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Apr 22, 2023 •600 likes •1.15k views

Toroidal azulenoids Nico Van Cleemput Nicolas.VanCleemput@UGent.be Research group CAAGT, Department of Applied Mathematics and Computer Science, Ghent University (Joint work with Gunnar Brinkmann, Olaf Delgado-Friedrichs and Edward Kirby)

  1. Toroidal azulenoids Nico Van Cleemput Nicolas.VanCleemput@UGent.be Research group CAAGT, Department of Applied Mathematics and Computer Science, Ghent University (Joint work with Gunnar Brinkmann, Olaf Delgado-Friedrichs and Edward Kirby) caagt Toroidal azulenoids – p.1/24

  2. Outline 1. Motivation 2. Translation to tiles 3. Tools 4. Methods 5. Results caagt Toroidal azulenoids – p.2/24

  3. Azulenoids Azulene caagt Toroidal azulenoids – p.3/24

  4. Azulenoids 4 n + 2 annulene with a bridging bond if a π -electron migrates towards the five membered ring then in principle two ’aromatic-sextets’ could be formed ⇒ aromatic behaviour might be expected within Huckel theory caagt Toroidal azulenoids – p.3/24

  5. Azulenoids Consistent with this view is that it has a small dipole moment, and does indeed show some aromatic properties, under milder conditions. caagt Toroidal azulenoids – p.3/24

  6. Question We don’t yet know whether and how the electron mobility might manifest itself among azulenes embedded within a fullerene-style network. How many variations of such networks are theoretically possible? Edward Kirby caagt Toroidal azulenoids – p.4/24

  7. Torus caagt Toroidal azulenoids – p.5/24

  8. Torus caagt Toroidal azulenoids – p.5/24

  9. Torus caagt Toroidal azulenoids – p.5/24

  10. Tiling a subdivision of the plane into faces (or tiles) everything is locally finite the intersections of two different tiles are points or lines or are empty. caagt Toroidal azulenoids – p.6/24

  11. Tiling a subdivision of the plane into faces (or tiles) everything is locally finite the intersections of two different tiles are points or lines or are empty. Periodic tiling ⇐ ⇒ up to symmetry there are only a finite set of tiles caagt Toroidal azulenoids – p.6/24

  12. Example tiling caagt Toroidal azulenoids – p.7/24

  13. Example tiling caagt Toroidal azulenoids – p.7/24

  14. Barycentric subdivision For each face: one point For each edge: one point For each vertex: one point ⇒ subdivision consists of triangles caagt Toroidal azulenoids – p.8/24

  15. Chamber system Define Σ = � σ 0 , σ 1 , σ 2 | σ 2 i = 1 � σ 0 : change the green point (vertex). σ 1 : change the red point (edge). σ 2 : change the black point (face). caagt Toroidal azulenoids – p.9/24

  16. Chamber system Define Σ = � σ 0 , σ 1 , σ 2 | σ 2 i = 1 � σ 0 : change the green point (vertex). σ 1 : change the red point (edge). σ 2 : change the black point (face). Chamber system C of T = barycentric subdivision together with Σ caagt Toroidal azulenoids – p.9/24

  17. Delaney/Dress graph The Delaney/Dress graph D of a periodic tiling is the set of equivalence classes of the chambers of the chamber sys- tem of the tiling under the symmetry group, together with the actions of Σ . caagt Toroidal azulenoids – p.10/24

  18. Example Delaney/Dress graph caagt Toroidal azulenoids – p.11/24

  19. Example Delaney/Dress graph caagt Toroidal azulenoids – p.11/24

  20. Example Delaney/Dress graph caagt Toroidal azulenoids – p.11/24

  21. Example Delaney/Dress graph caagt Toroidal azulenoids – p.12/24

  22. Example Delaney/Dress graph caagt Toroidal azulenoids – p.12/24

  23. Example Delaney/Dress graph ⇒ Delaney/Dress graph is not sufficient to distinguish be- tween tilings! caagt Toroidal azulenoids – p.12/24

  24. Delaney/Dress symbol Define functions m ij : D → N m 01 ( d ) is the size of the face of T that belongs to d . m 12 ( d ) is the number of faces that meet in the vertex that belongs to d . caagt Toroidal azulenoids – p.13/24

  25. Delaney/Dress symbol Define functions m ij : D → N m 01 ( d ) is the size of the face of T that belongs to d . m 12 ( d ) is the number of faces that meet in the vertex that belongs to d . Delaney/Dress symbol of the tiling is ( D ; m 01 , m 12 ) caagt Toroidal azulenoids – p.13/24

  26. Example Delaney/Dress symbol caagt Toroidal azulenoids – p.14/24

  27. Example Delaney/Dress symbol m 01 = 4 m 12 = 4 caagt Toroidal azulenoids – p.14/24

  28. Example Delaney/Dress symbol caagt Toroidal azulenoids – p.15/24

  29. Example Delaney/Dress symbol m 01 = 6 m 12 = 3 caagt Toroidal azulenoids – p.15/24

  30. Example Delaney/Dress symbol caagt Toroidal azulenoids – p.16/24

  31. Example Delaney/Dress symbol caagt Toroidal azulenoids – p.16/24

  32. Example Delaney/Dress symbol caagt Toroidal azulenoids – p.16/24

  33. Example Delaney/Dress symbol m 01 m 12 A 4 3 B 8 3 C 8 3 caagt Toroidal azulenoids – p.16/24

  34. Delaney/Dress symbol ( D ; m 01 , m 12 ) is the Delaney/Dress symbol of a periodic tiling of the plane iff. 1. D is finite 2. Σ works transitively on D 3. m 01 is constant on � σ 0 , σ 1 � orbits and ∀ d ∈ D : d ( σ 0 σ 1 ) m 01 ( d ) = d 4. m 12 is constant on � σ 1 , σ 2 � orbits and ∀ d ∈ D : d ( σ 1 σ 2 ) m 12 ( d ) = d 5. ∀ d ∈ D : d ( σ 0 σ 2 ) 2 = d 1 m 12 ( d ) − 1 1 6. � d ∈D ( m 01 ( d ) + 2 ) = 0 caagt Toroidal azulenoids – p.17/24

  35. Refined question How many variations of fullerene-style networks for which there exists a partition of the atoms into azulenes are the- oretically possible, assuming there is only one equivalence class of azulenes? caagt Toroidal azulenoids – p.18/24

  36. Translation Restrictions azulenoid: 1 equivalence class of azulenes every atom part of exactly one azulene caagt Toroidal azulenoids – p.19/24

  37. Translation Restrictions azulenoid: 1 equivalence class of azulenes every atom part of exactly one azulene Restrictions Delaney/Dress symbol: ∃ σ 0 σ 1 orbit O : m 01 ( O ) = 8 ∧ ∀ σ 1 σ 2 orbit V : O ∩ V � = ∅ caagt Toroidal azulenoids – p.19/24

  38. Translation Restrictions azulenoid: 1 equivalence class of azulenes every atom part of exactly one azulene Restrictions Delaney/Dress symbol: ∃ σ 0 σ 1 orbit O : m 01 ( O ) = 8 ∧ ∀ σ 1 σ 2 orbit V : O ∩ V � = ∅ ∀ σ 1 σ 2 orbit V : m 12 ( V ) = 3 caagt Toroidal azulenoids – p.19/24

  39. Translation Restrictions azulenoid: 1 equivalence class of azulenes every atom part of exactly one azulene Restrictions Delaney/Dress symbol: ∃ σ 0 σ 1 orbit O : m 01 ( O ) = 8 ∧ ∀ σ 1 σ 2 orbit V : O ∩ V � = ∅ ∀ σ 1 σ 2 orbit V : m 12 ( V ) = 3 1 m 12 ( d ) − 1 1 � ( m 01 ( d ) + 2) = 0 d ∈D caagt Toroidal azulenoids – p.19/24

  40. Method Octagon and the different vertex orbits caagt Toroidal azulenoids – p.20/24

  41. Method Octagon and the different vertex orbits caagt Toroidal azulenoids – p.20/24

  42. Method Octagon and the different vertex orbits Calculate and assign remaining m 01 values caagt Toroidal azulenoids – p.20/24

  43. Method Octagon and the different vertex orbits Calculate and assign remaining m 01 values Assign remaining σ 0 ’s caagt Toroidal azulenoids – p.20/24

  44. Method Octagon and the different vertex orbits Calculate and assign remaining m 01 values Assign remaining σ 0 ’s Replace octagon with azulene caagt Toroidal azulenoids – p.20/24

  45. Method Octagon and the different vertex orbits Calculate and assign remaining m 01 values Assign remaining σ 0 ’s Replace octagon with azulene caagt Toroidal azulenoids – p.20/24

  46. Visualisation caagt Toroidal azulenoids – p.21/24

  47. Visualisation caagt Toroidal azulenoids – p.21/24

  48. Visualisation caagt Toroidal azulenoids – p.21/24

  49. Results m 01 values # strings # symbols 1 4 4 4 4 4 6 24 24 21 6 2 4 4 4 4 4 8 12 24 42 42 3 4 4 4 4 4 8 16 16 21 48 4 4 4 4 4 4 10 10 20 21 0 5 4 4 4 4 4 12 12 12 7 44 6 4 4 4 4 6 6 8 24 105 0 7 4 4 4 4 6 6 12 12 54 2 8 4 4 4 4 6 8 8 12 105 12 9 4 4 4 4 8 8 8 8 10 160 10 4 4 4 6 6 6 6 12 35 6 11 4 4 4 6 6 6 8 8 70 38 12 4 4 6 6 6 6 6 6 4 25 caagt Toroidal azulenoids – p.22/24

  50. Results 383 symbols of tilings containing octagons caagt Toroidal azulenoids – p.22/24

  51. Results 383 symbols of tilings containing octagons ⇓ 1274 azulenoids caagt Toroidal azulenoids – p.22/24

  52. Translation only one orbit of azulenes under the subgroup of translations or all the azulenes have the same orientation caagt Toroidal azulenoids – p.23/24

  53. Translation only caagt Toroidal azulenoids – p.23/24

  54. End Thanks for your attention! caagt Toroidal azulenoids – p.24/24

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