster # Cluster with one element(t2) 18 5(t2, t3) 42 7(t2, t3, t4) 372 130(t2, t3, t4, t5) 1068 670(t2, t3, t4, t5, t6) 1747 1686(t2, t3, t4, t5, t6, t7) 1808 1804(t2, t3, t4, t5, t6, t7, t8) 1812 1812(t6, t7, t8) 1812 1812

N=8

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Single-step clustering of C60

tk # Cluster # Cluster with one elementk = 2 18 5k = 3 23 5k = 4 218 47k = 5 219 36k = 6 1233 845k = 7 1241 825k = 8 1784 1757k = 9 1781 1750k = 10 1807 1802k = 11 1810 1808k = 12 1812 1812

N=12

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Five special C60–IsomersFor k = 2 one gets 18 different values (= 18 clusters).Five of these clusters consist of one C60–Isomer only.

tr(A2) # C60–Isomers

60 164 166 368 1770 8672 25474 26276 41378 297

tr(A2) # C60–Isomers

80 19782 9684 5086 1988 1090 292 196 1

100 1

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Five special C60–Isomers: physical interpretation

Three least stable and two most stable isomers

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Relative Energy

Top: Highest relative energy and less spherical structure⇒ least stable isomers

Bottom: Smallest relative energy and more spherical structure⇒ most stable isomers

Compare the paper

R. Sure, A. Hansen, et al. Phys. Chem. Chem. Phys. (2017)

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Outlook

I Classification of isomers of Cn for n = 70,80,82, . . .I Search for potentially interesting isomers for large nI Dimension reduction: a formula for the least possible N

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ReferencesV. Andova, F. Kardos, and R. Skrekovski, Mathematical aspects of fullerenes, ArsMathematica Contemporanea 11, 2016, 252-279

A. Bille, V. Buchstaber, M. Schandar, E. Spodarev, Spectral clustering ofcombinatorial fullerene isomers based on their facet eigenstructure, Preprint,2018

V. Buchstaber, N. Erokhovets, Fullerenes, polytopes and toric topology, 67-178,Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 35, World Sci. Publ., 2018

F. Cataldo, A. Graovac, O. Ori, The Mathematics and Topology of Fullerenes,Springer, 2011

P. W. Fowler, D. E. Manolopoulos, An Atlas of Fullerenes, Dover Publ., NY, 2006

I. Gutman, D. Vidovic, and L. Popovic, Graph representation of organicmolecules, J. Chem. Soc., Faraday Trans., 1998, 94(7), 857–860

R. Sure, A. Hansen, P. Schwerdtfeger, S. Grimme, Comprehensive theoretical

study of all 1812 C60 isomers, Phys. Chem. Chem. Phys., The Royal Society of

Chemistry, 2017, 19, 14296