[PPT] - caagt Toroidal azulenoids p.1/30 Outline 1. Motivation 2. PowerPoint Presentation

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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)

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Outline

1. Motivation
2. Translation to tiles
3. Tools
4. Methods
5. Results

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Azulenoids

Azulene

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Azulenoids

4n + 2 annulene with a bridging bond if a π-electron migrate towards the five membered ring then in principle two ’aromatic-sextets’ could be formed ⇒ aromatic behaviour might be expected within Hückel theory

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Azulenoids

Consistent with this view is that it has a small dipole moment, and does indeed show some aromatic properties, under milder conditions.

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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

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Torus

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Torus

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Torus

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Tiling

Tiling T = set of tiles t1, t2, . . . with ti ⊂ E2, ti homeomorph to B(0, 1) that satisfy the following conditions: 1.

t∈T

t = E2

2. ∀ti, tj(i = j) ∈ T : t◦

i ∩ t◦ j = ∅ ∧ ti ∩ tj ∈

{∅, {points}, {lines}}.

3. ∀x ∈ E2 : x has a neighbourhood that only intersects a

finite number of tiles.

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Tiling

Tiling T = set of tiles t1, t2, . . . with ti ⊂ E2, ti homeomorph to B(0, 1) that satisfy the following conditions: 1.

t∈T

t = E2

2. ∀ti, tj(i = j) ∈ T : t◦

i ∩ t◦ j = ∅ ∧ ti ∩ tj ∈

{∅, {points}, {lines}}.

3. ∀x ∈ E2 : x has a neighbourhood that only intersects a

finite number of tiles. Periodic tiling ⇐ ⇒ symmetry group contains two independent translations

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Example tiling

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Example tiling

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Barycentric subdivision

For each face: one point For each edge: one point For each vertex: one point ⇒ subdivision consists of triangles

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Chamber system

Define Σ = σ0, σ1, σ2|σ2

i = 1

σ0 : change the green point (vertex). σ1 : change the red point (edge). σ2 : change the black point (face).

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Chamber system

Define Σ = σ0, σ1, σ2|σ2

i = 1

σ0 : change the green point (vertex). σ1 : change the red point (edge). σ2 : change the black point (face). Chamber system C of T = barycentric subdivision together with Σ

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Delaney/Dress graph

The Delaney/Dress graph D of a periodic tiling is the set of

rbits of the chambers of the chamber system of the tiling

under the symmetry group.

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Example Delaney/Dress graph

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Example Delaney/Dress graph

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Example Delaney/Dress graph

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Example Delaney/Dress graph

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Example Delaney/Dress graph

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Example Delaney/Dress graph

⇒ Delaney/Dress graph is not sufficient to distinguish be- tween tilings!

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Delaney/Dress symbol

Define functions rij : C → N; c → rij(c) with rij(c) the smallest number for which c(σiσj)rij(c) = c.

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Delaney/Dress symbol

Define functions rij : C → N; c → rij(c) with rij(c) the smallest number for which c(σiσj)rij(c) = c. r02 is a constant function with value 2. r01(c) is the size of the face of T that belongs to c. r12(c) is the number of faces that meet in the vertex that belongs to c.

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Delaney/Dress symbol

Define functions mij : D → N; d → mij(c) in such a manner that the following diagram is commutative: C rij

✲ N

D mij

✲ ✲

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Delaney/Dress symbol

Define functions mij : D → N; d → mij(c) in such a manner that the following diagram is commutative: C rij

✲ N

D mij

✲ ✲

Delaney/Dress symbol of the tiling is (D; m01, m12)

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Example Delaney/Dress symbol

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Example Delaney/Dress symbol

m01 = 4 m12 = 4

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Example Delaney/Dress symbol

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Example Delaney/Dress symbol

m01 = 6 m12 = 3

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Example Delaney/Dress symbol

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Example Delaney/Dress symbol

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Example Delaney/Dress symbol

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Example Delaney/Dress symbol

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Example Delaney/Dress symbol

m01 m12 A 4 B C

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Example Delaney/Dress symbol

m01 m12 A 4 B C

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Example Delaney/Dress symbol

m01 m12 A 4 B 8 C 8

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Example Delaney/Dress symbol

m01 m12 A 4 B 8 C 8

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Example Delaney/Dress symbol

m01 m12 A 4 3 B 8 3 C 8 3

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Delaney/Dress symbol

(D; m01, m12) is the Delaney/Dress symbol of a periodic tiling iff.

1. D is finite
2. Σ works transitively on D
3. m01 is constant on σ0, σ1 orbits and

∀d ∈ D : d(σ0σ1)m01(d) = d

4. m12 is constant on σ1, σ2 orbits and

∀d ∈ D : d(σ1σ2)m12(d) = d

5. ∀d ∈ D : d(σ0σ2)2 = d

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Delaney/Dress symbol

K(D) =

d∈D

( 1 m01(d) + 1 m12(d) − 1 2)

1. K(D) < 0
2. K(D) = 0
3. K(D) > 0

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Delaney/Dress symbol

K(D) =

d∈D

( 1 m01(d) + 1 m12(d) − 1 2)

1. K(D) < 0 → hyperbolic plane
2. K(D) = 0
3. K(D) > 0

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Delaney/Dress symbol

K(D) =

d∈D

( 1 m01(d) + 1 m12(d) − 1 2)

1. K(D) < 0 → hyperbolic plane
2. K(D) = 0 → euclidean plane
3. K(D) > 0

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Delaney/Dress symbol

K(D) =

d∈D

( 1 m01(d) + 1 m12(d) − 1 2)

1. K(D) < 0 → hyperbolic plane
2. K(D) = 0 → euclidean plane
3. K(D) > 0 → sphere iff.

4 K(D) ∈ N

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Representation

σ0 σ1 σ2 m01 m12

1 σ0(1) σ1(1) σ2(1) m01(1) m12(1)

2 σ0(2) σ1(2) σ2(2) m01(2) m12(2)

3 σ0(3) σ1(3) σ2(3) m01(3) m12(3)

. . . . . . . . . . . . . . . . . .

N σ0(N) σ1(N) σ2(N) m01(N) m12(N)

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Canonical form

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Canonical form

based on index priority depth-first traversal of graph

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Canonical form

based on index priority depth-first traversal of graph canonical relabelling when

m01(1) . . . m12(1) . . . σ0(1) . . . σ1(1) . . . σ2(1) . . .

is lexicographically smallest.

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Minimal symbol

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Minimal symbol

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Minimal symbol

add symmetry

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Minimal symbol

add symmetry ⇒ map orbits onto each other

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Minimal symbol

add symmetry ⇒ map orbits onto each other Choose two orbits c and d

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Minimal symbol

add symmetry ⇒ map orbits onto each other Choose two orbits c and d Is mij(c) = mij(d)?

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Minimal symbol

add symmetry ⇒ map orbits onto each other Choose two orbits c and d Is mij(c) = mij(d)? index priority depth-first traversal from c and d

σi(c) maps onto σi(d)

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Minimal symbol

add symmetry ⇒ map orbits onto each other Choose two orbits c and d Is mij(c) = mij(d)? index priority depth-first traversal from c and d

σi(c) maps onto σi(d) mij(·) = mij(·)?

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Example minimal symbol

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Example minimal symbol

σ0 σ1 σ2 m01 m12 A A B F 8 3 A B B A C 8 3 B C C D B 4 3 C D D C E 4 3 D E E F D 8 3 E F F E A 8 3 F

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Example minimal symbol

σ0 σ1 σ2 m01 m12 A A B F 8 3 A B B A C 8 3 B C C D B 4 3 a D D C E 4 3 a E E F D 8 3 E F F E A 8 3 F

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Example minimal symbol

σ0 σ1 σ2 m01 m12 A A B F 8 3 A B B A C 8 3 B C C D B 4 3 a D D C E 4 3 a E E F D 8 3 E F F E A 8 3 F

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Example minimal symbol

σ0 σ1 σ2 m01 m12 A A B F 8 3 A B B A C 8 3 b C C D B 4 3 a D D C E 4 3 a E E F D 8 3 b F F E A 8 3 F

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Example minimal symbol

σ0 σ1 σ2 m01 m12 A A B F 8 3 A B B A C 8 3 b C C D B 4 3 a D D C E 4 3 a E E F D 8 3 b F F E A 8 3 F

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Example minimal symbol

σ0 σ1 σ2 m01 m12 A A B F 8 3 c B B A C 8 3 b C C D B 4 3 a D D C E 4 3 a E E F D 8 3 b F F E A 8 3 c

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Example minimal symbol

σ0 σ1 σ2 m01 m12 A A B F 8 3 c B B A C 8 3 b C C D B 4 3 a D D C E 4 3 a E E F D 8 3 b F F E A 8 3 c

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Example minimal symbol

σ0 σ1 σ2 m01 m12 A A B F 8 3 c B B A C 8 3 b C C D B 4 3 a D D C E 4 3 a E E F D 8 3 b F F E A 8 3 c

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Example minimal symbol

σ0 σ1 σ2 m01 m12 a a a b 4 3 b b c a 8 3 c c a b 8 3

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Refined question

How many variations of fullerene-style networks for which there exists a partition of the atoms into azulenes are the-

retically possible, assuming there is only one orbit of azu-

lenes?

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Translation

Restrictions azulenoid: 1 orbit of azulenes every atom part of exactly one azulene

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Translation

Restrictions azulenoid: 1 orbit of azulenes every atom part of exactly one azulene Restrictions Delaney/Dress symbol: ∃σ0σ1 orbit O : r01(O) = 8 ∧ ∀σ1σ2 orbit V : O ∩ V = ∅

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Translation

Restrictions azulenoid: 1 orbit of azulenes every atom part of exactly one azulene Restrictions Delaney/Dress symbol: ∃σ0σ1 orbit O : r01(O) = 8 ∧ ∀σ1σ2 orbit V : O ∩ V = ∅ ∀σ1σ2 orbit V : r12(V ) = 3

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Translation

Restrictions azulenoid: 1 orbit of azulenes every atom part of exactly one azulene Restrictions Delaney/Dress symbol: ∃σ0σ1 orbit O : r01(O) = 8 ∧ ∀σ1σ2 orbit V : O ∩ V = ∅ ∀σ1σ2 orbit V : r12(V ) = 3

d∈D

( 1 m01(d) + 1 m12(d) − 1 2) = 0

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Method

Octagon and the different vertex orbits

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Method

Octagon and the different vertex orbits

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Method

Octagon and the different vertex orbits Calculate and assign remaining m01 values

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Method

Octagon and the different vertex orbits Calculate and assign remaining m01 values

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Method

Octagon and the different vertex orbits Calculate and assign remaining m01 values Assign remaining σ0’s

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Method

Octagon and the different vertex orbits Calculate and assign remaining m01 values Assign remaining σ0’s Replace octagon with azulene

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Method

Octagon and the different vertex orbits Calculate and assign remaining m01 values Assign remaining σ0’s Replace octagon with azulene

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Method

Octagon and the different vertex orbits Calculate and assign remaining m01 values Assign remaining σ0’s Replace octagon with azulene

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Method

Octagon and the different vertex orbits Calculate and assign remaining m01 values Assign remaining σ0’s Replace octagon with azulene

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Visualisation

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Visualisation

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Visualisation

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Results

m01 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 5 4 4 4 4 4 12 12 12 7 44 6 4 4 4 4 6 6 8 24 105 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

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Results

383 symbols of tilings containing octagons

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Results

383 symbols of tilings containing octagons ⇓ 1274 azulenoids

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Translation only

ne orbit of azulenes under the subgroup of translations
r

all the azulenes have the same orientation

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Translation only

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