<table>
  <tbody>
    <tr>
      <td>2.31 Representation of the molecule of corannulene</td>
      <td>58</td>
    </tr>
    <tr>
      <td>2.32 Cartographies of the parameters of the corannulene</td>
      <td>58</td>
    </tr>
    <tr>
      <td>2.33 Representation of the molecule fullerene C<sub>50</sub>-D<sub>3</sub></td>
      <td>59</td>
    </tr>
    <tr>
      <td>2.34 Cartographies of the parameters of fullerene C<sub>50</sub>-D<sub>3</sub></td>
      <td>59</td>
    </tr>
    <tr>
      <td>2.35 Representation of the molecule fullerene C<sub>50</sub>-D<sub>5h</sub></td>
      <td>60</td>
    </tr>
    <tr>
      <td>2.36 Cartographies of the parameters of fullerene C<sub>50</sub>-D<sub>5h</sub></td>
      <td>60</td>
    </tr>
    <tr>
      <td>2.37 Representation of the fullerenic molecule C<sub>50</sub>Cl<sub>10</sub>-D<sub>3</sub></td>
      <td>61</td>
    </tr>
    <tr>
      <td>2.38 Cartographies of the parameters of cholorofullerene D<sub>3</sub></td>
      <td>61</td>
    </tr>
    <tr>
      <td>2.39 Representation of the fullerenic molecule C<sub>50</sub>Cl<sub>10</sub>-D<sub>5h</sub></td>
      <td>61</td>
    </tr>
    <tr>
      <td>2.40 Cartographies of the parameters of cholorofullerene D<sub>5h</sub></td>
      <td>62</td>
    </tr>
    <tr>
      <td>2.41 Representation of the molecule C<sub>64</sub>-C<sub>3v</sub></td>
      <td>62</td>
    </tr>
    <tr>
      <td>2.42 Cartographies of the parameters of the fullerene C<sub>64</sub>-C<sub>3v</sub></td>
      <td>63</td>
    </tr>
    <tr>
      <td>2.43 Representation of the molecule C<sub>64</sub>-C<sub>2</sub></td>
      <td>63</td>
    </tr>
    <tr>
      <td>2.44 Cartographies of the parameters of the fullerene C<sub>64</sub>-C<sub>2</sub></td>
      <td>64</td>
    </tr>
    <tr>
      <td>2.45 Representation of the molecule C<sub>64</sub>-D<sub>2</sub></td>
      <td>64</td>
    </tr>
    <tr>
      <td>2.46 Cartographies of the parameters of the fullerene C<sub>64</sub>-D<sub>2</sub></td>
      <td>64</td>
    </tr>
    <tr>
      <td>2.47 Representation of the molecule C<sub>64</sub>-C<sub>s</sub></td>
      <td>65</td>
    </tr>
    <tr>
      <td>2.48 Cartographies of the parameters of the fullerene C<sub>64</sub>-C<sub>s</sub></td>
      <td>65</td>
    </tr>
    <tr>
      <td>2.49 Representation of the molecule C<sub>64</sub>Cl<sub>4</sub>-C<sub>3v</sub></td>
      <td>66</td>
    </tr>
    <tr>
      <td>2.50 Cartographies of the parameters of the fullerene C<sub>64</sub>Cl<sub>4</sub>-C<sub>3v</sub></td>
      <td>66</td>
    </tr>
    <tr>
      <td>2.51 Representation of the molecule C<sub>64</sub>Cl<sub>4</sub>-C<sub>s</sub></td>
      <td>66</td>
    </tr>
    <tr>
      <td>2.52 Cartographies of the parameters of the fullerene C<sub>64</sub>Cl<sub>4</sub>-C<sub>s</sub></td>
      <td>67</td>
    </tr>
    <tr>
      <td>2.53 Representation of the molecule C<sub>64</sub>Cl<sub>4</sub>-D<sub>2</sub></td>
      <td>67</td>
    </tr>
    <tr>
      <td>2.54 Cartographies of the parameters of the fullerene C<sub>64</sub>Cl<sub>4</sub>-D<sub>2</sub></td>
      <td>68</td>
    </tr>
    <tr>
      <td>2.55 Representation of the molecule C<sub>64</sub>Cl<sub>4</sub>-C<sub>2</sub></td>
      <td>68</td>
    </tr>
    <tr>
      <td>2.56 Cartographies of the parameters of the fullerene C<sub>64</sub>Cl<sub>4</sub>-C<sub>2</sub></td>
      <td>68</td>
    </tr>
    <tr>
      <td>2.57 Data of the pyramidalization angles and the hybridization of each molecule considering the carbon environment</td>
      <td>69</td>
    </tr>
    <tr>
      <td>2.58 Stages ring expansion calculated by DFT B3LYP-6-311G method [32]</td>
      <td>70</td>
    </tr>
    <tr>
      <td>2.59 Rearrangement routes of aromatic nitrile imines [32]</td>
      <td>71</td>
    </tr>
    <tr>
      <td>2.60 Conformation 8aA and classification of atoms according to their environment and hybridization</td>
      <td>72</td>
    </tr>
    <tr>
      <td>2.61 Cartographies and representation of the parameters of conformation 8aA</td>
      <td>72</td>
    </tr>
    <tr>
      <td>2.62 Set of values of parameters of conformation 8aA (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>72</td>
    </tr>
    <tr>
      <td>2.63 Cartographies and representation of the parameters of conformation V1</td>
      <td>73</td>
    </tr>
    <tr>
      <td>2.64 Set of values of parameters of conformation V1 (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>73</td>
    </tr>
    <tr>
      <td>2.65 Cartographies and representation of the parameters of conformation V2</td>
      <td>74</td>
    </tr>
    <tr>
      <td>2.66 Set of values of parameters of conformation V2 (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>74</td>
    </tr>
    <tr>
      <td>2.67 Cartographies and representation of the parameters of conformation V3</td>
      <td>75</td>
    </tr>
    <tr>
      <td>2.68 Set of values of parameters of conformation V3 (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>75</td>
    </tr>
    <tr>
      <td>2.69 Cartographies and representation of the parameters of conformation V4</td>
      <td>76</td>
    </tr>
    <tr>
      <td>2.70 Set of values of parameters of conformation V4 (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>76</td>
    </tr>
    <tr>
      <td>2.71 Cartographies and representation of the parameters of conformation TS9a</td>
      <td>77</td>
    </tr>
    <tr>
      <td>2.72 Set of values of parameters of conformation TS9a (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>77</td>
    </tr>
    <tr>
      <td>2.73 Cartographies and representation of the parameters of conformation V6</td>
      <td>78</td>
    </tr>
    <tr>
      <td>2.74 Set of values of parameters of conformation V6 (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>78</td>
    </tr>
    <tr>
      <td>2.75 Cartographies and representation of the parameters of conformation 10a</td>
      <td>79</td>
    </tr>
    <tr>
      <td>2.76 Set of values of parameters of conformation 10a (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>79</td>
    </tr>
    <tr>
      <td>2.77 Cartographies and representation of the parameters of conformation TS11a</td>
      <td>80</td>
    </tr>
    <tr>
      <td>2.78 Set of values of parameters of conformation TS11a (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>80</td>
    </tr>
    <tr>
      <td>2.79 Cartographies and representation of the parameters of conformation 12a</td>
      <td>81</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>2.80</td>
      <td>Set of values of parameters of conformation 12a (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>81</td>
    </tr>
    <tr>
      <td>2.81</td>
      <td>Graph displaying the angle of pyramidalization as a function of carbon atoms</td>
      <td>82</td>
    </tr>
    <tr>
      <td>2.82</td>
      <td>Graph displaying the angle of pyramidalization as a function of carbon atoms (taking into account the configurations)</td>
      <td>82</td>
    </tr>
    <tr>
      <td>2.83</td>
      <td>Graph displaying the hybridization as a function of carbon atoms</td>
      <td>83</td>
    </tr>
    <tr>
      <td>2.84</td>
      <td>Graph showing the hybridization and the energies as a function of transition states</td>
      <td>83</td>
    </tr>
    <tr>
      <td>2.85</td>
      <td>Graph showing the angular defect as a function of carbon atoms</td>
      <td>84</td>
    </tr>
    <tr>
      <td>2.86</td>
      <td>Graph representing the spherical curvature as a function of carbon atoms</td>
      <td>84</td>
    </tr>
    <tr>
      <td>2.87</td>
      <td>Representation of the reaction path of route (i) 2.58</td>
      <td>85</td>
    </tr>
    <tr>
      <td>3.1</td>
      <td>Representation of the molecule C<sub>11</sub>H<sub>8</sub> with the POAV1 (in green) and POAV2 (in red) vectors</td>
      <td>91</td>
    </tr>
    <tr>
      <td>3.2</td>
      <td>Representation of the molecule C<sub>11</sub>H<sub>8</sub> with the POAV1 (in pink) and POAV2 (in grey) vectors and the HOMO</td>
      <td>92</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>Representation of the HOMO and the nodal surface of the ethylene molecule in 3D</td>
      <td>92</td>
    </tr>
    <tr>
      <td>3.4</td>
      <td>Representation of the linear interpolation for the ethylene molecule</td>
      <td>93</td>
    </tr>
    <tr>
      <td>3.5</td>
      <td>Representation by linear interpolation of POAV1 and POAV2 (in blue the normal, in green the POAV1 and in red the POAV2)</td>
      <td>93</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Simplified energetic diagram of the benzene</td>
      <td>101</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Schematic representation of the distribution along a molecule γ</td>
      <td>102</td>
    </tr>
    <tr>
      <td>4.3</td>
      <td>Illustration of the 2 distributions: Hückel and Möbius-types [49]</td>
      <td>103</td>
    </tr>
    <tr>
      <td>4.4</td>
      <td>Illustrative representation of the Möbius ribbons [47]</td>
      <td>107</td>
    </tr>
    <tr>
      <td>4.5</td>
      <td>Scheme representing a Hückel and Möbius type annulene [47]</td>
      <td>107</td>
    </tr>
    <tr>
      <td>4.6</td>
      <td>Representation of a polyacene, a cyclacene and a Möbius cyclacene [47]</td>
      <td>108</td>
    </tr>
    <tr>
      <td>4.7</td>
      <td>Representation of a coronene and a Möbius coronene [47]</td>
      <td>108</td>
    </tr>
    <tr>
      <td>4.8</td>
      <td>Representation of Möbius annulene [47]</td>
      <td>108</td>
    </tr>
    <tr>
      <td>4.9</td>
      <td>Distribution of the energy levels of S<sub>N</sub><sup>1</sup>(φ<sub>M</sub>) in the case of π(S<sub>N</sub><sup>1</sup>) ≡ 0 [mod.4]</td>
      <td>111</td>
    </tr>
    <tr>
      <td>4.10</td>
      <td>Distribution of the energy levels of S<sub>N</sub><sup>1</sup>(0) in the case of π(S<sub>N</sub><sup>1</sup>) ≡ 0 [mod.4]</td>
      <td>112</td>
    </tr>
    <tr>
      <td>4.11</td>
      <td>Distribution of the energy levels of S<sub>N</sub><sup>1</sup>(φ<sub>M</sub>) in the case of π(S<sub>N</sub><sup>1</sup>) ≠ 0 [mod.4]</td>
      <td>114</td>
    </tr>
    <tr>
      <td>4.12</td>
      <td>Distribution of the energy levels of S<sub>N</sub><sup>1</sup>(0) in the case of π(S<sub>N</sub><sup>1</sup>) - 3 ≡ 0 [mod.4]</td>
      <td>115</td>
    </tr>
    <tr>
      <td>4.13</td>
      <td>Evolution of the orbitals along the circle - Möbius case</td>
      <td>119</td>
    </tr>
    <tr>
      <td>4.14</td>
      <td>Position of the C<sub>2</sub> axis - transverse to the plane (on the left) and in the plane (on the right)</td>
      <td>120</td>
    </tr>
    <tr>
      <td>4.15</td>
      <td>Representations of the λ<sub>1,+</sub> and λ<sub>1,-</sub> energy solutions</td>
      <td>126</td>
    </tr>
    <tr>
      <td>4.16</td>
      <td>Representations of the energy of a Möbius-C<sub>2</sub> invariant</td>
      <td>126</td>
    </tr>
    <tr>
      <td>4.17</td>
      <td>Representation of a symmetric Möbius configuration</td>
      <td>127</td>
    </tr>
    <tr>
      <td>4.18</td>
      <td>Representation of the energy E(a, 1) for a Möbius case</td>
      <td>128</td>
    </tr>
    <tr>
      <td>4.19</td>
      <td>Representation of the energy E(a, b) for a Möbius case</td>
      <td>129</td>
    </tr>
    <tr>
      <td>4.20</td>
      <td>Orbital distributions for N = 4 and k = 1, 2, 3, 4</td>
      <td>134</td>
    </tr>
    <tr>
      <td>4.21</td>
      <td>Representation of asymptotic angles</td>
      <td>135</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Comparison between the behavior of orbitals in Möbius ribbons and allenes (here a cyclobutadiene and a [2]-cumulene) [9]</td>
      <td>138</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Representation of a [n]-cumulene</td>
      <td>138</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>Representation of the hybrid systems at the origin of the formation of orbitals in helix (lower part : p<sub>x</sub> and p<sub>y</sub>-systems are represented schematically, the green and red lobes indicate the sign - higher part: the representation of the orbitals by DFT computations)</td>
      <td>139</td>
    </tr>
    <tr>
      <td>5.4</td>
      <td>Best fitted helix for distribution of equivalent [4]-cumulene version of Möbius system and n=1</td>
      <td>144</td>
    </tr>
    <tr>
      <td>5.5</td>
      <td>Best fitted helix for distribution of equivalent [4]-cumulene version of Möbius system and n=2</td>
      <td>144</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>5.6</td>
      <td>Schematic illustration of a molecule to define a Hamiltonian square matrix in the Löwdin partitioning technique [57] (the graph consists of two branch graphs a and b in cyan and a center graph c in black - the dashed edges in red represents the connections between them, with A and B being the branch-root nodes).</td>
      <td>149</td>
    </tr>
    <tr>
      <td>5.7</td>
      <td>Schematic representation of a linear chain depending on the z axis</td>
      <td>151</td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Representation of helical MOs of a [3]-cumulene at different states of rotation (0, 50°, 90° carried out by DFT computations) in the ground state</td>
      <td>156</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>Molecular orbitals and energies of a [3]-cumulene associated for rotations from 0° to 90° (carried out by DFT computations) in the ground state</td>
      <td>157</td>
    </tr>
    <tr>
      <td>6.3</td>
      <td>Representation of helical MOs of a [4]-cumulene at different states of rotation (0, 50°, 90° carried out by DFT computations) in the ground state</td>
      <td>158</td>
    </tr>
    <tr>
      <td>6.4</td>
      <td>Molecular orbitals and energies associated for rotations from 0° to 90° (carried out by DFT computations) in the ground state</td>
      <td>158</td>
    </tr>
    <tr>
      <td>6.5</td>
      <td>Helical molecular orbitals for a rotation of 10° (carried out by DFT computations) in the ground state</td>
      <td>159</td>
    </tr>
    <tr>
      <td>6.6</td>
      <td>Representation of helical MOs of a [3]-cumulene at different states of rotation (0, 50°, 90° carried out by DFT computations) in a triplet state.</td>
      <td>160</td>
    </tr>
    <tr>
      <td>6.7</td>
      <td>Representation of helical MOs of a [4]-cumulene at different states of rotation (0, 50°, 90° carried out by DFT computations) in a triplet state.</td>
      <td>161</td>
    </tr>
    <tr>
      <td>6.8</td>
      <td>Representation of helical MOs of [2]-hetero-cumulenes at 25° of torsion by DFT computations in the ground state</td>
      <td>163</td>
    </tr>
    <tr>
      <td>6.9</td>
      <td>Representation of helical MOs of [3]-hetero-cumulenes at 25° of torsion by DFT computations in the ground state</td>
      <td>164</td>
    </tr>
    <tr>
      <td>6.10</td>
      <td>Representation of the molecular orbitals of the B=N=B=N=B-cumulene (by DFT computations) for a rotation of 30° (charge 0 - doublet and quadruplet states)</td>
      <td>164</td>
    </tr>
    <tr>
      <td>6.11</td>
      <td>Representation of the molecular orbitals of the N=B=N=B=N-cumulene (by DFT computations) for a rotation of 30° (charge 1 - singlet and triplet states)</td>
      <td>165</td>
    </tr>
    <tr>
      <td>6.12</td>
      <td>Representation of the molecular orbitals of the B=N=B=N=B=N-cumulene (by DFT computations) for a rotation of 30° (charge 0, 1 and 2 - singlet, doublet and triplet states)</td>
      <td>165</td>
    </tr>
    <tr>
      <td>6.13</td>
      <td>Representation of the molecular orbitals of the DPBD molecule (by DFT computations)</td>
      <td>166</td>
    </tr>
    <tr>
      <td>6.14</td>
      <td>Representation of the molecular orbitals LUMO+3 of the tolanophane molecule (by DFT computations)</td>
      <td>167</td>
    </tr>
    <tr>
      <td>6.15</td>
      <td>Representation of the molecular orbitals HOMO and HOMO-5 of the rhenium complex [Re<sub>2</sub>H<sub>8</sub>]<sup>2-</sup> (by DFT computations)</td>
      <td>168</td>
    </tr>
    <tr>
      <td>a1.16</td>
      <td>Representation of the [3]-cumulene in the fundamental state at 0°.</td>
      <td>174</td>
    </tr>
    <tr>
      <td>a1.17</td>
      <td>Representation of the [4]-cumulene in the fundamental state at 0°.</td>
      <td>174</td>
    </tr>
    <tr>
      <td>a1.18</td>
      <td>Representation of the [11]-cumulene in the fundamental state at 0°.</td>
      <td>174</td>
    </tr>
    <tr>
      <td>a1.19</td>
      <td>Representation of the phosphaallene in the fundamental state at 25°.</td>
      <td>174</td>
    </tr>
    <tr>
      <td>a1.20</td>
      <td>Representation of the diphosphoaallene in the fundamental state at 25°.</td>
      <td>174</td>
    </tr>
    <tr>
      <td>a1.21</td>
      <td>Representation of the phosphaazaallene in the fundamental state at 25°.</td>
      <td>174</td>
    </tr>
    <tr>
      <td>a1.22</td>
      <td>Representation of the arsaallene in the fundamental state at 25°.</td>
      <td>175</td>
    </tr>
    <tr>
      <td>a1.23</td>
      <td>Representation of the arsaphosphaallene in the fundamental state at 25°.</td>
      <td>175</td>
    </tr>
    <tr>
      <td>a1.24</td>
      <td>Representation of the diarsaallene in the fundamental state at 25°.</td>
      <td>175</td>
    </tr>
    <tr>
      <td>a1.25</td>
      <td>Representation of the silaallene in the fundamental state at 25°.</td>
      <td>175</td>
    </tr>
    <tr>
      <td>a1.26</td>
      <td>Representation of the phosphasilaallene in the fundamental state at 25°.</td>
      <td>175</td>
    </tr>
    <tr>
      <td>a1.27</td>
      <td>Representation of the silaketene in the fundamental state at 25°.</td>
      <td>175</td>
    </tr>
    <tr>
      <td>a1.28</td>
      <td>Representation of the germaallene in the fundamental state at 25°.</td>
      <td>175</td>
    </tr>
    <tr>
      <td>a1.29</td>
      <td>Representation of the germaphosphaaallene in the fundamental state at 25°.</td>
      <td>176</td>
    </tr>
    <tr>
      <td>a1.30</td>
      <td>Representation of the phosphabutatriene in the fundamental state at 25°.</td>
      <td>176</td>
    </tr>
    <tr>
      <td>a1.31</td>
      <td>Representation of the diphosphabutatriene in the fundamental state at 25°.</td>
      <td>176</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>a1.32Representation of the arsabutatriene in the fundamental state at 25°.</td>
      <td>176</td>
    </tr>
    <tr>
      <td>a1.33Representation of the silabutatriene in the fundamental state at 25°.</td>
      <td>176</td>
    </tr>
    <tr>
      <td>a1.34Representation at 30° of rotation.</td>
      <td>176</td>
    </tr>
    <tr>
      <td>a1.35Representation at 30° of rotation.</td>
      <td>177</td>
    </tr>
    <tr>
      <td>a1.36Representation at 30° of rotation.</td>
      <td>177</td>
    </tr>
    <tr>
      <td>a1.37Representation at 30° of rotation.</td>
      <td>177</td>
    </tr>
    <tr>
      <td>a1.38Representation at 30° of rotation.</td>
      <td>177</td>
    </tr>
    <tr>
      <td>a1.39Representation at 30° of rotation.</td>
      <td>177</td>
    </tr>
    <tr>
      <td>a1.40Representation at 30° of rotation.</td>
      <td>177</td>
    </tr>
    <tr>
      <td>a1.41Representation of the rhenium complex.</td>
      <td>178</td>
    </tr>
  </tbody>
</table>
List of Tables

<table>
  <tbody>
    <tr>
      <td>2.1</td>
      <td>Average descriptor parameters for the large fullerene family</td>
      <td>45</td>
    </tr>
    <tr>
      <td>2.2</td>
      <td>Average descriptor parameters for a selection of the medium fullerene family</td>
      <td>45</td>
    </tr>
    <tr>
      <td>2.3</td>
      <td>Average descriptor parameters for a selection of the small fullerene family</td>
      <td>46</td>
    </tr>
    <tr>
      <td>2.4</td>
      <td>Set of minimum and maximum values of parameters of the 1,6-methano-[10]-annulene (C<sub>11</sub>H<sub>10</sub>, Pyr, Hyb, AngDef, SphCurv)</td>
      <td>49</td>
    </tr>
    <tr>
      <td>2.5</td>
      <td>Set of minimum and maximum values of parameters of the 1,5-methano-[10]-annulene (C<sub>11</sub>H<sub>10</sub>, Pyr, Hyb, AngDef, SphCurv)</td>
      <td>50</td>
    </tr>
    <tr>
      <td>2.6</td>
      <td>Set of minimum and maximum values of parameters of the molecule 7bH-cyclopenta-[cd]-indene (C<sub>11</sub>H<sub>8</sub>, Pyr, Hyb, AngDef, SphCurv)</td>
      <td>50</td>
    </tr>
    <tr>
      <td>2.7</td>
      <td>Set of minimum and maximum values of parameters of the molecule syn-1,6,8,13-bismethano-[14]-annulene (C<sub>16</sub>H<sub>14</sub>, Pyr, Hyb, AngDef, SphCurv)</td>
      <td>51</td>
    </tr>
    <tr>
      <td>2.8</td>
      <td>Set of minimum and maximum values of parameters of the molecule trans-15,16-diethyldihydropyrene (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>52</td>
    </tr>
    <tr>
      <td>2.9</td>
      <td>Set of minimum and maximum values of parameters of the bicyclo-[2.2.1]-hept-2-ene (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>53</td>
    </tr>
    <tr>
      <td>2.10</td>
      <td>Set of minimum and maximum values of parameters of the molecule C<sub>12</sub>H<sub>16</sub> (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>54</td>
    </tr>
    <tr>
      <td>2.11</td>
      <td>Set of minimum and maximum values of parameters of 9,9',10,10'-tetrahydrodianthracene (Pyra, Hyb, AngDef, SphCurv)</td>
      <td>54</td>
    </tr>
    <tr>
      <td>2.12</td>
      <td>Set of minimum and maximum values of parameters of sesquinorbornatriene (bridged down) (Pyr, Hybr, AngDef, SphCurv)</td>
      <td>55</td>
    </tr>
    <tr>
      <td>2.13</td>
      <td>Set of minimum and maximum values of parameters of the sesquinorbornatriene (bridged to the top) (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>56</td>
    </tr>
    <tr>
      <td>2.14</td>
      <td>Set of minimum and maximum values of parameters of the Trans-cyclooctene (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>57</td>
    </tr>
    <tr>
      <td>2.15</td>
      <td>Set of minimum and maximum values of parameters of the molecule C<sub>24</sub>H<sub>24</sub> (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>57</td>
    </tr>
    <tr>
      <td>2.16</td>
      <td>Set of minimum and maximum values of parameters of corannulene (Pyr, Hy, AngDef, SphCurv)</td>
      <td>58</td>
    </tr>
    <tr>
      <td>2.17</td>
      <td>Set of minimum and maximum values of parameters of the molecule fullerene C<sub>50</sub>-D<sub>3</sub> (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>59</td>
    </tr>
    <tr>
      <td>2.18</td>
      <td>Set of minimum and maximum values of parameters of the molecule fullerene C<sub>50</sub>-D<sub>5h</sub> (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>60</td>
    </tr>
    <tr>
      <td>2.19</td>
      <td>Set of minimum and maximum values of parameters of the fullerenic molecule C<sub>50</sub>Cl<sub>10</sub>-D<sub>3</sub> (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>61</td>
    </tr>
    <tr>
      <td>2.20</td>
      <td>Set of minimum and maximum values of parameters of the fullerenic molecule C<sub>50</sub>Cl<sub>10</sub>-D<sub>5h</sub> (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>62</td>
    </tr>
    <tr>
      <td>2.21</td>
      <td>Set of minimum and maximum values of parameters of the molecule C<sub>64</sub>-C<sub>3v</sub> (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>62</td>
    </tr>
    <tr>
      <td>2.22</td>
      <td>Set of minimum and maximum values of parameters of the molecule C<sub>64</sub>-C<sub>2</sub> (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>63</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>2.23</td>
      <td>Set of minimum and maximum values of parameters of the molecule C<sub>64</sub>-D<sub>2</sub> (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>64</td>
    </tr>
    <tr>
      <td>2.24</td>
      <td>Set of minimum and maximum values of parameters of the molecule C<sub>64</sub>-C<sub>s</sub> (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>65</td>
    </tr>
    <tr>
      <td>2.25</td>
      <td>Set of minimum and maximum values of parameters of the molecule C<sub>64</sub>Cl<sub>4</sub>-C<sub>3v</sub> (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>66</td>
    </tr>
    <tr>
      <td>2.26</td>
      <td>Set of minimum and maximum values of parameters of the molecule C<sub>64</sub>Cl<sub>4</sub>-C<sub>s</sub> (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>67</td>
    </tr>
    <tr>
      <td>2.27</td>
      <td>Set of minimum and maximum values of parameters of the molecule C<sub>64</sub>Cl<sub>4</sub>-D<sub>2</sub> (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>67</td>
    </tr>
    <tr>
      <td>2.28</td>
      <td>Set of minimum and maximum values of parameters of the molecule C<sub>64</sub>Cl<sub>4</sub>-C<sub>2</sub> (Pyr, Hyb, AngDef, SphCurv)</td>
      <td>68</td>
    </tr>
    <tr>
      <td>2.29</td>
      <td>Set of environment and energy of conformation 8aA.</td>
      <td>72</td>
    </tr>
    <tr>
      <td>2.30</td>
      <td>Set of environment and energy of conformation V1</td>
      <td>73</td>
    </tr>
    <tr>
      <td>2.31</td>
      <td>Set of environment and energy of conformation V2</td>
      <td>74</td>
    </tr>
    <tr>
      <td>2.32</td>
      <td>Set of environment and energy of conformation V3</td>
      <td>75</td>
    </tr>
    <tr>
      <td>2.33</td>
      <td>Set of environment and energy of conformation V4</td>
      <td>76</td>
    </tr>
    <tr>
      <td>2.34</td>
      <td>Set of environment and energy of conformation TS9a</td>
      <td>77</td>
    </tr>
    <tr>
      <td>2.35</td>
      <td>Set of environment and energy of conformation V6</td>
      <td>78</td>
    </tr>
    <tr>
      <td>2.36</td>
      <td>Set of environment and energy of conformation 10a</td>
      <td>79</td>
    </tr>
    <tr>
      <td>2.37</td>
      <td>Set of environment and energy of conformation TS11a</td>
      <td>80</td>
    </tr>
    <tr>
      <td>2.38</td>
      <td>Set of environment and energy of conformation 12a</td>
      <td>81</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Overview of the angles (in degrees) of the 4 examples</td>
      <td>134</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Examples of angles values (in degrees) along a molecular chain</td>
      <td>135</td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Set of studied [n]-hetero-cumulenes</td>
      <td>162</td>
    </tr>
  </tbody>
</table>
Nomenclature

$L_N(\phi)$ Arrangement of atoms in line with a $\phi$ distribution

$S_N^1(\phi)$ Arrangement of atoms in circle with a $\phi$ distribution

$x \equiv y \pmod{4}$ $x$ is congruent to $y$ - modulo 4

Ang Def: Angular Defect

AO: Atomic Orbital

CAS: Complete Active Space

CASPT2: Complete Active Space Perturbation Theory

CASSCF: Complete Active Space Self-Consistent Field

CI: Configuration Interaction

COF: Covalent Organic Framework

DFT: Density Functional Theory

DPBD: Diphenylbutadiyne

FVP: Flash Vacuum Pyrolysis

GGA: Generalized Gradient Approximation

GTO: Gaussian Type Orbitals

HF: Hartree-Fock

HOMO: Highest Occupied Molecular Orbital

HSE: Heyd Scuseria Ernzerhof

Hyb: Hybridization

IEPA: Independent Electron Pair Approximation

KS: Kohn-Sham

LCAO: Linear Combination of Atomic Orbitals

LDA: Local Density Approximation

LUMO: Lowest Unoccupied Molecular Orbital

MAD: Mean Absolute Deviation

MCSF: Multi Configurational Self Consistent field
MO: Molecular Orbital

MOF: Metal-Organic Framework

NBO: Natural Bond Orbital

PBE: Perdew Burke Ernzerhof

PES: Potential Energy Surface

POAV: π-Orbital Axis Vector

Pyr: Pyramidalization angle

RHF: Restricted Hartree-Fock

Sph Curv: Spherical Curvature

STO: Slater Type Orbitals
Index

$C_2$ invariance, 119, 123

$C_{3v}$ symmetry, 32, 185

$L_N(\phi)$ molecules, 123

$Reg \star (A)$, 29

$S_N^1(\phi)$, 104

$W_\sigma$, 90

[11]-cumulene, 159

[2]-hetero-cumulenes, 162

[3]-cumulene, 156, 160

[3]-hetero-cumulenes, 163

[4]-cumulene, 157

$[B = N]$ or $[N = B]$-cumulenes, 164

$\tilde{n}$, 35

$\kappa(A)$, 38

$\phi$ distribution, 123

$\pi$-orbital, 32, 35, 88, 89

$\pi$-system, 129, 139

$\sigma$-bonds, 31

$\sigma$-orbital, 35, 88, 89

$\star(A)$, 27, 48

$sp$-hybridized, 138

$sp^2$-hybridized, 138

$u_\pi$ vector, 88

[n]-cumulene, 155, 156

[n]-hetero-cumulenes, 162

Admissible molecule, 35, 36, 182

Allene, 102

Angle distribution, 99

Angular defect, 26, 40, 45, 47, 182

Annulene, 101, 102

Anti-aromatic, 101

Aromaticity, 10, 101

Aromaticity criteria, 99, 101

Atomic orbital, 15, 30

B3LYP, 18, 21

Basis set, 19

Born-Oppenheimer, 19

Cartography, 47

CAS, 22

Characteristic polynomial, 152

Chiral, 140, 153

Chlorofullerene, 48

Circular distribution, 145

Complete Active Space Self Consistent Field, 22

Computational techniques, 173

Configuration interaction, 22

Correlation factor, 143

Covalent Organic Frameworks, 94

Criteria of existence, 99

Cumulated angle, 142

Cumulene, 99, 138, 139

Curie's principle, 153

Curvature, 26, 37, 38

Cyclic molecule, 102

Database of Tománek and Frederick, 44

Degeneracies, 111, 187

DFT, 10, 18–20

Distribution, 102

Distribution of a molecule, 18

Distribution of angles, 119, 141

DPBD, 155, 165, 166

Effective Hamiltonian, 150

Eigenfunction, 16, 17

Eigenvalue, 16, 17, 111, 125–128

Eigenvector, 16

Electrohelicity, 140

Electronic saturation, 110, 111, 114

Electronic structure, 109

Electrostatic energy, 20

Energy, 126, 128

Energy equations, 99

Energy proof, 110

Equivalent representation, 104

Ethylene, 152

Even case, 110, 140, 187

Exchange-correlation functional, 21

Excited state, 160
Extended Hückel method, 13

Family of angles, 132

Fullerene, 26, 45–48

Gaussian, 18

Gaussian curvature, 40

Generalized Gradient Approximation, 21

Global weight, 33

Hückel annulenes, 107

Hückel distribution, 144

Hückel matrices, 105

Hückel matrix, 13, 105, 106, 125, 127, 128, 151

Hückel model, 99

Hückel rules, 101

Hückel systems, 101

Hamiltonian, 15, 16

Hamiltonian matrix, 150

Hartree-Fock, 19, 22

HEL, 143

Helical distribution, 142

Helical frontier orbital, 139

Helical molecular orbital, 99

Helical orbital, 10, 155

Helical orbital criterion, 157

Helical state, 99, 149

Helicogenic axis, 152

Hetero-cumulene, 155

HOMO, 92

Homotopy, 102

HSE, 21

Hybrid orbitals, 30, 31

Hybridization, 26, 30, 32–34, 45, 47, 181

Hybridization coefficients, 32, 45, 181

Hybridization numbers, 34, 45, 89, 186

Infinite chain, 134

Isosurface, 92

Kinetic energy, 20

Löwdin partitioning, 149

LCAO method, 19

Linear allene system, 137

Linear chain, 102

Linear interpolation, 92

Linear system, 89, 138

Local curvature, 43

Local density approximation, 21

Möbius, 10

Möbius annulenes, 107

Möbius case, 110

Möbius cyclic system, 137

Möbius distribution, 103, 127

Möbius energy, 111

Möbius model, 99

Möbius molecule, 106

Möbius systems, 137

MAD, 142

Mechanical constraint, 140

Metal Organic Frameworks, 94

Metal-cumulenes, 167

Molecular orbital, 15

Moller-Plesset, 19

Multi configurational self consistent field, 22

Nitrile imines, 70

Nodal surface, 92

Non-fullerenic compounds, 26

Non-fullerenic molecule, 48

Non-planar system, 30

Non-planarity, 39

Non-saturated case, 112, 113

Non-trivalent case, 29

Normalization, 15, 32, 90, 180, 186

Number of electrons available, 17

Odd case, 110, 140

Optimal distribution, 124

Orbital constraints, 121

Orthogonality, 88, 184

Orthogonality condition, 31

Osculating sphere, 38, 39

Overlap, 30

Overlap integral, 14

Parallel extremity, 140

PBE0, 21

Perpendicular extremity, 140

Perturbation theory, 19

Planar molecule, 129

Planarity, 91

POAV, 30

POAV1, 9, 26, 32, 180, 181

POAV2, 9, 25, 31, 87, 88, 91

Post-Hartree-Fock, 19

Pychemcurv, 43, 47

Pyramidalization angle, 9, 26, 30, 35, 40, 45–47, 91, 180, 182

Reactivity, 25

Regularized star, 29, 180

Relative weight, 33

Restricted Hartree-Fock, 22

Ribbon case, 110, 113
Rotation, 99, 120

Rules of Hückel, 99

Schrödinger equation, 15, 18

Secular determinant, 147

Self-consistent field, 19, 21

Shape of a helix, 141

Sigma hybridization, 35

Simple Hückel method, 13

Slater, 22

Spherical curvature, 25, 37, 38, 45, 47, 182

Symmetric Möbius configuration, 127

Symmetrical matrix, 105

Symmetry group, 151

Synthesis, 139

Taylor expansion, 41

Tolanophane, 155, 165, 166

Torsion distribution, 122

Total energy of a molecule, 17, 179

Total overlap, 16

Trigonometry reminder, 109

Triplet state, 160

Trivalent regular case, 26

Twisted molecule, 118

Twisted ribbon, 103

Variational method, 20

Wave functions, 15, 19

Y, 35

Zero distribution, 103
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<figure><img src="image_168.png" /></figure>
ECOLE DOCTORALE :
Sciences Exactes et leurs Applications de l'Université de Pau et des Pays de l'Adour

LABORATOIRES :
IPREM Institut des Sciences Analytiques et de Physico-Chimie pour l'Environnement et les Matériaux
LMAP Laboratoire de Mathématiques et de leurs Applications

ABSTRACT

This thesis is the result of 3 years of work based on the subject: "Structural and Electronic Chemistry: Role of Symmetries and Curvature".

The originality of the topic consists in the multidisciplinary nature of the subject which involves both theoretical chemists and mathematicians. An important part of the work is based on the understanding of theoretical chemistry and its use by modeling.

One of the objectives attributed to this work, is the understanding of molecular systems by geometrical tools mainly initiated by Robert C. Haddon in 1997. We first propose a clarification and a generalization of these tools based on carbon materials such as fullerenes whose study of topological characteristics is largely exploited in this manuscript.

All the theoretical points and tools developed (the pyramidalization angle, the spherical curvature, the angular defect, and the hybridization) are widely illustrated with cartographies by the use of programming tools and softwares which provide the study of the deformation, the curvature and the investigation of the chemical and physical properties of the systems.

The use of the tools resulting from this work will facilitate the simulation of molecules without size limitation and for systems whose modelisation by ab-initio or DFT calculations remains inaccessible.

We also tried to show the relation which could exist between the geometry of the systems and the orbital information.
We were interested in many systems as examples most of them are carbon-based and potentially aromatic.

Consequently, we propose the study of the criteria of aromaticity according to the theories stated among others by Hückel. By this study, we focus on less known linear systems: the [n]-cumulenes which present the curiosity of helical orbitals. We established the criteria for the existence of this type of helices according to the molecular symmetry.

The aim *in fine* was to establish the link between the topological characteristics and the chemical reactivity of these molecules and materials.

CONTACT

**Didier BEGUE**
Professeur
IPREM, Pôle CAPT, Pau
didier.begue@univ-pau.fr

**Jacky CRESSON**
Professeur
IPRA, LMAP, Pau
jacky.cresson@univ-pau.fr

**Julia SABALOT-CUZZUBBO**
Doctorante
IPREM, Pôle CAPT, Pau
sabalot.julia@univ-pau.fr