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Spectral classification of fullerenes
Joint work with Artur Bille (Skoltech/UUlm) andVictor Buchstaber (Steklov Math. Inst./Skoltech)
Evgeny Spodarev | Ulm University | 3.12.2018
Seite 2 Spectral classification of fullerenes | Evgeny Spodarev | Ulm University
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
Seite 3 Spectral classification of fullerenes | Evgeny Spodarev | Ulm University
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)
Seite 4 Spectral classification of fullerenes | Evgeny Spodarev | Ulm University
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:
Seite 5 Spectral classification of fullerenes | Evgeny Spodarev | Ulm University
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?
Seite 6 Spectral classification of fullerenes | Evgeny Spodarev | Ulm University
What are isomers and how many do exist?
Two combinatorially nonequivalent fullerenes with the samenumber of hexagons are called combinatorial isomers.
Seite 7 Spectral classification of fullerenes | Evgeny Spodarev | Ulm University
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)
Seite 8 Spectral classification of fullerenes | Evgeny Spodarev | Ulm University
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?
Seite 9 Spectral classification of fullerenes | Evgeny Spodarev | Ulm University
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
Seite 10 Spectral classification of fullerenes | Evgeny Spodarev | Ulm University
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
Seite 11 Spectral classification of fullerenes | Evgeny Spodarev | Ulm University
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
Seite 12 Spectral classification of fullerenes | Evgeny Spodarev | Ulm University
Five special C60–Isomers: physical interpretation
Three least stable and two most stable isomers
Seite 13 Spectral classification of fullerenes | Evgeny Spodarev | Ulm University
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)
Seite 14 Spectral classification of fullerenes | Evgeny Spodarev | Ulm University
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
Seite 15 Spectral classification of fullerenes | Evgeny Spodarev | Ulm University
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