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# Structural and electronic chemistry: role of symmetry and curvature

Julia Sabalot-Cuzzubo

► To cite this version:

Julia Sabalot-Cuzzubo. Structural and electronic chemistry: role of symmetry and curvature. Theoretical and/or physical chemistry. Université de Pau et des Pays de l'Adour, 2022. English. NNT: 2022PAUU3007. tel-03936415

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https://theses.hal.science/tel-03936415

Submitted on 12 Jan 2023

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THÈSE

UNIVERSITE DE PAU ET DES PAYS DE L'ADOUR

École doctorale des Sciences Exactes et leurs Applications (ED 211 SEA)

Présentée et soutenue le 06 Avril 2022

par Julia SABALOT-CUZZUBBO

pour obtenir le grade de docteur
de l'Université de Pau et des Pays de l'Adour

Spécialité : Chimie Physique

CHIMIE STRUCTURALE ET ÉLECTRONIQUE : RÔLE DES SYMÉTRIES ET DE LA COURBURE

MEMBRES DU JURY

RAPPORTEURS

• Yannick CARISSAN

• Sergey KRANOSHCHEKOV

Maître de conférences - Université Paul Cézanne - Aix-Marseille III

Professeur - Moscow State University

EXAMINATEURS

• Karine COSTUAS

• Alfonso SAN MIGUEL FUSTER

• Corinne NARDIN

• Panagiotis KARAMANIS

Directrice de recherches - Université de Rennes 1

Professeur - Université Claude Bernard Lyon 1

Professeur - Université de Pau et des Pays de l'Adour

Chargé de recherches - Université de Pau et des Pays de l'Adour

DIRECTEURS

• Didier BEGUE

• Jacky CRESSON

Professeur - Université de Pau et des Pays de l'Adour

Professeur - Université de Pau et des Pays de l'Adour
<figure><img src="image_2.png" /></figure>
# Contents

Remerciements 7

General introduction 9

I Reminders 11

0.1 Problem of geometry 13

0.1.1 Geometrical-chemical structure of molecules 13

0.1.2 The skeleton of a molecule 13

0.2 Problem of chemistry 13

0.2.1 Hückel methods 13

0.2.2 Geometry of molecules 14

0.2.3 Interaction with the atomic orbitals 14

0.2.4 Hückel matrices 15

Schrödinger equation, wave functions and atomic orbitals 15

The Hamiltonian: Matrix **H** 15

Interaction hypotheses, geometry and **H** matrix 16

Eigenvalues and energy of molecular orbitals 17

Distribution of a molecule 18

0.3 Quantum computing methods 18

0.3.1 Description of electronic structure of molecules 18

0.3.2 Density Functional Theory (DFT) 20

0.3.3 The exchange-correlation functionals 21

0.3.4 Multi reference methods 22

II π-Orbital Axis Vectors theory 23

Introduction to the study of the Robert C. Haddon's concepts 25

1 The π-Orbital Axis Vector 1 (POAV1) theory 26

1.1 The π-Orbital Axis Vector (POAV1) of Robert C. Haddon 26

1.1.1 Pyramidalization angle and π-Orbital Axis Vector (POAV1) 26

Regular trivalent case 26

Non-regular trivalent case 29

1.2 Relationship with chemistry 30

1.2.1 Hybridization 30

Orthogonality and normalization conditions 31

Relation between hybridization and the POAV1 32

Relative weight and hybridization numbers 33

1.3 Limitations of the pyramidalization angle and the POAV1 35

Admissible molecules 36

1.4 Haddon's spherical Curvature 37
<table>
  <tbody>
    <tr>
      <td>1.4.1</td>
      <td>Spherical curvature</td>
      <td>37</td>
    </tr>
    <tr>
      <td>1.4.2</td>
      <td>Spherical curvature in non regular case</td>
      <td>38</td>
    </tr>
    <tr>
      <td>1.4.3</td>
      <td>Limitations of the spherical curvature</td>
      <td>39</td>
    </tr>
    <tr>
      <td>1.5</td>
      <td>Angular defect</td>
      <td>40</td>
    </tr>
    <tr>
      <td>1.6</td>
      <td>Conclusion</td>
      <td>41</td>
    </tr>
    <tr>
      <td><b>2</b></td>
      <td><b>Study of POAV1 over fullerenic and non-fullerenic compounds</b></td>
      <td><b>43</b></td>
    </tr>
    <tr>
      <td>2.1</td>
      <td>Methods of implementation of the concepts</td>
      <td>43</td>
    </tr>
    <tr>
      <td>2.2</td>
      <td>POAV1 analysis of the Tománek and Frederick database</td>
      <td>44</td>
    </tr>
    <tr>
      <td>2.2.1</td>
      <td>Database analysis</td>
      <td>44</td>
    </tr>
    <tr>
      <td>2.2.2</td>
      <td>Analysis following the POAV1 notions</td>
      <td>45</td>
    </tr>
    <tr>
      <td></td>
      <td>Fullerenes visualization-Cartographies</td>
      <td>47</td>
    </tr>
    <tr>
      <td>2.3</td>
      <td>Application to other compounds</td>
      <td>48</td>
    </tr>
    <tr>
      <td>2.3.1</td>
      <td>Non-fullerenic molecules</td>
      <td>48</td>
    </tr>
    <tr>
      <td>2.3.2</td>
      <td>Fullerenic molecules</td>
      <td>59</td>
    </tr>
    <tr>
      <td>2.4</td>
      <td>Application to reactivity</td>
      <td>70</td>
    </tr>
    <tr>
      <td>2.4.1</td>
      <td>Presentation of the ring expansion reaction path of a Nitrile Imine</td>
      <td>70</td>
    </tr>
    <tr>
      <td>2.4.2</td>
      <td>Results</td>
      <td>72</td>
    </tr>
    <tr>
      <td>2.4.3</td>
      <td>Conformation 8aA - Allenic Benzonitrile Imine</td>
      <td>72</td>
    </tr>
    <tr>
      <td></td>
      <td>Characterics of V1</td>
      <td>73</td>
    </tr>
    <tr>
      <td></td>
      <td>Characterics of V2</td>
      <td>74</td>
    </tr>
    <tr>
      <td></td>
      <td>Characterics of V3</td>
      <td>75</td>
    </tr>
    <tr>
      <td></td>
      <td>Characterics of V4</td>
      <td>76</td>
    </tr>
    <tr>
      <td>2.4.4</td>
      <td>Conformation 2 : transition state TS9a</td>
      <td>77</td>
    </tr>
    <tr>
      <td></td>
      <td>Characterics of V6</td>
      <td>78</td>
    </tr>
    <tr>
      <td>2.4.5</td>
      <td>Conformation 3 - 10a - bicyclo[4.1.0]hepta-2,4,7-triene</td>
      <td>79</td>
    </tr>
    <tr>
      <td></td>
      <td>Characterics of TS11a</td>
      <td>80</td>
    </tr>
    <tr>
      <td>2.4.6</td>
      <td>Conformation 4 - 12a cycloheptatetraene</td>
      <td>81</td>
    </tr>
    <tr>
      <td>2.4.7</td>
      <td>Comparative graphs</td>
      <td>82</td>
    </tr>
    <tr>
      <td>2.5</td>
      <td>Conclusion</td>
      <td>85</td>
    </tr>
    <tr>
      <td><b>3</b></td>
      <td><b>Beyond the π-Orbital Axis Vector: the POAV2 theory</b></td>
      <td><b>87</b></td>
    </tr>
    <tr>
      <td>3.1</td>
      <td>Study of POAV2 theory</td>
      <td>87</td>
    </tr>
    <tr>
      <td>3.1.1</td>
      <td>σ and π-orbitals</td>
      <td>88</td>
    </tr>
    <tr>
      <td>3.1.2</td>
      <td>Existence, uniqueness of POAV2 and orthogonality conditions</td>
      <td>88</td>
    </tr>
    <tr>
      <td>3.1.3</td>
      <td>Hybridization numbers</td>
      <td>89</td>
    </tr>
    <tr>
      <td>3.2</td>
      <td>Comparison of the POAV1 and POAV2</td>
      <td>91</td>
    </tr>
    <tr>
      <td>3.2.1</td>
      <td>Relative variation of POAV2 versus POAV1</td>
      <td>91</td>
    </tr>
    <tr>
      <td>3.2.2</td>
      <td>Quality of POAV1 versus POAV2 to predict the direction of the π-system</td>
      <td>92</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>To reach further...</td>
      <td>94</td>
    </tr>
    <tr>
      <td>3.4</td>
      <td>Conclusion</td>
      <td>95</td>
    </tr>
    <tr>
      <td colspan="2"><b>Conclusion of the part</b></td>
      <td><b>96</b></td>
    </tr>
    <tr>
      <td colspan="2"><b>III Aromaticity, helical states and applications</b></td>
      <td><b>97</b></td>
    </tr>
    <tr>
      <td colspan="2"><b>Introduction</b></td>
      <td><b>99</b></td>
    </tr>
  </tbody>
</table>
<table><tr><td>4 Criteria of aromaticity - theory</td><td>100</td></tr><tr><td>4.1 Hückel aromaticity</td><td>101</td></tr><tr><td>4.1.1 Hückel rules</td><td>101</td></tr><tr><td>4.1.2 Two visions of annulenes: Hückel and Möbius</td><td>102</td></tr><tr><td>Geometrical construction</td><td>102</td></tr><tr><td>Optimal structures</td><td>104</td></tr><tr><td>Equivalent structures</td><td>104</td></tr><tr><td>4.2 Reviews of mathematical tools with Hückel matrices</td><td>105</td></tr><tr><td>4.2.1 Hückel matrices for molecules as S<sub>N</sub><sup>1</sup>(φ), L<sub>N</sub>(φ)</td><td>105</td></tr><tr><td>Molecules S<sub>N</sub><sup>1</sup>(φ)</td><td>105</td></tr><tr><td>Molecules L<sub>N</sub>(φ)</td><td>106</td></tr><tr><td>4.3 Determination of energy equations Möbius - Hückel</td><td>106</td></tr><tr><td>4.3.1 Reviews of Möbius molecules</td><td>106</td></tr><tr><td>4.3.2 Cases of the molecules S<sub>N</sub><sup>1</sup>(0) and S<sub>N</sub><sup>1</sup>(φ<sub>M</sub>)</td><td>109</td></tr><tr><td>Trigonometry reminder</td><td>109</td></tr><tr><td>Energy proof</td><td>110</td></tr><tr><td>Energy level structure and conditions on N</td><td>110</td></tr><tr><td>4.3.3 Energetic comparison of structures S<sub>N</sub><sup>1</sup>(0) and S<sub>N</sub><sup>1</sup>(φ<sub>M</sub>) - stability</td><td>116</td></tr><tr><td>The π(S<sub>N</sub><sup>1</sup>) ≡ 0 [mod.4] case</td><td>116</td></tr><tr><td>The π(S<sub>N</sub><sup>1</sup>) ≠ 0 [mod.4] case</td><td>117</td></tr><tr><td>4.3.4 Molecule S<sub>N</sub><sup>1</sup> twisted by a distribution and a C<sub>2</sub> invariance</td><td>118</td></tr><tr><td>Geometry of a twisted molecule S<sub>N</sub><sup>1</sup></td><td>118</td></tr><tr><td>C<sub>2</sub> invariance of a twisted molecule</td><td>119</td></tr><tr><td>4.3.5 Molecules L<sub>N</sub>(0) and L<sub>N</sub>(φ)</td><td>123</td></tr><tr><td>Geometry of the molecules L<sub>N</sub>(φ)</td><td>123</td></tr><tr><td>Twisted C<sub>2</sub> invariant L<sub>N</sub>(φ) molecules</td><td>123</td></tr><tr><td>4.3.6 Optimal distribution of S<sub>N</sub><sup>1</sup>(φ) molecules</td><td>124</td></tr><tr><td>Optimal distributions C<sub>2</sub> invariant</td><td>124</td></tr><tr><td>Optimal Möbius and symmetric Möbius distributions</td><td>126</td></tr><tr><td>4.3.7 Distribution of an L<sub>N</sub>(φ) molecule equivalent to a S<sub>N</sub><sup>1</sup>(φ<sub>M</sub>) molecule</td><td>129</td></tr><tr><td>Notion of equivalent molecule</td><td>129</td></tr><tr><td>4.3.8 Equivalence between S<sub>N</sub><sup>1</sup>(φ<sub>M</sub>) and L<sub>N</sub>(φ)</td><td>130</td></tr><tr><td>Properties of the coefficients a<sub>n</sub>:</td><td>130</td></tr><tr><td>Torsion angle and properties:</td><td>131</td></tr><tr><td>Some examples with N = 4 and k = 1, 2, 3, 4</td><td>133</td></tr><tr><td>4.3.9 Asymptotic distribution of angles along a chain</td><td>134</td></tr><tr><td>Orbital cases of the form (b<sub>n</sub>, a<sub>n</sub>)</td><td>134</td></tr><tr><td>Case of orbitals of the form (b<sub>n</sub>, -a<sub>n</sub>)</td><td>135</td></tr><tr><td>4.4 Conclusion</td><td>135</td></tr></table>

<table><tr><td>5 Beyond aromaticity criteria - Helical states</td><td>137</td></tr><tr><td>5.1 Möbius systems according to the study of Roald Hoffmann</td><td>137</td></tr><tr><td>5.1.1 [n]-cumulenes: linear case</td><td>138</td></tr><tr><td>5.1.2 Helical molecular orbitals</td><td>139</td></tr><tr><td>Helical molecular orbitals in even and odd cases</td><td>140</td></tr><tr><td>5.2 Helical states</td><td>141</td></tr><tr><td>5.2.1 Properties of helical states</td><td>141</td></tr><tr><td>General case</td><td>141</td></tr><tr><td>Distributions of angles</td><td>141</td></tr><tr><td>Distribution of angles for twisted [n]-cumulenes</td><td>143</td></tr><tr><td>5.3 Criteria for the existence of helices</td><td>149</td></tr><tr><td>5.3.1 By a physical approach: Löwdin partitioning technique</td><td>149</td></tr></table>
<table><tr><td>5.3.2 By chemistry: using molecular symmetries</td><td>151</td></tr><tr><td>5.3.3 In the non-planar linear molecules</td><td>153</td></tr><tr><td>5.4 Conclusion</td><td>154</td></tr></table>

# 6 Applications and visualizations 155

## 6.1 Results on [n]-cumulenes 156

### 6.1.1 Fundamental states 156

[3]-cumulene 156

[4]-cumulene 157

[11]-cumulene 159

### 6.1.2 Excited states 160

[3]-cumulene 160

[4]-cumulene 161

## 6.2 Molecules with hetero-elements and more examples 162

### 6.2.1 [n]-hetero-cumulenes 162

[2]-hetero-cumulenes 162

[3]-hetero-cumulenes 163

### 6.2.2 [B = N] or [N = B]-cumulenes 164

### 6.2.3 Other examples 165

DPBD molecule 166

Tolanophane molecule 166

Molecules with metals 167

## 6.3 Conclusion 168

# Conclusion 169

# IV General conclusion 170

# V Appendices 172

## Appendix 1: Chemistry part 173

### 1-Computational techniques 173

### 2-Representation of the molecules used for DFT computations in 6 173

[n]-cumulenes 174

[3]-cumulene 174

[4]-cumulene 174

[11]-cumulene 174

[2]-heterocumulenes 174

[3]-heterocumulenes 176

[B = N] or [N = B]-cumulenes 176

DPBD molecule 177

Tolanophane molecule 177

Rhenium complex [$Re_2H_8]^{2-}$ 178

## Appendix 2: Technical and proofs part of the results used 179

### 1-Proofs of characteristic polynomial of H 179

### 2-Proofs of total energy of a molecule 179

### 3-Proofs of POAV$_{\epsilon}$(A) 180

### 4-Proofs of condition of normalization 180

### 5-Proofs of hybridization coefficients and POAV1 181

### 6-Proofs of sp$^3$ hybridization conditions 181

### 7-Proofs of admissible molecules 182
<table>
  <tbody>
    <tr>
      <td>8-Proofs of non-linear relationship between spherical curvature and pyramidalization angle</td>
      <td>182</td>
    </tr>
    <tr>
      <td>9-Proofs of spherical curvature in non-regular case</td>
      <td>182</td>
    </tr>
    <tr>
      <td>10-Proofs of relationship angular defect and pyramidalization angle</td>
      <td>182</td>
    </tr>
    <tr>
      <td>11-Proofs of conditions of orthogonality</td>
      <td>184</td>
    </tr>
    <tr>
      <td>12-Proofs of relation between the angles θ<sub>i,j</sub></td>
      <td>184</td>
    </tr>
    <tr>
      <td>13-Proofs of C<sub>3v</sub> symmetry</td>
      <td>185</td>
    </tr>
    <tr>
      <td>14-Proofs of components of the u<sub>π</sub> vector solutions of linear system</td>
      <td>185</td>
    </tr>
    <tr>
      <td>15-Proofs of linear system M.u<sub>π</sub> = 0</td>
      <td>185</td>
    </tr>
    <tr>
      <td>16-Proofs of hybridization numbers n<sub>1</sub>, n<sub>2</sub>, n<sub>3</sub></td>
      <td>186</td>
    </tr>
    <tr>
      <td>17-Proofs of sp<sup>3</sup> normalization.</td>
      <td>186</td>
    </tr>
    <tr>
      <td>18-Proofs of study of the degeneracies of energy levels in even case π(S<sub>N</sub><sup>1</sup>) ≡ 0 [mod.4] for the Möbius case.</td>
      <td>187</td>
    </tr>
    <tr>
      <td>19-Proofs of study of the degeneracies of energy levels in even case π(S<sub>N</sub><sup>1</sup>) ≡ 0 [mod.4] for the ribbon case</td>
      <td>187</td>
    </tr>
    <tr>
      <td>20-Proofs of structure of molecules S<sub>N</sub><sup>1</sup> C<sub>2</sub> invariant, property of torsion distribution</td>
      <td>187</td>
    </tr>
    <tr>
      <td>21-Proofs of torsion angle and properties - φ<sub>1,n</sub> relation.</td>
      <td>188</td>
    </tr>
    <tr>
      <td>22-Proofs of torsion angle and properties - φ<sub>l,l+1</sub></td>
      <td>188</td>
    </tr>
    <tr>
      <td>23-Proofs of orbital cases of the form (b<sub>n</sub>, a<sub>n</sub>): φ<sub>∞,n</sub></td>
      <td>189</td>
    </tr>
    <tr>
      <td>24-Proofs of A<sub>N,n,+,,z</sub> angle between ψ<sub>+,n</sub>(0) and ψ<sub>+,n</sub>(z)</td>
      <td>189</td>
    </tr>
    <tr>
      <td>25-Proofs of Hückel distribution coefficients in the case θ = 0</td>
      <td>190</td>
    </tr>
    <tr>
      <td><b>VI Abstract</b></td>
      <td><b>191</b></td>
    </tr>
    <tr>
      <td><b>List of figures</b></td>
      <td><b>193</b></td>
    </tr>
    <tr>
      <td><b>List of tables</b></td>
      <td><b>198</b></td>
    </tr>
    <tr>
      <td><b>Nomenclature</b></td>
      <td><b>200</b></td>
    </tr>
    <tr>
      <td><b>Index</b></td>
      <td><b>202</b></td>
    </tr>
    <tr>
      <td><b>Bibliography</b></td>
      <td><b>205</b></td>
    </tr>
  </tbody>
</table>
<figure><img src="image_3.png" /></figure>
# Remerciements

Je remercie l'ensemble des membres du jury et rapporteurs pour avoir bien voulu évaluer ce travail de thèse.

Je souhaite tout d'abord remercier mes encadrants de thèse, Pr Didier Bégué et Pr Jacky Cresson, pour m'avoir donné ma chance (surtout une seconde chance) et de m'avoir fait confiance tout au long de ces trois années. Je vous remercie pour nos longues réunions (plus ou moins préparées), nos discussions formelles ou informelles et nos échanges de mails (à toutes heures du jour et de la nuit!) qui nous ont permis d'avancer dans nos travaux. Je vous remercie d'avoir investi votre temps et d'avoir toujours été enthousiastes pour mes travaux de thèse.

Je remercie énormément aussi Dr Germain Salvato-Vallverdu, pour toute son aide, son investissement très précieux pour notre projet et sa patience (notamment en programmation!).

Merci également aux Dr William Lafrague-Dit-Hauret et Pr Michel Rerat pour leur réflexion et aide pour notre publication sur les états hélicoïdaux.

Je remercie également toute l'équipe de chimie théorique de l'IPREM pour son aide tout au long de ces trois années. Aussi, je remercie Dr Delphine Bessières et Pr Corinne Nardin pour avoir gentiment suivi mon travail durant tous mes comités de thèse. Merci Corinne pour tous vos conseils et votre écoute dans ma période de doutes pendant mon "faux départ".

Merci Jacky pour ta patience et ta pédagogie quand j'avais besoin d'explications sur certaines démonstrations mathématiques pendant ces trois années.

Merci Didier pour ta patience au début de la thèse quand le lancement de calculs n'était pas évident pour moi et pour nos discussions quotidiennes autour de sujets variés d'actualités ou bien que ce soit sur nos goûts musicaux, les ennuis de santé (le kiné quotidien!) ou le Covid!

Pour terminer, je tiens à remercier mes parents, sans qui, je ne serai certainement pas arrivée là. Merci d'avoir été là pour moi, d'avoir cru en moi durant toutes ces années d'études, merci pour votre soutien, votre présence et votre amour sans faille malgré les difficultés. Nona, je regrette ton absence pour la fin de ma thèse, tu aurais été fière *di tua piccola*.

Enfin, merci à toi aussi Benjamin pour ton soutien quotidien, ton écoute, ton optimisme et pour avoir su comprendre mes doutes pendant ce moment de ma vie.
<figure><img src="image_4.png" /></figure>
# General introduction

The research theme of this manuscript is part of a dynamic context of research of **new materials**. The current research needs in new materials can be explained by a growing desire to reduce the use of fossil resources to advance in terms of **energy transition**. Our research projects are based on an interdisciplinary collaboration involving **chemistry** and **mathematics**, and are particularly interested in fullerenes molecules, mainly carbon compounds with high potential, whose discovery has generated great interest. In order to exploit the complete potential of these materials, it is essential to understand the relationships between the structural arrangements and the physico-chemical behaviors, in other words their properties and reactivity.

Several challenges emerged when we chose to focus on these topics and we are at the very beginning of the theme project. It fundamental to set up a systematic and rigorous methodology to classify the different structural arrangements and the resulting properties. From classification and by exploiting the mathematical results, in the mid-term objective of the project it would be possible to isolate 3D structures with interesting electrical and mechanical properties. The long term objective is to develop a database that will allow us to propose materials of interest for the improvement of batteries [1] or the optimization of seawater purification membranes [2], to mention only two examples.

This manuscript represents a theoretical overview of **three years of work** and **initiates** investigations on these topics in our research team.

A first approach to these issues, prior to this thesis topic was to research and understand the possible deformations of graphene. Actually, it is known that deformations in the form of waves can modify the intrinsic molecule properties. The first simulations performed in 2018 by the CAPT research group of IPREM were able to show that deformations involving curvature influence the reactivity or conductivity of the material.

After a complete **reminder** in part I, the **second** major part of the work was focused on the study of **concepts** developed by **Robert C. Haddon** [3], a pioneer in the research of the relationship between **curvature** and **reactivity** of materials. He presents the question in both **geometrical and chemical terms**, and shows that the way the object is curved seems to play a major role in its reactivity, in particular on $C_{60}$ fullerene. He proposes the construction of a **pyramidalization angle** associated with the curvature. However, his concept shows **limitations**, the angle cannot be defined in all possible situations. Based on these limitations, we have implemented an answer as generalist as possible, redefining most of R. C. Haddon's concepts and bringing a clarification to the **POAV1** ($\pi$-Orbital Axis Vector 1) theory. This preliminary work resulted in the publication of our first article [4] on this topic. This first article is illustrated by concrete examples from D. Tománek and Frederick's database [5]. Following this initial work, the study of other concepts developed by R. C. Haddon [6] became obvious. The notion of **POAV2** ($\pi$-Orbital Axis Vector 2) is a more generalized continuity of POAV1 and shows that the notions established by R. C. Haddon are not clear and require further and improved definitions. To allow the solving of some constraints, new and more precise definitions related to the reactivity of fullerenes are presented in this manuscript.
We can resume our first works such as the geometry of a system is not only a problem inherent to the geometry of the object. There is in fact a **coupling** between the **geometry and the orbital information**. It is then, possible to associate a discrete object with an object that can be called continuous "geometrical-chemical".

A **third** major part of this manuscript concerns the study of **aromaticity criteria** according to the theories stated by the researcher Erich Hückel. The properties of some carbon materials such as graphene are based on π-conjugation, aromaticity and therefore on the sp² hybridization of the constituent carbons. However, other visions in terms of aromaticity exist for cyclic molecules with sp² hybridized carbons. The study of these different notions, developed in the 1960's by Edgar Heilbronner [7] and Howard Zimmerman [8], are proved to be of fundamental importance in our developments. Based on this work, which is well accepted in the theoretical chemistry community, we have re-established the general **energy equations** of the so-called Möbius systems as a function of the number of electrons π. The logical continuation of the work was the study of π-orbitals of **Möbius** type systems. Recent works of the research team of the famous Nobel prize laureate Roald Hoffmann [9], have particularly interested us. In effect, this work shows a relation between the orbitals of cyclic molecules called "Möbius" and the orbitals of molecules called [n]-cumulenes. Our work presented here, provides an explanation and a mathematical justification of the existence of this curiosity discovered by R. Hoffmann et al. [9] which is called: the **helical orbitals**.

Mathematical developments were complemented by **quantum calculations** performed by **DFT** (Density Functional Theory) and by multi-configuration calculations of the CASPT2 (Theory of Perturbation of the Complete Active Space) type aiming at clarifying and confirming the DFT results (in particular those previously carried out by the team of R. Hoffmann [9]).

From a more **applicative** point of view, our work presents numerous and varied possibilities. First in part II, we have performed precise cartographies of the properties of a large family of **fullerenes**, and **non-fullerenic** molecules, using the notions of pyramidalization angle, curvature, hybridization, and angular defect. It should be noted that the implementation of our illustrations is possible, with the support of a visualization program [10] available for free online, that was specially developed for the project by our research group at IPREM.

Then, in order to improve our understanding of the behavior of the curiosities represented by the **helical orbitals**, we have carried out an important visualization work on **[n]-cumulenes**, various molecules with and without hetero-atoms.

**Publication and preprints associated to the thesis**

* Relating the molecular topology and local geometry: Haddon's pyramidalization angle and the Gaussian curvature (J. Sabalot-Cuzzubbo, G. Salvato-Vallverdu, D. Bégué, and J. Cresson. *Relating the molecular topology and local geometry: Haddon's pyramidalization angle and the Gaussian curvature*. Journal of Chemical Physics, 152:244310, 2020)

* Etude des structures optimales et équivalentes de quelques molécules - le cas linéaire et le cycle pour la distribution nulle ou de Möbius J. Sabalot-Cuzzubbo, J. Cresson, D. Bégué. 2020. *Preprint*

* Generation of helical states - Breaking of symmetries, Curie's principle, and excited states (J. Sabalot-Cuzzubbo, D. Bégué, J. Cresson. 2021. *Preprint hal-03360966f*)

* Haddon's POAV2 versus POAV theory for non-planar molecules J. Sabalot-Cuzzubbo, J. Cresson, D. Bégué. 2021. *Preprint*
# Part I

## Reminders
*This first part gives the necessary reminders for the understanding of the manuscript: reminders of geometry and chemistry.*

*The geometry reminder gives an overview of the geometric-chemical structure of the molecules according to the available information but also to give the criteria of a molecular skeleton.*

*The chemistry reminder includes the main points such as: the two Hückel methods with their characteristics, the definition of the geometry of a molecule from a chemical point of view, the interactions between atomic orbitals, and the context of Hückel matrices.*

*This last point gives a review of the:*

* Schrödinger equation,

* wave functions,

* atomic orbitals,

* Hamiltonian matrix,

* interaction assumptions,

* eigenvalues and energies of molecular orbitals.

*Finally, we present the necessary reminders concerning the methods of quantum computation.*
# 0.1 Problem of geometry

## 0.1.1 Geometrical-chemical structure of molecules

Molecules can be described according to different information: geometrical, chemical and electronic.

* The geometrical data concerns the curve on which the atoms are arranged. This open or closed curve is of limited length in $R^3$.

* The chemical data relates to the orbital structures specific to the molecule.

* The electronic data is about the electrons available in the molecule to form chemical bonds.

From a geometrical point of view we studied the cases of N atoms arranged either on straight line segments $L_N$ we will discuss about a linear chain, or circles which will be assimilated to cycles $S_N$.

*It should be noted that the geometrical point of view can be confusing. Indeed, the part we are interested in is related to the symmetry group of the molecule. The use of a curve to describe the arrangement of the atoms simplifies the study, but is not clearly representative of the molecule.*

## 0.1.2 The skeleton of a molecule

For a molecule M made of N atoms noted $A_i$ with $i = 1, ..., N$. We can characterize a molecule by a set of points $\mathcal{A}$ in $R^3$ which will be represented by the position of the atoms. There exists a matrix $\mathcal{R}$ such that $R_{i,j} = 1$ if $A_i$ and $A_j$ form a bond and $R_{i,j} = 0$ if there is no binding. We then associate a molecule to $M = (\mathcal{A}, \mathcal{R})$.

We associate for a molecule M a geometrical representation called skeleton of the molecule. By definition, we define the skeleton as the geometrical object in $R^3$ obtained by the plot of a segment between atoms $A_i$ and $A_j$ if $R_{i,j} = 1$. We note $S(M)$ the skeleton of M.

# 0.2 Problem of chemistry

From a chemical point of view, to correlate the adjacent matrices of the graphs associated with the molecules seen previously, we consider the Hückel matrices, then Hückel methods.

## 0.2.1 Hückel methods

The study of **Hückel matrices** provides a first approximation to study only the molecular interatomic distances but not the geometry, which requires further investigation of the Hückel method approach. A hierarchy between the matrices is organized according to the consideration of the neighboring atoms in the molecules. These matrices are associated with the graph theory.

A distinction is made between the **simple Hückel method** and the so-called **extended method** [11]. First, in the simple method, it is the interactions between atoms of the same nature that are of interest. In this case, only one orbital is involved, and finally allows the study of interatomic distances between first neighbors.

The simple method leads only to relations with parameters $\alpha, \beta$. Where $\alpha$ is the coulombic integral, a negative parameter which represents the energy of the atomic orbital (AO) before interaction and $\beta$ the negative resonance integral which is proportional to the overlap between two AOs and depends on the distance between atoms. A non-zero value of the $\beta$ parameter shows the influence of the overlap, so the resonance integral must be used when two AOs are involved. It is then very complicated to solve the system. This method is therefore preferred in cases where only identical atoms are present and that only one AO is involved and that the interactions between neighbors are constant. The example of $\pi$-systems of hydrocarbons is to be preferred in this case. The advantage of the extended Hückel method is that it is applicable to all molecules and does not make any other approximation
than the monoelectronic Hamiltonian, but has some limitations. Indeed, the limitations are based on the simplification of the Hamiltonian, where, for example the differentiation of the energies of the distributions of two electrons between 2 degenerate molecular orbitals (MO) is not possible.

### 0.2.2 Geometry of molecules

As indicated at the beginning of the work 0.1.1, we consider molecules whose geometric structure is carried by a curve in $\mathbb{R}^3$ parameter such that: $\gamma: [0, l] \to \mathbb{R}^3$, where $l > 0$ is the length of the curve. We have then: $\Gamma = \gamma([0, l])$ the curve. Note that we use the curvilinear coordinate tools.

The atoms are uniformly positioned on the curve, we remind that N is the number of atoms whose position is given as :

$$t_i = (i - 1) \left( \frac{l}{N - 1} \right) \qquad (1)$$

with $i = 1, ..., N$

The first and the last atom constitute the beginning and the end of the curve.

Note that the N atoms are arranged in $\mathbb{R}^3$, so if an atom has a bond or an interaction with another atom then it is connected by a segment. A polygonal curve is obtained which sometimes be replaced by a curve, a line or a circle.

We tried to describe the structure of the MOs of these molecules, so we used the Hückel methods as in 0.2.1.

### 0.2.3 Interaction with the atomic orbitals

Let us consider an atom and the family of atoms interacting with it. The interaction takes place between two AOs such that $\chi_a$ and $\chi_b$. Two AOs interact to form an MO if their **overlap integral** is non-zero [11].

$$S_{a,b} = \langle \chi_a | \chi_b \rangle \qquad (2)$$

This result is obtained from the formula of Wolsberg-Helmhotz described in [11].

The literature studies mostly simple geometries of molecules where the interactions between AOs are essentially between neighboring atoms arranged on the curve. In general, we consider an atom $A_i$ and its interactions with $A_{i-1}$ and $A_{i+1}$. Finally, the effective geometry does not play a role since everything occurs locally.

The existence of overlap between orbitals is also related to the molecular geometry.

Let us illustrate this with a helix where the N atoms are arranged. For a circular helix the parameters are:

$$\left\{ \begin{array}{l} x(t) = r \cos(t) \\ y(t) = \epsilon r \sin(t) \\ z(t) = bt \end{array} \right. \qquad (3)$$

where $\epsilon$ is +1 or -1 which corresponds to the direction of the helix and $b > 0$ the pitch of the helix $2\pi b$.

The interaction between atoms is given by the length $l_i$ which is the distance at which two atoms interact through their AO. The neighboring atoms of the curve will obviously interact and then we consider the first, second neighbors etc. It is then possible to filter the structure to take into account the longer interactions. The structure of the helix induces that if $2\pi b < l_i$ then, there will be interaction
with an atom positioned further on the curve. This, will have some consequence on the shape of the Hückel matrices for helical curves.

### 0.2.4 Hückel matrices

To be interested in Hückel matrices, we need to study the Schrödinger equation.

#### Schrödinger equation, wave functions and atomic orbitals

The **Schrödinger equation** associated to the **Hamiltonian** $H$ is:

$$ H.[\Psi] = E.\Psi \qquad (4) $$

$\Psi$ being the **MO** and solutions of the equation. Moreover, the wave functions are eigenfunctions of the Hamiltonian operator $H$.

The set of **wave functions** is defined as a vector space in $\mathbb{C}$, denoted by $\mathcal{O}$ and a scalar product defined for all $\phi$ and $\psi$ such that:

$$ \langle \phi, \psi \rangle = \int_{\Delta} \phi * \psi dx \qquad (5) $$

where $dx$ is the volume element, $\Delta$ the space in which we work and $\phi* = \bar{\phi}$ the complex conjugate of $\phi$.

Note $\chi_i$ ($i = 1, ..., n$) are the **AOs** associated to atoms $A_i$.

They verify:

$$ \langle \chi_i, \chi_i \rangle = 1 \qquad (6) $$

and

$$ \langle \chi_i, \chi_i \rangle = S_{i,j} \qquad (7) $$

with $i, j = 1, ..., N$ and where $S_{i,j}$ is the overlap integral of the AOs $\chi_i$ and $\chi_j$.

Hückel's hypothesis consists in using the AOs as the basis of the vector space $\mathcal{O}$, i.e. to neglect the overlap between 2 distinct orbitals. Any wave function $\Psi \in \mathcal{O}$ is written as a linear combination of the AOs with coefficients in $\mathbb{C}$:

$$ \Psi = \sum_{i=1}^{N} c_i \chi_i \qquad (8) $$

with $c_i \in \mathbb{C}$.

The notion of **normalization** of $\Psi$ is given by $\langle \Psi, \Psi \rangle = 1$ is:

$$ \sum_{i=1}^{N} c_i^2 = 1 \qquad (9) $$

#### The Hamiltonian : Matrix H

We also associate to the Hamiltonian operator $H$ a matrix $\mathbf{H}$ using the orthonormal basis $\mathcal{O}$. We note this basis: $\phi_i, i = 1, ..., N$ and define such that:

$$ \mathbf{H} = \begin{pmatrix} H_{11} & H_{12} & \dots & H_{1N} \\ H_{21} & H_{22} & \dots & H_{2N} \\ \dots & \dots & \dots & \dots \\ H_{N1} & H_{N2} & \dots & H_{NN} \end{pmatrix} \qquad (10) $$
where column i is the vector $H[\phi_i]$.

The coefficients $H_{ij}$ are obtained by the scalar product between $\phi_i$ and $H[\phi_i]$:

$$H_{ij} = \langle \phi_i | H | \phi_j \rangle = \langle \phi_i, H[\phi_j] \rangle \quad (11)$$

From a vector point of view the Schrödinger equation is written:

$$H.\Psi = E.\Psi \quad (12)$$

The **eigenvalues** of the matrix **H** correspond to the eigenvalues of the operator **H**. The **eigenvectors** of **H** are the coefficients of the **eigenfunctions** of **H** in the basis $\{\phi_i\}_{i=1,...,N}$.

The matrix **H** has some properties. If only one type of atom is involved in the molecule, i.e. if it is homogeneous, we have for all i:

$$H_{ii} = \alpha \quad (13)$$

where $\alpha$ depends only on the atom considered.

The mentioned parameter $\alpha$ (in 0.2.1) represents the **energy** of an electron in the considered AO and we have:

$$H_{ij} = H_{ji} \quad (14)$$

Thus, the matrix is symmetric. As a consequence, the eigenvalues are real and the eigenvectors orthogonal.

## Interaction hypotheses, geometry and H matrix

Let us focus on the coefficients of the **H** matrix. The hypotheses are presented by Y. Jean and F. Volatron in [11]:

* if two AOs are not carried by bounded atoms we have: $H_{ij} = 0$,

* if two AOs are carried by bound atoms we have:

$$H_{ij} = \mu_{ij}\beta \quad (15)$$

where $\beta$ is the **total overlap** of the two AOs and $\mu_{ij}$ is $0 < \mu_{ij} \le 1$ which depends of the overlap of the two AOs.

Set:

$$\mu_{ij} = \cos(\phi_i) \quad (16)$$

where $\phi_i$ is the distribution of the orbitals.

The matrix **H** is written as:

$$H = \alpha \mathrm{Id}_N + \beta \mathbf{A}_{\phi} \quad (17)$$

The matrix $\mathbf{A}_{\phi}$ is:

$$\mathbf{A}_{\phi} = \begin{pmatrix} 0 & \cos(\phi_1) & 0 & \dots & \dots & 0 & \cos(\phi_N) \\ \cos(\phi_1) & 0 & \cos(\phi_2) & 0 & \dots & \dots & 0 \\ 0 & \cos(\phi_2) & 0 & \cos(\phi_3) & 0 & \dots & 0 \\ \vdots & & & & & & \vdots \\ 0 & \dots & \dots & 0 & \cos(\phi_{N-2})\beta & 0 & \cos(\phi_{N-1}) \\ \cos(\phi_N) & 0 & \dots & \dots & 0 & \cos(\phi_{N-1}) & 0 \end{pmatrix} \quad (18)$$
We note $\nu_1(\phi) \le \dots \le \nu_N(\phi)$ the real ordered eigenvalues of $A_\phi$:

$$ \lambda_i = \alpha + \beta\nu_i(\phi) \qquad (19) $$

$A_\phi$ is of zero trace:

$$ \sum_{i=1}^{N} \nu_{i}(\phi) = 0 \qquad (20) $$

and

$$ \sum_{i=1}^{N} \lambda_{i} = N\alpha. \qquad (21) $$

### Eigenvalues and energy of molecular orbitals

To find the eigenfunctions of the Schrödinger equation it is necessary to determine the solutions of the system: $H\Psi = E\Psi$.

Consider then for all $E \in \mathbb{R}$ the matrix:

$$ M(E) = H - EId \qquad (22) $$

where $Id$ is the identity matrix of $\mathbb{R}^N$.

An eigenfunction of $H$ is equivalent to the existence of a non-zero solution of the linear system:

$$ E(E)\Psi = 0 \qquad (23) $$

A necessary and sufficient condition is that the characteristic polynomial of $H$ is zero in $E$:

$$ P_{H}(E) = | M(E) | = 0 \qquad (24) $$

where $|\cdot|$ represents the determinant.

Indeed, in this case the matrix $M(E)$ has a non-zero kernel and therefore non-zero solutions.

The matrix $M(E)$ is symmetric, its eigenvalues are real, so we can order them:

$$ \lambda_1 \le \lambda_2 \le \dots \le \lambda_N \qquad (25) $$

Also, the polynomial $P_H(E)$ can be written by factorization:

$$ P_{H}(E) = (E - \lambda_{1}) \dots (E - \lambda_{N}) \qquad (26) $$

For more details, about the characteristic polynomial of $H$, refer to Appendix 2 section 1.

We can then determine the total energy $E$ of the molecule such that:

$$ E = \sum_{i=1}^{N} g_{i}E_{i} \qquad (27) $$

where $g_i$ is the quantum number of the orbital $\Psi_i$ which by the principle of exclusion of Pauli can take the values 0, 1, or 2 since each orbital can at most 2 electrons.

Note $\pi(S)$ the number of electrons available to make bonds:

$$ \sum_{i=1}^{N} g_{i} = \pi(S) \qquad (28) $$

We can deduce the value of $g_i$ according to the parity of $\pi(S)$:
* if $\pi(S) = 2m$ then $g_i = 2$ for $i = 1, \dots, \frac{\pi(S)}{2}$ and $g_i = 0$ if $i = \frac{\pi(S)}{2} + 1, \dots, N$

* if $\pi(S) = 2m + 1$ then $g_i = 2$ for $i = 1, \dots, \frac{\pi(S)-1}{2}$, $g_i = 1$ for $i = \frac{\pi(S)+1}{2}, \dots, N$ and $g_i = 0$ for $i = \frac{\pi(S)+1}{2} + 1, \dots, N$.

Thus,

$$E_i = \alpha + \beta \nu_i(\phi) \qquad (29)$$

We have:

$$E = \pi(S)\alpha + \beta \sum_{i=1}^{N} g_i \nu_i(\phi) \qquad (30)$$

For more details, about the total energy of a molecule, refer to Appendix 2 section 2.

## Distribution of a molecule

Associated to the notion of Hamiltonian, it is necessary to define the notion of distribution of a molecule which we will use in the following.

The distribution of a molecule is determined following steps:

* the Hückel matrix must be written,

* calculate the scalar products $H_{ij}$ which are decomposed according to the formula 15.

The quantity corresponding to the distribution is then an angle $\phi_i$, which is associated to $H_{i,i+1}$. As the angle is not clearly identified, we have to take into account the basis of the wave functions used with a vector basis in $\mathbb{R}^3$.

## 0.3 Quantum computing methods

In all the work presented in this manuscript, we have carried out calculations using ab-initio and density functional theory (DFT) implemented in the Gaussian code [12], we used in particular the DFT Hamiltonian type B3LYP [13, 14]. We mainly used Gaussian which is a popular and widely used quantum chemistry software. It is a gaussian based software ideally used for closed-shell systems and can be used with different methods (DFT, Hartree-Fock, Moller-Plesset,...). In order to obtain reliable energies on multi references systems calculations were also carried out in both CASSCF and CASPT2 levels of theory. These calculations were performed using Molpro program packages [15].

### 0.3.1 Description of electronic structure of molecules

When one wants to determine electronic properties of atoms and molecules by the point of view of molecular modeling, we have to remember that in the “quantum world” the Schrödinger equation is the master equation that describes the behavior of all the bodies in submicroscopic scale. Based on the De Broglie’s observations on the wave-particle behavior of the matter, the Schrödinger equation is based on the fact that all bodies behave as waves with a certain wavelength. This equation has some special characteristics:

* it is best solved as an algebraic partial differential equation, not as an analytical one

* it is an equation of eigenvalues, as the ones in algebraic mathematics and, for this, it needs linear operators and functions that describes vector in an algebraic space
* it needs the function to give the same function back, i.e., it needs a “guess” function to find this same function itself after a cycle of calculation.

The last item states the basis of what is called the **self-consistent field**. The guess function is used to solve Schrödinger equation, yielding eigenvalues and a function in a “better shape”. In a realistic world, solving this equation is not possible for a system with more than two electrons. It is in this point that “quantum chemistry” started to be developed more strongly with the introduction of approximations that allow one to solve the electronic structure of multi-electronic atoms and molecules. These approximations can be divided into physical-mathematical approximations and physical approximations. The **first physical approximation** is the **Born-Oppenheimer**, that states that as the nuclei of the atoms are much heavier than the electrons, they are much slower, so, they can be considered as statistic while the electrons move. This results in the decoupling of partial equations that would have needed to be calculated on both spatial coordinates (nuclei and electrons) before, and now, with such approximation, they need to be calculated only on the nuclei coordinates **or** on the electron coordinates.

By the third decade of XXth century, novel approaches have been proposed to determine the electronic structure of molecular systems and solids. The first one was proposed by D. R. Hartree and Vladimir A. Fock. The so-called **Hartree-Fock** method consists in treating the electrons and their wave-function as a determinant called Slater's determinant. This mathematics allowed the calculation to take place considering that the multi-electronic problem could be resolved as “multi” problems of one electron. This was introduced by an effective potential operator (Fock operator) which states a general field of electrons. This means that one electron is sensible to the electromagnetic field of the nuclei and the other N-1 electrons. This also does not take into account any relativistic effect that may happen.

This one-electron picture could then be implemented in a computational code to be solved iteratively (based on the self-consistent field scheme). The electrons could be translated into mathematical functions called **basis set**. These electrons could then be associated to specific orbitals (wave-functions) respecting the Pauli's exclusion theorem. The composition of the molecular orbitals was performed on the basis of a **linear combination of atomic orbitals** (LCAO method). The basis set can then be decorticated and designed to accommodate the needs of the physical problem being studied (number of core functions, polarization effects, felxibility to accomodate ionic clouds, etc).

Because of the one-electron picture, the Hartree-Fock method lacks of description of the electronic correlation in a many-body system. In this way, metallic systems and systems where electrons can delocalize over a region of the space are poorly-described by this method. Many others, called **post-Hartree-Fock** methods have tried, with different degrees of success, to incorporate these effects on the Hartree-Fock scheme. This, has been accomplished by **Perturbation theory**, as it is the case on $n^{th}$ order **Moller-Plesset perturbation theory** (MPn), coupled-cluster, configuration of interactions, multi-configuration methods, and semi-empirical ones.

A common point between these methods, exception made for the semi-empirical ones, is the limitation on the number of basis functions. These methods, although quite efficient on the description of the electronic structure of molecules, mainly, scale quickly with the number of atoms and in this way, are not practical for every system being studied, even though they would be very appropriate because the description of both exchange and correlation would be the appropriate in many cases. The semi-empirical methods, by their turn, can be employed for large systems, with a very low accuracy though. This is due to the fact that multi-center integrals are not calculated "on the fly" but are replaced by experimentally-obtained (or calculated with a higher level of theory method for model systems) values. This has been bypassed in parts by the introduction of **Density Functional Theory (DFT)** and this is discussed in the next section.
### 0.3.2 Density Functional Theory (DFT)

The use of **Density Functional Theory (DFT)** became a common-place in the molecular modeling of materials in the nanometric scale. In this method, the energy of a system is described as a functional of the electronic density, skipping the direct wave-function calculation in the process. While wave-function-based methods, such as Hartree-Fock (HF), solve a Schrödinger-like equation (the Fock equation), DFT is based on the assumption that there is a relationship between the total electronic energy and the overall electronic density.

*Note that for the Fock equation, the same is valid for Perturbation Theory based methods, such as MP2, and Coupled-Cluster one.*

The first basis of this idea was already present in the Thomas-Fermi model, however the breakthrough came by the Hohenberg-Kohn theorem in 1964, showing that the ground-state energy and other properties of a system were uniquely defined by the electron density. In other words, the energy E is hence a functional of the density $ρ(\vec{r})$ at the point $\vec{r}$. Based on the definition of the electric potential, one can write this as:

$$E[\rho(\vec{r})] = \int V_{\text{ext}}(\vec{r})\rho(\vec{r})d\vec{r} + F[\rho(\vec{r})] \quad (31)$$

The first term $V_{\text{ext}}(\vec{r})$ arises from the interaction of the electrons with the external potential produced by the Coulomb interaction with the nuclei. On the other hand, $F[\rho(\vec{r})]$ is the sum of the kinetic energy of the electrons and the contribution from inter-electronic interactions.

The **variational method** can be applied in the solving of this equation, in which the minimum value of $E[\rho(\vec{r})]$ represents the energy of the electron density of the ground-state. From this, it is a common-sense that DFT is intrinsically incapable of describing excited-states without further physical-mathematical considerations.

Then, in 1965, with the **Kohn-Sham (KS) theorem**, the term $F[\rho(\vec{r})]$ is suggested to be approximated as the sum of three terms in the form:

$$F[\rho(\vec{r})] = E_{KE}[\rho(\vec{r})] + E_{H}[\rho(\vec{r})] + E_{XC}[\rho(\vec{r})] \quad (32)$$

where $E_{KE}[\rho(\vec{r})]$ is the kinetic energy, $E_{H}[\rho(\vec{r})]$ the electron-electron Coulombic energy (also known as Hartree electrostatic energy), and $E_{XC}[\rho(\vec{r})]$ a term responsible for describing the exchange and correlation contributions.

The first term is further defined as the **kinetic energy** of a system of non-interacting electrons with the same density $ρ(\vec{r})$ as the real system. The second term associates the **electrostatic energy** as the classical interaction between two charged densities over all possible pairwise interactions. It is interesting to note that this term is classical and do not take into account the quantum many-body behavior of the electrons, i.e., as they are correlated to each other and the Heisenberg uncertainty principle, for instance.

Finally, the last term is then responsible for accounting for this quantum behavior of electrons and it is one of the master pieces of DFT, known as the exchange-correlation functional. As no analytical form of this functional is known, several approximations are proposed to treat this quantum mechanical problem and based on this, DFT cannot be considered always as an *ab initio* method, since the definition of this functional can take into account empirical parameters. However, even simple approximations to it can give favorable results and this is the key success of DFT.

Numerically, the electronic structure of the system is solved in a self-consistent way, in which a guess of an initial electronic density is fed into the above equations. The application of the variational approach allows one to reach the ground-state electronic density and, consequently, energy by a so-called
**self-consistent field.**

Calculating these specific energies for a multi-electronic system takes the form of multi-centered multi-index integrals, what is responsible for a ready increase in the computation time (often in the order of $O(N^4)$, where N is the number of atoms). In this way, treating large molecular systems becomes almost non-practical.

Moreover, the combination of these two equations bears a striking resemblance to those of Hartree-Fock theory. The difference between them is found on the fact that the HF has no correlation being calculated whereas KS is a correlated method. To go further to the latter, one only needs to replace the exchange energy by the exchange-correlation one.

### 0.3.3 The exchange-correlation functionals

The fact that the **exchange-correlation functional** is not known gives rise to the appearance of several difference ready to be employed by the user. Choosing the most appropriate demands testing it against the family of materials being studied and the nature of the physical-chemical issue.

The first functionals are based on two different approximations: the **Local Density Approximation** (LDA) and **Generalized Gradient Approximation** (GGA). For the former, the electron density varies locally whereas for the latter it also depends on gradient of it. In well-behaved systems, the electron density is smooth, almost constant, thus, it does not depend on its gradient, justifying the use of LDA for these cases. For heteroatomic systems, this may not be true and GGA may become needed.

To refine even more the quality of the results, mainly the atomization energies, bond lengths and vibrational frequencies, functionals called **hybrid** have been developed over the three last decades. These functionals incorporate a portion of the exact exchange in HF theory with exchange and correlation from other sources such as LDA and GGA. These functionals are constructed by a linear combination of the HF exact exchange functional and any other exchange-correlation functional. The weight of participation of each component is typically found by fitting the predictions done by these functionals to experimental results.

The most used **hybrid functionals** are **B3LYP** [16, 17], **PBE0** [18, 19], **HSE** [20] and meta-hybrid **GGA** (as the Minnesota functionals) [21, 22]. Specifically, the B3LYP functional which has been used throughout this thesis, is written in the form of:

$$E_{XC}^{B3LYP} = E_{X}^{LDA} + a_{0}(E_{X}^{HF} - E_{X}^{LDA}) + a_{x}(E_{X}^{GGA} - E_{X}^{LDA}) + E_{C}^{LDA} + a_{c}(E_{C}^{GGA} - E_{C}^{LDA}) \quad (33)$$

Where $E_{XC}^{B3LYP}$ stands for the exchange-correlation energy in the B3LYP scheme, $E_{X}^{LDA}$ the exchange energy in the LDA scheme, $E_{X}^{HF}$ the exchange energy of HF, $E_{X}^{GGA}$ the exchange energy in GGA scheme, $E_{C}^{LDA}$ the correlation energy in LDA scheme, and $E_{C}^{GGA}$ the correlation energy in GGA scheme. The parameters $a_0$, $a_x$ and $a_c$ are the weight of participation of each component and assume the values of 0.20, 0.72 and 0.81, respectively. These parameters were determined beforehand by A. Becke by fitting the analogous B3PW91 functional to a set of molecular parameters [13].

This functional has been widely used for description of ground state properties of molecular systems. However, a serious low accuracy appears when trying to describe excited state properties and where the self-interaction problem becomes pathological.

Note that the self-interaction problem consists on the fact that a electron can interact with itself. This leads to problem of over-delocalization of the electron cloud since this electron tends to repulse itself.
This drawback has limited the use of B3LYP functional for charge-transfer systems and problems evolving the energies of the excited states for example.

### 0.3.4 Multi reference methods

Unlike the **Restricted Hartree-Fock (RHF)** solution which corresponds to the description of a **Slater** determinant made up of $N/2$ doubly occupied orbitals the exact solution that we are trying to describe by taking into account the correlation is in fact made up composed of an infinite number of configurations that can be thought of as excitations with respect to the N-electron RHF determinant in an infinitely large base. Of course, it is impossible to work in a complete or infinite basis. Various methods are then proposed by theoretical chemists to evaluate as accurately as possible this correlation. Three types of approaches are used:

* the methods aiming at solving variably the Schrödinger equation in the eigenstates by using a basis of Slater determinants or configurations. These are **Configuration Interaction (C.I.)** type approaches,

* perturbation methods requiring the definition of a minimal basis and a partition of the Hamiltonian operator adapted to ensure the convergence of the perturbation series in a Rayleigh-Schrödinger type approach,

* pairwise methods which consist in evaluating the correlation energy of pairs of electrons assumed without interactions (Independent Electron Pair Approximation - IEPA) or in interaction (Coupled Pair Theory).

These approaches have all in common that they build the basis for the development of the desired function from single-electron functions optimized at the **Hartree-Fock level** (generally monodeterminal ground state).

On the contrary, in the **Multi Configurational Self Consistent Field (MCSCF)** methods, the objective is to optimize with the use of a variational method both the monoelectronic functions used and the multiconfigurational development of the eigenfunction of the electronic Hamiltonian.

Among the numerous developments proposed, the **Complete Active Space Self Consistent Field (CASSCF)** approaches lead to the development of the multiconfigurational wave function on the totality of the configurations generated by the set of all possible excitations in a restricted space of occupied and virtual molecular orbitals (active space).

The purpose of this paragraph is not to present the mathematical details of all these computations method, but rather to set out the general considerations for the use of these approaches when one has to describe one or more electronic states of more or less strongly correlated electrons. Thus, in a system where static correlation is not involved, the RHF determinant remains a relatively good approximation for the description of the wave function. The addition of a method taking into account the dynamic correlation is done with this determinant (as for the DFT, methods ...). The RHF determinant is considered as the reference determinant. On the contrary, in the case of a system of strongly correlated electrons several Slater determinants must be considered to describe this state. The wave function used is most often a CAS wave function and the dynamic correlation is taken into account on the latter. This CAS wave function being made of several determinants, the system has several references. The term Multi-Reference (MR) thus translates the fact of considering the dynamic correlation on several reference determinants.
<figure><img src="image_5.png" /></figure>
# Part II

## π-Orbital Axis Vectors theory
<figure><img src="image_6.png" /></figure>
We dedicate this part to the theory of π-orbital axis vectors, as we are interested in the π-orbitals of molecules. Their interest, is that they are involved in molecular reactivity.

But how to get interested in them?

* By using DFT calculation methods, however it is heavy and long in preparation and calculation time, and it is not possible for large molecules.

* Therefore, we are interested in the work of Robert C. Haddon's research team which proposes simple calculation tools but not sufficiently generalized.

Thus, in this part of the manuscript we propose a detailed review, the extension and the interpretation of these different tools.
# Introduction: Study of the Robert C. Haddon's concepts

The interest of this part concerns the issue of **reactivity** which is a fundamental point for the study of molecular structures. The goal is to carry out **cartographies** representing the reactivity. Many current topics depend on the knowledge of this reactivity as for instance in the case of hydrogen storage, or in some porous systems.

Generally, the obtention of these cartographies are performed by quantum calculations as DFT but they are sometimes complex in the case of molecules with a large number of atoms, or by molecular dynamics which is not sufficiently accurate.

To find the solution to these limitations, we must refer to the fundamental notions of reactivity. Chemically, the reactivity is connected to the π-system, it is therefore necessary to find one or more **tools** in order to determine this system, without quantum calculations, in an explicit and algorithmic way and in a not expensive way.

In the first chapter, we answer these questions by referring to the work of Robert C. Haddon, a pioneer on these topics. Initially, R. C. Haddon tried to identify the **planarity** of molecules, to answer this, he proposes a set of tools and notions.

The first notion is the **pyramidalization angle** well-known by chemists. First described by R. C. Haddon in regular cases in his article "$C_{60}$: sphere or polyhedron?" [3], we have chosen in this manuscript to work also in less regular cases.

Then, it is the notion of **π-Orbital Axis Vector 1 (POAV1)** illustrated by R. C. Haddon and L. T. Scott [23] related to the pyramidalization angle that we investigate. The POAV1 is the vector tool that best describes the direction of the π-system of a molecule. It provides a more precise access to the notion of system planarity even if the pyramidalization angle gives a first knowledge of it.

**Hybridization**, a well-known tool, is also studied in detail, in order to relate hybridization to the POAV1. However, despite all these tools, there are some limitations. Tools from mathematics, and more particularly from differential geometry, are useful. Concepts of **curvature**, and especially **spherical curvature** are given. In addition, we also consider the **angular defect**.

We study precisely all the concepts developed by the researcher and consider some of his concepts in a more general way with physical, geometrical and chemical meanings.

In the second chapter, we first explain the **methods of implementation** of the concepts that we have used throughout this work. Then, we expose the **application methods** that we have developed with the different concepts presented. We present among other things, our visualization program based on the concepts applied to a database of fullerene. Finally, we present a large number of our results according to all the notions on non-fullerenic molecules and along a reaction path of nitrile imine molecule (energy, minimum and maximum values of pyramidalization angle, hybridization, angular defect, spherical curvature, and the corresponding cartographies).

The third chapter, provides an opportunity to explore beyond the first concepts with a reflexion on the **POAV2** concept proposed by R. C. Haddon [6] as a complement to POAV1.

We conclude the chapter with an overview of possible applications of the concepts for the characterization of geometries, topologies, and reactivity of organic covalent molecules.

Then, we give a comparison between the two tools **POAV1** and **POAV2**, in order to show which is the most appropriate since there is no other method to characterize the π-system and more specifically the reactivity.
# Chapter 1

# The π-Orbital Axis Vector 1 (POAV1) theory

In this chapter, we consider and study 3 main points:

1. the clarification of the notions of pyramidalization angle (Pyr), π-Orbital Axis Vector (POAV1), hybridization and curvature,

2. the extension to the notion of angular defect,

3. the properties of POAV1 and associated tools based on fullerenes and other non-fullerenic compounds.

We used, among others [3][23][6], and results published in our first paper [4].

## 1.1 The π-Orbital Axis Vector (POAV1) of Robert C. Haddon

### 1.1.1 Pyramidalization angle and π-Orbital Axis Vector (POAV1)

Robert C. Haddon developed in 1990s the concept of **pyramidalization angle** (Pyr) and **π-Orbital Axis Vector (POAV1)** ([3, 23, 6]). In this series of articles, R. C. Haddon describes the concept for molecules such as fullerenes and more particularly the $C_{60}$ fullerene. He introduces a **local concept**, i.e. specific to an atom. So from a geometrical point of view with respect to a vertex, since the fullerene (illustrated in first approximation as a soccer ball by Kroto et al.[24]) can be apprehended as a discrete sphere made of 20 hexagons and 12 polygons. We will then distinguish the use of the *Pyr* and π-Orbital Axis Vector (POAV1) in **regular trivalent** cases and in **non-regular trivalent** cases.

#### Regular trivalent case

The **trivalent regular case** is the simplest case that we can observe. It is the case studied by R. C. Haddon in [3]. In fact, R. C. Haddon studies the specific case of $C_{60}$, and assumes that all the carbon-carbon bonds are identical thus that they are of the same length and that the angles between bonds are equal.

The following figure illustrates the angle $\theta_{AB}$ associated to calculation of *Pyr* and the normal vector ($n_a$) in the specific case of $C_{60}$:
<figure><img src="image_7.png" /><figcaption>Figure 1.1: Representation of the pyramidalization angle in a regular case</figcaption></figure>

Trivalent regular case means that all the bonds are of equal lengths and that each atom of the molecule is connected to three atoms.

In the case of any fullerene, the carbon atom studied are then always be bounded by 3 neighboring carbon atoms.

For the construction of *Pyr* we consider an atom and its three neighbors as a polyhedron with vertex A and the star of A ($\star(A)$) containing the three neighbors B. We can calculate the **angle of pyramidalization** in A such that :

$$ \text{Pyr}(A) = \theta_{AB} - \frac{\pi}{2} \qquad (1.1) $$

It should be noted that if *Pyr(A)* = 0 the molecule is considered as planar and if *Pyr(A)* ≠ 0 the molecule is non-planar.

The *Pyr* is by definition obtained for a given atom A in a molecule associated with the existence of the particular vector called **POAV1(A)** vector which has the property to make a constant angle $\theta_{AB}$ with each neighbors B connected to A [25]. Where, B ∈ $\star(A)$ such that $\frac{\pi}{2} < \theta_{AB} < \pi$.

Thus, the vector POAV1(A) is the normal vector to the plane formed by the neighboring B atoms at the vertex A.

To prove the previous statements, let A be a given atom of the molecule M. As the molecule M is trivalent, we have three connected atoms $B_1, B_2, B_3$ connected to A. These three atoms define a plane $P(A)$. As the three bonds are equivalents, the atom A belongs to the intersection of each mediating plane, then in particular to the line passing trough A and the intersection of the mediating line in the triangle defined by $B_1, B_2$ and $B_3$. Let us denote by I the intersection point of all mediating line in the plane $P(A)$. We define POAV1(A) as the unique unitary vector directed from I to A.
<figure><img src="image_8.png" /><figcaption>Figure 1.2: Construction in a trivalent regular case [4]</figcaption></figure>

It remains to prove that POAV1(A) is such that the angle with each bond is constant. We first remark that the vector POAV1(A) is normal to the plane $P(A)$. Indeed, the line (IA) belongs to the intersection of two intersecting planes which are normal to the plane $P(A)$. It follows that POAV1(A) is a normal vector in A.

Let B be an atom of $\star(A)$ in $P(A)$. Let us denote by $l$ the bond length between A and atoms of $\star(A)$ and $z_A$ the height IA of A with respect to the plane $P(A)$. We denote by $\theta_{AB}$ the angle between POAV1(A) and the bond AB.

<figure><img src="image_9.png" /><figcaption>Figure 1.3: Angle between a bond and the POAV1(A) vector [4]</figcaption></figure>

The angle $\theta_{AB}$ satisfies:

$$ \cos(\pi - \theta_{AB}) = \frac{z_A}{l} \qquad (1.2) $$

and,

$$ \sin(\pi - \theta_{AB}) = \frac{IB}{l} \qquad (1.3) $$

As $\theta_{AB}$ satisfies $\frac{\pi}{2} \le \theta_{AB} < \pi$, we have $0 \le \pi - \theta_{AB} < \frac{\pi}{2}$ and the value of the arccos is sufficient to determine $\pi - \theta_{AB}$ completely. As seen in the formula, the angle $\theta_{AB}$ depends only on $z_A$ and $l$ which are quantities independent of B. As a consequence, $\theta_{AB}$ is a quantity only depending on A and the length of the bond but independent of B. As a result, that the angle between the vector POAV1(A) and each bond is constant.
To finish, from a more mathematical point of view we can refer to the works of Liu's team [26] and Romon's manuscript [27]. Indeed, the existence of the vector POAV1(A) is associated with the existence of a cone in A, which is tangent to the faces containing A. This is a mesh property which can be denoted as cone mesh. All meshes whose vertices are trivalent are conical. Romon [27] also shows that a mesh whose vertices are of order 4 is conical. We then understand that in such a case the POAV1 can be determined.

### Non-regular trivalent case

As we have just seen, R. C. Haddon defines the *Pyr* only in a certain case but not in a **non-regular trivalent case** [3, 23, 6]. The construction in a trivalent non-regular case will depend on the existence of a **regularized star** of a given atom in a molecule. Note that the POAV1 vector can be defined even for non-regular trivalent molecule.

In fact, for a trivalent molecule M, and A the considered atom of M. We define for each neighboring atom $B \in \star(A)$, a $Reg_{\epsilon}(B)$ with $\epsilon > 0$, a point of AB. We have then $Reg_{\star}(A)$ i.e a $\epsilon$-regularized $\star(A)$ as in the following figure 1.4.

<figure><img src="image_10.png" /><figcaption>Figure 1.4: Representation of the regularized $\star(A)$ [4]</figcaption></figure>

$Reg_{\epsilon}(B)$ is defined by the relation:

$$ \overrightarrow{AReg_{\epsilon}(B)} = \frac{\epsilon}{AB} \overrightarrow{AB} \quad \forall \epsilon \in \star(A) \qquad (1.4) $$

with

$$ \epsilon(A) = \min_{B \in \star(A)} AB \qquad (1.5) $$

which corresponds to the minimum of the bound lengths starting in A.

Finally, the regularization of the star of a given atom is **extendable** to the *Pyr*.
By definition, for a trivalent molecule M and A an atom of M, let $\epsilon > 0$, the $\epsilon$-pyramidalization angle of M in A is defined as the *Pyr* associated with A and the $\epsilon$-regularized star of A.
This definition depends only on $\epsilon$, and this is not sufficient.
We have then, for the same type of trivalent molecule, a $\epsilon > 0$ and POAV$_{\epsilon}$(A) be the vector defined in 1.1.1. The POAV$_{\epsilon}$(A) is **independent** of $\epsilon$.

The proof of this definition is developed in Appendix 2 section 3.

We can then deduce an essential point of the $\epsilon - Pyr$ : for M a trivalent molecule and for $\epsilon > 0$, the $\epsilon - Pyr$ in A is **independant** of $\epsilon$.

This property can be used to compute the *Pyr* in concrete cases where the bond lengths are not equals.

Note that this result is stated in the work of R. C. Haddon [3] p.1798.

## 1.2 Relationship with chemistry

The *Pyr* definition explained (1.1.1), is a **pure geometrical construction**. In fact, R. C. Haddon and L. T. Scott [23] expressed that the *Pyr* provides a "convenient index of the degree of non-planarity" of a given atom of a molecule. The major issue of R. C. Haddon is **chemistry** [3][23][6] and especially the chemical properties of the molecules related to the geometry. The need is to interpret the *Pyr* in this framework. As we seen in the previous section, the *Pyr* is related to the **POAV1** but also with others properties as the **hybridization** of atomic orbitals, and the measure of the **local distortion** of a given π-electron system in a given molecule.

### 1.2.1 Hybridization

The pyramidalization angle as defined previously, depends only on geometrical data and refers to the **non-planarity** of molecules. It is thus also related to the **hybridization** of AO.

We are interested in non-planar systems, so **non-planar conjugate systems**. We have the system of hybrid σ-orbitals. These hybrid orbitals ($h_1, h_2, h_3$) are linear combinations of the $s$ and $p_x, p_y, p_z$ orbitals. They are defined as:

$$h_i = c_{i,1}s + c_{i,2}p_x + c_{i,3}p_y + c_{i,4}p_z \quad (1.6)$$

The aim is to obtain the maximum **overlap** between orbitals, we must use $p_i$ vectors allowing this condition.

We define the orbitals such that:

$$p_i = c_{i,2}p_x + c_{i,3}p_y + c_{i,4}p_z \quad (1.7)$$

with $i = 1, 2, 3$ and where $c_i$ is a vector $c_i = (c_{i,2}, c_{i,3}, c_{i,4})$ and each $p_i$ is directed along a internuclear axis between A and an atom B of $\star(A)$.

More generally, let :

$$h_i = c_{i,1}s + \lambda_i p_i \quad (1.8)$$

with $i = 1, 2, 3$.

Based on the work of R. C. Haddon and L. T. Scott [23] and the approach of Radziszewski et al. [28], we describe the **hybrid π-orbitals** as:

$$h_{\pi} = c_{\pi}s + \lambda_{\pi} p_{\pi} \quad (1.9)$$
**Hybrid π-orbitals** are related to the concept of **POAV1** (for reminder POAV1 makes equal angles with the σ-bonds of the neighboring atoms). The connection between POAV1 and hybridization is made by the angle $\theta_{i,\pi}$ (with $i = 1, 2, 3$), which is the angle between $p_\pi$ and $p_i$ such that:

$$ \theta_{1,\pi} = \theta_{2,\pi} = \theta_{3,\pi} \qquad (1.10) $$

We take then the first relation of the *Pyr* (eq. 1.1):

$$ \mathrm{Pyr}(A) = \theta_{\sigma,\pi} - \frac{\pi}{2} \qquad (1.11) $$

If we refer to the work of R. C. Haddon [6], we also show that the orbitals of the **σ-bonds** diverge from the internuclear axis when the angles between the **σ-bonds** are inferior or equal to 100°. In the case of **π-orbitals** this is valid only when the angles between bonds are close. However when the **σ-bonds** are really different R. C. Haddon suggests the use of another tool that he calls **POAV2**, which takes into account the orthogonality between **σ** and **π-orbitals**. The POAV1 matches the POAV2 only if the bonds are **equal**.

To construct the geometrical model we have just described, we need to place it in a suitable reference frame. We also need to consider the POAV1, and $p_\pi$ collinear to $p_z$ as in eq. 1.9.

Thus, by indicating that the plane $\mathcal{P}$ is orthogonal to $p_z$, the **σ-orbitals** $h_1, h_2$ and $h_3$ make an angle made by *Pyr*($\mathcal{A}$) with the plane $\mathcal{P}$ as : $\frac{\pi}{2} + \mathrm{Pyr}(\mathcal{A})$.

If we refer to the work of R. C. Haddon and L. T. Scott [23], what we have just explained, can be summarized as an **intermediate hybridization** between $sp^2$ and $sp^3$ depending on the angle of *Pyr*. Then, we choose a reference frame to have the internuclear axis in the $(x, z)$ plane such as:

$$ h_i = c_{i,1}s + c_{i,2}p_x + c_{i,3}p_y + c_{i,4}p_z \qquad (1.12) $$

with $i = 1, 2, 3$.

## Orthogonality and normalization conditions

We define a condition of **orthogonality** between the **hybrid orbitals** described previously (section 1.2.1). In fact, the hybrid orbitals $h_1, h_2, h_3$, and $h_\pi$ are orthogonal to each other.

The **orthogonality condition** is:

$$ c_{i,1}c_{\pi} + c_{i,4}\lambda_{\pi} = 0 \qquad (1.13) $$

We also specify :

$$ c_{i,1} = \mu_i \lambda_{\pi} \qquad (1.14) $$

and

$$ c_{i,4} = -\mu_i c_{\pi} \qquad (1.15) $$

with $i = 1, 2, 3$.

The angle between $p_z$ and each $h_i$ is the same, it implies:

$$ \mu_1 = \mu_2 = \mu_3 = \mu \qquad (1.16) $$
The expressions of the σ-orbitals are:

$$
\begin{align}
h_1 &= \mu(\lambda_\pi s - c_\pi p_z) + c_{1,2}p_x \nonumber \\
h_2 &= \mu(\lambda_\pi s - c_\pi p_z) + c_{2,2}p_x + c_{2,3}p_y \tag{1.17} \\
h_3 &= \mu(\lambda_\pi s - c_\pi p_z) + c_{3,2}p_x + c_{3,3}p_y \nonumber
\end{align}
$$

In order to **normalize** the **π-orbitals** we have:

$$
c_{\pi}^{2} + \lambda_{\pi}^{2} = 1 \tag{1.18}
$$

In the same way the normalization according to *s* and *p<sub>*z*</sub>*:

$$
\begin{align}
h_1 &= \frac{1}{\sqrt{3}}(\lambda_\pi s - c_\pi p_z) + c_{1,2}p_x \nonumber \\
h_2 &= \frac{1}{\sqrt{3}}(\lambda_\pi s - c_\pi p_z) + c_{2,2}p_x + c_{2,3}p_y \tag{1.19} \\
h_3 &= \frac{1}{\sqrt{3}}(\lambda_\pi s - c_\pi p_z) + c_{3,2}p_x + c_{3,3}p_y \nonumber
\end{align}
$$

The detailed proofs of the normalization are available in Appendix 2 section 4.

Relation between hybridization and the POAV1

We can determine the relationship between the **hybridization coefficients** presented just previously
and the **POAV1** :

$$
c_{\pi} = \sqrt{3} c_{1,2} \tan(\mathrm{Pyr}(A)) \tag{1.20}
$$

and

$$
\lambda_{\pi} = \sqrt{1 - 3c_{1,2}^2 \tan^2(\mathrm{Pyr}(A))} \quad (1.21)
$$

We give the explicit formula of $c_\pi$ and $\lambda_\pi$ determining the **POAV1** vector. These coefficients are related by the normalization condition and their values depend on the *Pyr*.

The proofs of the development of the coefficients is given in Appendix 2 section 5.

We note that the works of R. C. Haddon and L. T. Scott [23] show the existence of $c_\pi$ in the case of
molecules with a $C_{3v}$ symmetry.

We proven what R. C. Haddon and L. T. Scott [23] expose in terms of hybridization in the precise case of a **C<sub>3v</sub>** **symmetry**.

This **specific symmetry** requires an adapted reference frame:

$$
\begin{align}
c_{2,2} &= c_{3,2} < 0 \nonumber \\
c_{2,3} &= -c_{3,3} \tag{1.22}
\end{align}
$$

In this specific symmetry, there are three σ<sub>v</sub> planes of symmetry in the plane (*p*<sub>x</sub>, *p*<sub>y</sub>) :

$$
c_{2,2}^2 + c_{2,3}^2 = c_{1,2}^2 \tag{1.23}
$$

This equality corresponds to the fact that we have an equilateral triangle with $c_{1,2} > 0$.
The σ-orbitals are:

$$h_1 = \frac{1}{\sqrt{3}}(\lambda_\pi s - c_\pi p_z) + c_{1,2}p_x$$

$$h_2 = \frac{1}{\sqrt{3}}(\lambda_\pi s - c_\pi p_z) + c_{2,2}p_x + c_{2,3}p_y \qquad (1.24)$$

$$h_3 = \frac{1}{\sqrt{3}}(\lambda_\pi s - c_\pi p_z) + c_{2,2}p_x - c_{2,3}p_y$$

If we normalize according to the sp³ **hybridization** conditions:

$$h_1 = \frac{1}{\sqrt{3}}(\lambda_\pi s - c_\pi p_z) + \frac{\sqrt{2}}{\sqrt{3}}p_x$$

$$h_2 = \frac{1}{\sqrt{3}}(\lambda_\pi s - c_\pi p_z) - \frac{1}{\sqrt{6}}p_x - \frac{1}{\sqrt{2}}p_y \qquad (1.25)$$

$$h_3 = \frac{1}{\sqrt{3}}(\lambda_\pi s - c_\pi p_z) - \frac{1}{\sqrt{6}}p_x + \frac{1}{\sqrt{2}}p_y$$

The proofs of these sp³ **hybridization** conditions is given in Appendix 2 section 6.

## Relative weight and hybridization numbers

Knowing the hybridization of an atom, we have interested in the weight of each component of an orbital, this is called the **relative weight**. We consider an orbital of type:

$$h = a_s s + a_x p_x + a_y p_y + a_z p_z \qquad (1.26)$$

with $a_s, a_x, a_y, a_z$ the real coefficients and $p_i$ the weight of the AO $p_i \in (s,x,y,z)$.

The aim is to compare the contribution of each orbital therefore, the relative weight between 2 given orbitals.

We consider the **relative weight** of the AO of s in MO h compared to the AO x in MO h as:

$$w_{s,x}(h) = \frac{w_s(h)}{w_x(h)} \qquad (1.27)$$

In the others cases we have the same approach:

$$w_{s,y}(h) = \frac{w_s(h)}{w_y(h)} \qquad (1.28)$$

$$w_{s,z}(h) = \frac{w_s(h)}{w_z(h)} \qquad (1.29)$$

The **global weight** denoting by $p = a_x p_x + a_y p_y + a_z p_z$, is in this case the (s,p)-relative weight:

$$w_{(s,p)} = \frac{w_s(h)}{w_x(h) + w_y(h) + w_z(h)} \qquad (1.30)$$

The important property of the relative weight is that there are **invariant** when the orbital is multiplied by a scalar $\mu \in \mathbb{R}$. This notion of weight for a given orbital can be extended to a finite family of orbitals
representing a particular local geometry.

The relative weight of a system of orbitals is given by:

$$w_{s,\star}(\mathfrak{h}) = \sum_{i=1}^{4} w_{s,\star}(h_i) \qquad (1.31)$$

with $\mathfrak{h} = (h_i)_{i=1,...,4}$ a family of orbitals, $(s, \star)$ the relative weight of $\mathfrak{h}$ and $\star \in \{x,y,z\}$.

Still based on the work of R. C. Haddon and L. T. Scott [23], we are interested in **hybridization numbers** that determine the weight of each orbital in a molecular system.

We define:

$$m = \frac{1}{\tilde{\lambda}_{\pi}^{2}} \qquad (1.32)$$

and

$$n = \tilde{\lambda}_{1}^{2} + \tilde{\lambda}_{2}^{2} + \tilde{\lambda}_{3}^{2} \qquad (1.33)$$

with $\tilde{\lambda}^2 = \frac{\lambda_{\pi}^2}{c_{\pi}^2}$.

We have the set of orbitals:

$$h_{\pi} = N_{\pi}(s + \tilde{\lambda}_{\pi}p_{\pi}) \qquad (1.34)$$

$$h_i = N_i(s + \lambda_i p_i) \qquad (1.35)$$

where $\| p_i \| = 1$ for $i = 1,2,3$.

Finally, the **hybridization numbers** **m** and **n** correspond to the relative weight of orbitals:

$$m = w_{s,z}(h_{\pi}) = \frac{w_{s}(h_{\pi})}{w_{z}(h_{\pi})} = \frac{1}{\tilde{\lambda}_{\pi}^{2}} = \frac{c_{\pi}^{2}}{\tilde{\lambda}_{\pi}^{2}} \qquad (1.36)$$

$$n = w_{s,z}(\mathfrak{h}_{\sigma}) + w_{x}(\mathfrak{h}_{\sigma}) + w_{y}(\mathfrak{h}_{\sigma}) \qquad (1.37)$$

$$m = \frac{2 \tan^2(Pyr(A))}{1 - 2 \tan^2(Pyr(A))} = \frac{c_{\pi}^2}{\tilde{\lambda}_{\pi}^2} \qquad (1.38)$$

and,

$$w_x(\mathfrak{h}_\sigma) = w_y(\mathfrak{h}_\sigma) = 1, \qquad (1.39)$$

then,

$$n = 3m + 2 \qquad (1.40)$$

From a computational point of view, the hybridization numbers can be used to describe the **hybridization** of the **σ-orbital**:

$$s^{\lambda_{\pi}^2} p_{x,y}^{n-3m} p_{z}^{c_{\pi}^2} \qquad (1.41)$$

and the **π-orbitals** hybridization:

$$s^{c_{\pi}^2} p_z^{\lambda_{\pi}^2} \qquad (1.42)$$

We are interested in the 2 classical cases of sp² and sp³ **hybridizations**.
In the sp² case we have:

$$(sp_{x,y}^2 p_z^0)_\sigma(s^0 p_z^1)_\pi = (sp^2)_\sigma(p_z^1)_\pi$$

And in the sp³ case, we have $\lambda_\pi = 0.866$ and $c_\pi = \frac{1}{2}$:

$$(s^{0.75} p_{x,y}^2 p_z^{0.75})_\sigma = (s^{0.75} p^{2.25})_\sigma (s^{0.25} p^{0.75})_\pi$$

To be more applicative, we will show in the continuation of the manuscript the illustrations of all the parameters described on fullerenes and non-fullerenic molecules.

From Robert C. Haddon's point of view [29], the **hybridization** of the $\pi$-orbitals will depend on the parameter m and the proportion of the weight of the s orbital to the $p_z$ orbital as:

$$s_{\pi}^{c_{\pi}^2} p_z^{\lambda_{\pi}^2} \iff s_{\lambda_{\pi}^2}^{\frac{c_{\pi}^2}{2}} p_z \quad (1.43)$$

If we consider the equation 1.38, again we find the parameter m: $s^m p$

In the case of **hybridization** of $\sigma$-orbitals, we fix $\tilde{n}$:

$$\tilde{n} = \frac{2}{\lambda_{\pi}^2} + m \quad (1.44)$$

Then, we have:

$$\tilde{n} = 3m + 2 \quad (1.45)$$

Thus, considering this new parameter, the average **hybridization** of the $\sigma$-orbitals is given by $sp^{\tilde{n}}$

We also define another parameter **Y** called **sigma hybridization** in relation to m which indicates the **deviation** of the conjugation of a system but also the contribution of the hybridization of the $p_z$ orbital in each $h_i$ orbital which will take into account the topology of the system such as the *Pyr(A)*:

$$Y = m \frac{\lambda_{\pi}^2}{3} = \frac{c_{\pi}^2}{3} \quad (1.46)$$

It is therefore possible to rewrite the hybridization of a system according to this new parameter, which will not change anything from a chemical point of view.

## 1.3 Limitations of the pyramidalization angle and the POAV1

As we have seen previously, the **pyramidalization angle** and the **POAV1** are closely related. The definition of the *Pyr* is connected to the existence of the POAV1 vector. Indeed, both are determined via an atom considered as a vertex in the molecule.

If we consider the case of the $C_{60}$ fullerene proposed by R. C. Haddon [3], he places his work in a particular and well defined situation where it is admitted that the atoms are all identical, the bond lengths and the angles equal. However, it is obvious that the reality is different. We have therefore seen that the *Pyr* could not be defined for arbitrary molecules.

More precisely, the pyramidalization angle can not be extend outside the family of **admissible molecules**.
The existence of a POAV1(A) vector implies that POAV1(A) has to be normal to any plane $P_ε(A)$. In fact, if POAV1(A) exists and is different from $n_A$, the unique unitary vector to all plane $P_ε(A)$, then it makes a given angle $α > 0$ with $n_A$. The POAV1(A) vector then generate a cone based in A by rotation around the axes $n_A$ corresponding to all the vectors $v$ such that $v$ makes an angle $α$ with the normal axe. As a consequence, for each bond corresponds a different vector, except when $α = 0$ corresponding to the normal vector (figure 1.5).

<figure><img src="image_11.png" /><figcaption>Figure 1.5: Representation of the pyramidalization angle [4].</figcaption></figure>

From now, we assume that POAV1(A) is normal to any $P_ε(A)$. For each triple choice T of elements in $Reg_ε(star(A))$, the explicit construction of the POAV1(A) vector (section 1.1.1) gives a unique vector POAV1(A) which is colinear to the vector $IA$ where $I_T$ is the intersection of the mediating line in the triangle defined by T. In general, other choice of triples will lead to different $I_T$. As A must belong to each line perpendicular to $P_ε(A)$ and passing trough $I_T$ for each T, this is not possible in general unless all the $I_T$ coincide, meaning that all the atoms of $Reg_ε(★(A))$ belong to a circle.

The two characteristics correspond to the definition of **admissible molecules**.

As a consequence, one must think to other characterizations of the local geometry of a molecule in order to study molecules of order $≥ 4$ in some atoms.

## Admissible molecules

We define an **admissible molecule** as:

a molecule M is said to be admissible if for each atom A of M and $ε > 0$, there exists a plane $P_ε(A)$ such as $Reg_ε(★(A)) ⊂ P_ε(A)$.

The geometry of the previous conditions is resumed in figure 1.6. If the atom A does not belong to the same plane as the atoms of $Reg_ε(★(A))$, then it belongs to the intersection of all the mediating plane of the line segment between two arbitrary atoms of $Reg_ε(★(A))$.
<figure><img src="image_12.png" /><figcaption>Figure 1.6: Geometry of an admissible molecule [4].</figcaption></figure>

As the intersection of two planes define a line, it gives huge constraints on the positioning of each atom of $Reg_\epsilon(\star(A))$, in particular, they belong to a circle $C_\epsilon$ in the plane $P_\epsilon(A)$. If we denote by $O_\epsilon$ the center of $C_\epsilon$, then A belongs to the line passing through $O_\epsilon$ and normal to $P_\epsilon(A)$.

The previous remark can be used to define the POAV1(A) vector in this general situation : let M be an admissible molecule and A be an atom of M. Let $\epsilon > 0$ be given. We denote by $C_\epsilon$ the circle in $P_\epsilon(A)$ such that $Reg_\epsilon(\star(A)) \subset C_\epsilon$. We denote by $O_\epsilon$ the center of $C_\epsilon$. We denote by POAV1(A) the normal unitary vector to $P_\epsilon(A)$ defined by:

$$POAV1(A) = \frac{\vec{O_\epsilon A}}{O_\epsilon A} \qquad (1.47)$$

As usual, the POAV1 vector is defined through quantities depending on $\epsilon$ so that it is not a priori trivial that it gives a well-defined quantity. However, by construction, the set of points $Reg_\epsilon(\star(A))$ and $Reg_\epsilon'(\star(A))$ for two different $\epsilon$, and $\epsilon'$ are homothetics as $Reg_\epsilon$ is a homothety of center A for all $\epsilon$. In particular, homotheties are affine transformations that send a line to a parallel line.

As $Reg_\epsilon(\star(A))$ belongs to a plane $P_\epsilon(A)$, $Reg_\epsilon'(\star(A))$ belongs to a plane $P_\epsilon'(A)$ parallel to $P_\epsilon(A)$ and has the same normal vector. By normalization, we obtain a unique vector POAV1(A) normal to all the plane $P_\epsilon(A)$ for $\epsilon > 0$.

Having the POAV1 vector, we can directly generalize the notion of *Pyr* as: let M be an admissible molecule and A be a given atom of M. The *Pyr* in A denoted by *Pyr(A)* is the angle between the vector POAV1(A) and each bond AB, $B \in \star(A)$.

*The proofs of this definition is given in Appendix 2 section 7.*

# 1.4 Haddon's spherical Curvature

We note that the term **curvature** is well-defined in classical differential geometry for smooth (continuous) surfaces. On the other hand, in the discrete case, i.e. for polyhedral surfaces, the definition is more problematic. That is why, we are interested in the work of R. C. Haddon [3], especially concerning the **spherical curvature** which is related to the **pyramidalization angle**.

## 1.4.1 Spherical curvature

To represent the shape of a molecule, the notions of *Pyr* and curvature are often **confused**, that is
why R. C. Haddon expresses in his work [3] the notion of **spherical curvature**.

The simple notion of **curvature** is given for a smooth curve in $\mathbb{R}$ and related to the osculating circle. The curvature is given by the parameter $\frac{1}{R}$ where R is the radius of the **osculating circle**. The objective is to approximate locally up to order 2 the curve by an arc of a circle. In $\mathbb{R}^3$ it is necessary to approximate the surface by a portion of sphere but it is not so simple.

If we propose a definition:

for an osculating sphere, let M be a regular molecule trivalent. For each atom A, there exists a unique sphere $S(A)$ such as $A \in S(A)$ and for all $B \in \star(A)$, $B \in S(A)$. The sphere $S(A)$ is named the **osculating sphere** to M in A, the following figure 1.7 represents the notion. Thus, by definition, the **spherical curvature** is the quantity $\kappa(A)$:

$$ \kappa(A) = \frac{1}{R_A} \qquad (1.48) $$

where $R_A$ is the radius of the osculating sphere to M in A.

<figure><img src="image_13.png" /><figcaption>Figure 1.7: Representation of the osculating sphere and the spherical curvature [4]</figcaption></figure>

We can therefore relate the spherical curvature to the *Pyr* by a **non-linear** relationship:

$$ \kappa(A) = \frac{2 \sin(Pyr(A))}{a} \qquad (1.49) $$

The proofs of this relation is given in Appendix 2 section 8.

We have then, when the *Pyr* is small the previous relation which became **linear**:

$$ \kappa(A) \approx \frac{2}{a} Pyr(A) \qquad (1.50) $$

### 1.4.2 Spherical curvature in non regular case

We can compute the spherical curvature in more **general cases**. The following figure describes the situation:
<figure><img src="image_14.png" /><figcaption>Figure 1.8: Spherical curvature in non-regular case [4]</figcaption></figure>

In the case presented here, the osculating sphere is of center $O_z$ ($O_z = (0, 0, z)$) such as:

$$z = \frac{L^2 + z_A^2 - l^2}{2z_a} \qquad (1.51)$$

where $z_A$ is the coordinate of A, $l = OB = OC = OD$ and $L = O_z A$.

Therefore, the spherical curvature is given by:

$$\kappa(A) = \frac{1}{\sqrt{l^2 + \frac{(L^2 + z_A^2 - l^2)^2}{4z_A^2}}} = \frac{1}{\sqrt{l^2 + \frac{(OA^2 - l^2)^2}{4z_A^2}}} \qquad (1.52)$$

The proofs of the relation in given in Appendix 2 section 9.

The spherical curvature gives access to the **non-planarity** of a molecule. However, we are always in the case of trivalent molecules.

### 1.4.3 Limitations of the spherical curvature

In the classical trivalent case, the **spherical curvature** provides a nice way to characterize the **non-planarity** of a given molecule. However, the generalization to molecules which are not trivalent leads to severe difficulties and is in general **impossible**.

If we consider a molecule which has atoms of order 4. Let A be an atom of this type. Then $\star(A)$ is made of 4 points B, C, D and E. In order to generalize the notion of spherical curvature, one has to construct a sphere interpolating the 5 points. The geometric construction of the osculating sphere made in section 1.4.2 for trivalent molecules determine a unique sphere interpolating 4 atoms. As a consequence, selecting any triple of points in $\star(A)$ denoted by T, one construct a unique **osculating sphere** in A with respect to T denoted by $S(A)_T$. In general, all these spheres are different and there exists no osculating sphere to the molecule in A.

Moreover, the spherical curvature describes the surface locally as a **dome** i.e a portion of a sphere. By consequence, this quantity is **not** sufficient to capture the **local geometry** of molecules that admit a
smooth representation, which have a negative **Gaussian curvature**, and that must be distinguished from a molecule with a positive Gaussian curvature.

## 1.5 Angular defect

The **angular defect** is a well-known notion in discrete differential geometry. This notion is always definable whereas the *Pyr* and the spherical curvature are not necessarily definable.

The angular defect is useful because it makes it possible to compute the **local shape** of a molecule.

To begin, we focus on the **Gaussian curvature** which can be defined as: a circle of radius *r* around *p*, which implies that for a given metric *d* on Σ, and a set of point *x* ∈ Σ as *d(x, p) = r* and *P(r)* the circumference of the circle.

*Note for more details that in terms of discrete curvature, the Gaussian curvature defines the local curvature for any type of polyhedron with a number of neighbors superior to 3. It gives a measure of how far the surface deviates from a plane surface at a vertex.*

To determine the Gaussian curvature *G(p)*, we need to compare the circumference of a circle in the plane given by 2π*r* to the value of *P* on Σ.

The **Gaussian curvature** is given as:

$$P(r) = 2\pi r - G(p)\pi \frac{r^3}{3} + \dots \qquad (1.53)$$

We consider the following quantity *K(p)* to generalize what is mentioned above:

$$K(p) = 2\pi - \frac{P(r)}{r} \qquad (1.54)$$

Related to the Gaussian curvature, the **angular defect** is defined as:

$$K(A) = 2\pi - \sum_{T \in \mathcal{T}(A)} \alpha_T \qquad (1.55)$$

where $\alpha_T$ is the angle at the vertex A of the face $T \in \mathcal{T}(A)$, $\mathcal{T}(A)$ is the family of triangles $T_i$ obtained by connecting the vertices in $\star(A)$ of two consecutive edges of A.

In reference to Romon's work [27], we know that the angular defect is related to the Gaussian curvature. Note that $K(A) = 0$ if and only if the surface is plane locally at a point A. If $K(A) < 2\pi$ the vertex A is spherical.

As for the other concepts presented in this manuscript, we show the relation between the **angular defect** and the *Pyr*.

Once again in the case of a **regular** molecule M and with *l* the length of the bond. We consider A an atom of the molecule, a triangle $AB_1B_2$ (where $B_1$ and $B_2 \in \star(A)$) and $\alpha_1(A)$, $\alpha_2(A)$, $\alpha_3(A)$ respectively the angles at the vertex A of triangles $AB_1B_2$, $B_2AB_3$ and $B_3AB_1$.

We have by definition:

$$\alpha_1(A) + \alpha_2(A) + \alpha_3(A) = 2\pi - K(A) \qquad (1.56)$$
And, the relationship between the *Pyr* and the angular defect is given by:

$$ \cos(Pyr(A)) \sum_{i=1}^{3} \sin\left(\frac{\theta_i(A)}{2}\right) = \sum_{i=1}^{3} \sin\left(\frac{\alpha_i(A)}{2}\right) \qquad (1.57) $$

$$ \cos \alpha_F(A) = \cos^2 Pyr(A) \cos \theta_F(A) + \sin^2 Pyr(A) \qquad (1.58) $$

and

$$ \theta_1(A) + \theta_2(A) + \theta_3(A) = 2\pi \qquad (1.59) $$

with $(\alpha_i(A))_{i=1,2,3}$, $(\theta_i(A))_{i=1,2,3}$ and we note by $\mathcal{P}(A)$ the plane defined by $\star(A)$, by $O$ the intersection of the median line of the triangle defined by $\star(A)$, by $\theta_1(A)$ the angle at vertex $O$ of the triangle $B_1OB_2$, and similarly by $\theta_2(A)$ and $\theta_3(A)$ the angle at vertex $O$ of the triangle $B_2OB_3$ and $B_3OB_1$.

The proofs of this relationship is given in Appendix 2 section 10.

To complete these relations, we know that: $\alpha_F(A) = \theta_F(A)$ if and only if $Pyr = 0$. Furthermore, we show the relationship between $\kappa(A)$ and $Pyr$ (eq. 1.57). The connection between the two notions is quite complex, using the **Taylor expansion** up to order 3, we can have an approximation of the relation:

$$ \cos(Pyr(A)) \left( \pi - \frac{1}{48} \sum_{i=1}^{3} \theta_i(A)^3 \right) \approx \frac{1}{2} K(A) - \frac{1}{48} \sum_{i=1}^{3} \alpha_i(A)^3 \qquad (1.60) $$

## 1.6 Conclusion

This chapter provides a **complete** presentation (detailed, clarified, generalized and proven) of tools stated in many works of R. C. Haddon. We present the tools of **pyramidalization angle** and **POAV1** first in the **regular** (perfect) case which is limited to **trivalent** cases which is not necessarily realistic and rather limiting. And then, in the **non-regular** case (i.e. with different bond lengths) but still trivalent. It opens to more cases but still remains limiting. The **regularization** method gives the possibility to generalise a bit more.

The *Pyr* is a **geometrical** tool and the POAV1 gives access to the privileged direction of the $\pi$-orbitals within the system, which gives an indication on the probable deformation of the system, on the **local distortion**, hence the relation with chemistry and more particularly with **reactivity**.

We associate these tools with **hybridization**, it gives the characterization of $\sigma$ and $\pi$-hybrid orbitals according to orthogonalization and normalization conditions and to highlight the intermediate hybridization between $sp^2$ and $sp^3$.

The study being complete: the **relative weights** and **hybridization numbers** (m and n) are described which give the composition of the hybrid orbitals involved in the planar or deformed systems. Finally, the **sigma hybridization** which is introduced, gives indications on the conjugation deviation of the system.

Obviously, these tools have some **limitations**, in particular we have defined the **admissible molecules** where these tools are clearly applicable.
These notions are associated with deformation, curvature, and in particular with **spherical curvature** which we prove in the non-regular cases which are the most close to reality but not totally since it remains in trivalent cases.

Finally, we showed that the **angular defect** tool is more universal since it is definable in most cases but has meaning only when it is associated with the *Pyr*. In effect, the *Pyr* gives access to the POAV1 which best represents the local reactivity.

All these tools provide access to information on the characterization of systems by their **deformation** and **reactivity**.
<figure><img src="image_15.png" /></figure>
Chapter 2

Study of POAV1 over fullerenic and
non-fullerenic compounds

In this second chapter, we expose how the different tools presented in the first chapter are used to carry
out a complete and detailed study on various compounds:

1. fullerenic molecules,

2. non-fullerenic molecules.

We also present how the tools are used for an application on a reaction path of a ring expansion of a
nitrile imine.

Then, a complete conclusion about these studies is given to explain more clearly the results.

To obtain our results we used a software "Pychemcurv" especially developed for the project thesis [10].

We used data from the database of Tománek and Frederick [5], and results from our first paper [4], but also results from publications of R. C. Haddon, A. R. Khamatgalimov et al. and C. Wentrup and al. [23, 6, 30, 31, 32].

**2.1 Methods of implementation of the concepts**

To apply the notions developed in chapter 1, we used a software developed by our research team at the Université de Pau et des Pays de l'Adour (UPPA) in the Institut des Sciences Analytiques et de Physico-Chimie pour l'Environnement et les Matériaux (IPREM) and the Institut Pluridisciplinaire de Recherches Appliquées (IPRA), under the supervision of Dr Germain Salvato-Vallverdu.

This collaboration with Dr Germain Salvato-Vallverdu allowed the effective implementation of the
POAV1 theory concepts, on a set of fullerenic or more non-fullerenic molecules.

The **software** developed called *Pychemcurv* [10] is a python program for structural analyzes of
molecular systems or solid state materials focusing on the **local curvature** at an atomic scale. The
hybridization of molecular orbitals are obtained by the computation of the local curvature (calculated
and mapped data). The software is easy to use since only the **cartesian** coordinates (`.xyz` file) of
the systems are required. The program is therefore applicable to most molecular systems. When the
`.xyz` files are loaded into the program, it gives access to all the local geometrical and hybridization
properties : *Pyr*, spherical curvature, angular defect, $c_\pi^2$, $\lambda_\pi^2$, *m*, *n*, hybridization, hybridization sigma.
Note that the website dedicated to the software is available, which includes all the parameters accessible
in an explicative way [10].

In the rest of the chapter we illustrate our results obtained from the software.

Notice that all the computational techniques used to carry out our results are given in Appendix 1.

## 2.2 POAV1 analysis of the Tománek and Frederick database

Following our interest in the work of R. C. Haddon [3] on the $C_{60}$ fullerene, we chose to work on fullerenic molecular systems. We worked on a **set of molecules** without limiting only on the typical $C_{60}$ fullerene.

To apply the different notions associated to POAV1, we used a very extensive **database** of $C_n$ fullerenes [5] (with $n$ being the number of atoms in the molecule). The database provides the cartesian coordinates and the energy (in electronvolts eV) of the fullerenes admissible from $C_{20}$ to $C_{720}$ according to their symmetries. The database is developed by Tománek and Frederick [5].

We remind that the associated energies of the database are defined according to the work of Guan et al. [33].

The database shows **three** families of fullerenes that we call: **small** (between the $C_{20}$ and $C_{60}$), **medium** (between $C_{60}$ and $C_{100}$) and **large** (between $C_{100}$ and $C_{720}$).

We have focused our analysis on different fullerenes including the $C_{32}$, $C_{40}$, $C_{60}$, and $C_{80}$ which are part of the **small** and **medium** fullerenes family.

### 2.2.1 Database analysis

The database of Tománek and Frederick [5] presents **2487** isomers of fullerenes according to their symmetries. The following figure represents the set of isomers of the database:

<figure><img src="image_16.png" /><figcaption>Figure 2.1: Representation of the number of atoms in function of the number of isomers in fullerenes [4]</figcaption></figure>

We observe that $C_{100}$ and $C_{52}$ have the highest number of isomers followed by $C_{50}$, $C_{98}$, $C_{46}$, $C_{94}$.
On the contrary, the $C_{20}$, $C_{24}$, $C_{26}$, $C_{60}$, $C_{70}$, $C_{72}$, $C_{74}$, $C_{180}$, $C_{240}$, $C_{260}$, $C_{320}$, $C_{500}$, $C_{540}$, $C_{720}$ present only one type of symmetry.

### 2.2.2 Analysis following the POAV1 notions

From a general point of view on the **complete database**, we note for our average parameters : a *Pyr* of **10.49°**, an angular defect (*Ang Def*) of **10.42°**, a spherical curvature (*Sph Curv*) of **0.262**, the hybridization coefficients $c_\pi^2$ and $\lambda_\pi^2$ are respectively of **0.074** and **0.926** and the hybridization numbers *m* and *n* are of **0.084** and **2.252**.

It allows to determine the hybridization (*Hyb*) as: $(s^{0.926}p_{x,y}^{2.000}p_z^{0.074})_\sigma (s^{0.074}p_z^{0.926})_\pi$.

As mentioned in the section 0.3 and in the Appendix 1, we compute all the following results by using **DFT**, implemented in the Gaussian code with the Hamiltonian type B3LYP with the orbital base 6-31G*.

The following tables display all the **average parameters** (after geometrical optimization) that we have calculated for the fullerenes with only one symmetry.

Firstly for the **large** family of the database:

<table><caption>Table 2.1: Average descriptor parameters for the large fullerene family</caption><thead><tr><th>Fullerene</th><th>Pyr(A)</th><th>Ang Def</th><th>Sph Curv</th><th>c<sup>2</sup><sub>π</sub></th><th>λ<sup>2</sup><sub>π</sub></th><th>m</th><th>n</th><th>Hybridization</th></tr></thead><tbody><tr><td>C<sub>180</sub></td><td>6.68</td><td>4.03</td><td>0.167</td><td>0.027</td><td>0.972</td><td>0.028</td><td>2.084</td><td>(s<sup>0.0972</sup>p<sup>2.000</sup><sub>x,y</sub>p<sup>0.027</sup><sub>z</sub>)<sub>σ</sub> (s<sup>0.027</sup>p<sup>0.972</sup><sub>z</sub>)<sub>π</sub></td></tr><tr><td>C<sub>240</sub></td><td>5.78</td><td>3.03</td><td>0.145</td><td>0.021</td><td>0.979</td><td>0.021</td><td>2.063</td><td>(s<sup>0.979</sup>p<sup>2.000</sup><sub>x,y</sub>p<sup>0.021</sup><sub>z</sub>)<sub>σ</sub> (s<sup>0.021</sup>p<sup>0.979</sup><sub>z</sub>)<sub>π</sub></td></tr><tr><td>C<sub>260</sub></td><td>5.56</td><td>2.80</td><td>0.140</td><td>0.019</td><td>0.981</td><td>0.019</td><td>2.058</td><td>(s<sup>0.981</sup>p<sup>2.001</sup><sub>x,y</sub>p<sup>0.019</sup><sub>z</sub>)<sub>σ</sub> (s<sup>0.019</sup>p<sup>0.981</sup><sub>z</sub>)<sub>π</sub></td></tr><tr><td>C<sub>320</sub></td><td>5.02</td><td>2.29</td><td>0.126</td><td>0.015</td><td>0.985</td><td>0.016</td><td>2.047</td><td>(s<sup>0.985</sup>p<sup>1.999</sup><sub>x,y</sub>p<sup>0.015</sup><sub>z</sub>)<sub>σ</sub> (s<sup>0.015</sup>p<sup>0.985</sup><sub>z</sub>)<sub>π</sub></td></tr><tr><td>C<sub>500</sub></td><td>4.03</td><td>1.49</td><td>0.101</td><td>0.010</td><td>0.990</td><td>0.010</td><td>2.030</td><td>(s<sup>0.990</sup>p<sup>2.000</sup><sub>x,y</sub>p<sup>0.010</sup><sub>z</sub>)<sub>σ</sub> (s<sup>0.010</sup>p<sup>0.990</sup><sub>z</sub>)<sub>π</sub></td></tr><tr><td>C<sub>540</sub></td><td>3.89</td><td>1.38</td><td>0.098</td><td>0.009</td><td>0.991</td><td>0.009</td><td>2.028</td><td>(s<sup>0.991</sup>p<sup>2.001</sup><sub>x,y</sub>p<sup>0.009</sup><sub>z</sub>)<sub>σ</sub> (s<sup>0.009</sup>p<sup>0.991</sup><sub>z</sub>)<sub>π</sub></td></tr><tr><td>C<sub>720</sub></td><td>3.38</td><td>1.07</td><td>0.086</td><td>0.007</td><td>0.993</td><td>0.007</td><td>2.021</td><td>(s<sup>0.993</sup>p<sup>2.000</sup><sub>x,y</sub>p<sup>0.007</sup><sub>z</sub>)<sub>σ</sub> (s<sup>0.007</sup>p<sup>0.993</sup><sub>z</sub>)<sub>π</sub></td></tr></tbody></table>

For the **medium** family:

<table><caption>Table 2.2: Average descriptor parameters for a selection of the medium fullerene family</caption><thead><tr><th>Fullerene</th><th>Pyr(A)</th><th>Ang Def</th><th>Sph Curv</th><th>c<sup>2</sup><sub>π</sub></th><th>λ<sup>2</sup><sub>π</sub></th><th>m</th><th>n</th><th>Hybridization</th></tr></thead><tbody><tr><td>C<sub>70</sub></td><td>10.79</td><td>10.42</td><td>0.270</td><td>0.073</td><td>0.927</td><td>0.080</td><td>2.238</td><td>(s<sup>0.927</sup>p<sup>1.998</sup><sub>x,y</sub>p<sup>0.073</sup><sub>z</sub>)<sub>σ</sub> (s<sup>0.073</sup>p<sup>0.927</sup><sub>z</sub>)<sub>π</sub></td></tr><tr><td>C<sub>72</sub></td><td>10.70</td><td>10.38</td><td>0.268</td><td>0.073</td><td>0.927</td><td>0.080</td><td>2.239</td><td>(s<sup>0.927</sup>p<sup>1.999</sup><sub>x,y</sub>p<sup>0.073</sup><sub>z</sub>)<sub>σ</sub> (s<sup>0.073</sup>p<sup>0.927</sup><sub>z</sub>)<sub>π</sub></td></tr><tr><td>C<sub>74</sub></td><td>10.48</td><td>9.81</td><td>0.262</td><td>0.069</td><td>0.931</td><td>0.074</td><td>2.222</td><td>(s<sup>0.931</sup>p<sup>2.000</sup><sub>x,y</sub>p<sup>0.069</sup><sub>z</sub>)<sub>σ</sub> (s<sup>0.069</sup>p<sup>0.931</sup><sub>z</sub>)<sub>π</sub></td></tr><tr><td>C<sub>80</sub></td><td>10.16</td><td>9.40</td><td>0.246</td><td>0.066</td><td>0.934</td><td>0.071</td><td>2.213</td><td>(s<sup>0.934</sup>p<sup>2.000</sup><sub>x,y</sub>p<sup>0.066</sup><sub>z</sub>)<sub>σ</sub> (s<sup>0.066</sup>p<sup>0.934</sup><sub>z</sub>)<sub>π</sub></td></tr></tbody></table>
And for the **small** family:

<table><caption>Table 2.3: Average descriptor parameters for a selection of the small fullerene family</caption><thead><tr><th>Fullerene</th><th>Pyr(A)</th><th>Ang Def</th><th>Sph Curv</th><th>c<sup>2</sup><sub>π</sub></th><th>λ<sup>2</sup><sub>π</sub></th><th>m</th><th>n</th><th>Hybridization</th></tr></thead><tbody><tr><td>C<sub>20</sub></td><td>20.90</td><td>36.01</td><td>0.513</td><td>0.292</td><td>0.708</td><td>0.413</td><td>3.238</td><td>(s<sup>0.708</sup>p<sup>1.999</sup><sub>x,y</sub>p<sup>0.292</sup><sub>z</sub>)<sub>σ</sub> (s<sup>0.292</sup>p<sup>0.708</sup><sub>z</sub>)<sub>π</sub></td></tr><tr><td>C<sub>24</sub></td><td>18.90</td><td>30.77</td><td>0.465</td><td>0.244</td><td>0.756</td><td>0.340</td><td>3.021</td><td>(s<sup>0.756</sup>p<sup>2.001</sup><sub>x,y</sub>p<sup>0.244</sup><sub>z</sub>)<sub>σ</sub> (s<sup>0.244</sup>p<sup>0.756</sup><sub>z</sub>)<sub>π</sub></td></tr><tr><td>C<sub>26</sub></td><td>18.13</td><td>28.13</td><td>0.447</td><td>0.218</td><td>0.782</td><td>0.286</td><td>2.857</td><td>(s<sup>0.782</sup>p<sup>1.999</sup><sub>x,y</sub>p<sup>0.218</sup><sub>z</sub>)<sub>σ</sub> (s<sup>0.218</sup>p<sup>0.782</sup><sub>z</sub>)<sub>π</sub></td></tr><tr><td>C<sub>40</sub></td><td>14.73</td><td>19.45</td><td>0.352</td><td>0.145</td><td>0.855</td><td>0.177</td><td>2.531</td><td>(s<sup>0.855</sup>p<sup>2.000</sup><sub>x,y</sub>p<sup>0.145</sup><sub>z</sub>)<sub>σ</sub> (s<sup>0.145</sup>p<sup>0.855</sup><sub>z</sub>)<sub>π</sub></td></tr><tr><td>C<sub>60</sub></td><td>11.64</td><td>12.00</td><td>0.281</td><td>0.085</td><td>0.915</td><td>0.093</td><td>2.278</td><td>(s<sup>0.915</sup>p<sup>1.999</sup><sub>x,y</sub>p<sup>0.085</sup><sub>z</sub>)<sub>σ</sub> (s<sup>0.085</sup>p<sup>0.915</sup><sub>z</sub>)<sub>π</sub></td></tr></tbody></table>

**Pyramidalization angle:** On the entire database, the **average** *Pyr* is 10.49°. We notice that the *Pyr* varies between 3.38° and 20.90°.

The largest *Pyr* values are measured in the case of **small** fullerenes, for *n* < 60.

Actually, in the case of small fullerenes the constraints are more important which induces a more important curvature on the different vertices.

We note a clear difference between the family of large and medium fullerenes.

In the case of **large** fullerenes the *Pyr* values are lower and not close to the average value of *C*<sub>60</sub> which can be used as a reference. These large fullerenes are less constrained and therefore have fewer vertices with strong pyramidalization.

The following figure displays the distribution of the *Pyr* values as a function of the number of atoms present in the fullerenes (for *n* = 20 to 100 on the basis of the isomers). We note in dashed line the value of the *Pyr* in the case of *C*<sub>60</sub>, and the *sp*<sup>2</sup>, *sp*<sup>3</sup> hybridization situations.

<figure><img src="image_17.png" /><figcaption>Figure 2.2: Boxplot representation of the pyramidalization angle (in degrees) in function of the number of atoms [4]</figcaption></figure>

As mentioned by the graph, we observe that the *Pyr* angle decreases when the number of atoms increases. In the case of the family of small fullerenes there is a greater standard deviation of the *Pyr* than for the large family, this situation illustrates the presence of more isomers for the small fullerenes.
**Angular defect, spherical curvature and pyramidalization angle:** The angular defect follows the same behavior as the *Pyr*.

In the case of the **large** fullerene family, the *Ang Def* is relatively small and decreases with the increasing of the number of atoms.

This is also the case for the family of **medium** fullerenes.

For fullerenes with $n < 60$ the *Ang Def* is more important which is in accordance with the constraints imposed by the geometry of small fullerenes.

The **spherical curvature** shows the same trend and a **linear relationship** between the *Pyr* and the *Sph Curv* as previously proven in eq. 1.50.

**Hybridization and pyramidalization angle:** We observed that on the 3 families, the *Pyr* is positioned in the situations commonly encountered between $sp^2$ and $sp^3$.

But some fullerenes have a hybridization **beyond** $sp^3$ for *Pyr* > 19°.

### Fullerenes visualization-Cartographies

As presented in the previous section 2.1, the software *Pychemcurv* [10] provides cartographies with an adaptable color scale for each of the parameters.

We report here the **cartographies** that we carried out for the 3 families of fullerenes (small with $C_{20}$, medium with $C_{70}$ and large with $C_{180}$, $C_{60}$ is used as a reference) of the database. We observe the parameters presented in chapter 1: the pyramidalization angle, the angular defect, the spherical curvature and the hybridization.

<figure><img src="image_18.png" /><figcaption>Figure 2.3: Cartographies of the pyramidalization angle for the fullerenes $C_{20}$, $C_{60}$, $C_{70}$, $C_{180}$</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Figure 2.4: Cartographies of the angular defect for the fullerenes $C_{20}$, $C_{60}$, $C_{70}$, $C_{180}$</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Figure 2.5: Cartographies of the spherical curvature for the fullerenes $C_{20}$, $C_{60}$, $C_{70}$, $C_{180}$</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Figure 2.6: Cartographies of the hybridization for the fullerenes $C_{20}$, $C_{60}$, $C_{70}$, $C_{180}$</figcaption></figure>

## 2.3 Application to other compounds

In this section, we used **non-fullerenic molecules** proposed by Robert C. Haddon in two of his works from 1986 [6] and 1990 [30]. We also studied molecules of a recent publication of the research team of A. R. Khamatgalimov concerning the **fullerenes** $C_{50}$ [31]. We are also focusing on the interesting work of Han et al. [34] on **chlorofullerenes**.

In fact, all the POAV1 theory parameters presented in chapter 1 are applicable to both fullerenic and non-fullerenic molecules as we illustrate in the following by 25 examples.

The molecules have been studied following the same method, by DFT B3LYP/6-311G* calculations and visualizations of the properties of Robert C. Haddon's concepts as in the previous section.

*For more details on computation techniques see the Appendix 1 section 1.*

It should be noted that we are positioned in the so-called cases of the $\star(A)$, i.e. the atom we are interested in has **only 3 neighbors**. However, in some of the cases studied here we are in situations beyond 3 neighbors which does not define correctly the $\star(A)$. Our programs allow us to extract the data of the studied properties are based on 3 neighbors in the case of *Pyr*, hybridization and spherical curvature. Excluding the angular defect which permits to get free of the three neighbors situations.

### 2.3.1 Non-fullerenic molecules

*Note that a geometry optimization was necessary for each of the molecules and the visualizations were carried out by the software VESTA [35].*

*Notice that the number in brackets corresponds to each of the molecules in abscissa on the figure 2.57.*

* Molecule $C_{11}H_{10}$: 1,6-methano-[10]-annulene (n°1) [6]
The molecule presents 2 isomers. The first isomer (of *Cs* symmetry) has the following geometry (figure 2.7) including 2 bridged hexagons.

<figure><img src="image_22.png" /><figcaption>Figure 2.7: Representation of the molecule 1,6-methano-[10]-annulene ($C_{11}H_{10}$)</figcaption></figure>

<table><caption>Table 2.4: Set of minimum and maximum values of parameters of the 1,6-methano-[10]-annulene ($C_{11}H_{10}$, *Pyr*, *Hyb*, *AngDef*, *SphCur*)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td>-425.23</td><td></td></tr><tr><td>Pyr (°)</td><td>0.55</td><td>5.06</td></tr><tr><td>Hybridization</td><td>2.00</td><td>2.05</td></tr><tr><td>Angular defect (°)</td><td>-72.66</td><td>2.31</td></tr><tr><td>Spherical curvature</td><td>0.01</td><td>0.12</td></tr></tbody></table>

<figure><img src="image_23.png" /><figcaption>Figure 2.8: Cartographies of the parameters of 1,6-methano-[10]-annulene ($C_{11}H_{10}$)</figcaption></figure>

We notice that the *Pyr* are higher on the carbons at the junction between the 2 hexagons than on the carbons constituting the hexagons. The hybridization is uniform in the whole molecule around 2.00. The angular defect gives access to situations where 4 neighboring atoms are present. Here we observe especially a very negative value on the central atom. Finally, the spherical curvature is also more pronounced at the junctions between hexagons and relatively weaker on the carbons of the remaining skeleton.

* Molecule $C_{11}H_{10}$: 1,5-methano-[10]-annulene (n°2) [6]

The second isomer has the geometry illustrated in the figure 2.9, with 2 different "geometrical" patterns of the molecule 1,6-methano-[10]-annulene:

<figure><img src="image_24.png" /><figcaption>Figure 2.9: Representation of the molecule 1,5-methano-[10]-annulene ($C_{11}H_{10}$)</figcaption></figure>
<table><caption>Table 2.5: 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*)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-425.20</td></tr><tr><td>Pyr (°)</td><td>0.06</td><td>8.24</td></tr><tr><td>Hybridization</td><td>2.00</td><td>2.13</td></tr><tr><td>Angular defect (°)</td><td>-72.31</td><td>6.08</td></tr><tr><td>Spherical curvature</td><td>0.002</td><td>0.20</td></tr></tbody></table>

This isomer of *C*<sub>1</sub> geometry, has an energy of lower than the isomer 1,6-methano-[10]-annulene.

<figure><img src="image_25.png" /><figcaption>Figure 2.10: Cartographies of the parameters of 1,5-methano-[10]-annulene (*C*<sub>11</sub>*H*<sub>10</sub>)</figcaption></figure>

Once again here, we remark that the *Pyr* values are higher at the carbons involved in the bridge than in the rest of the molecular skeleton. The hybridization is on the entire molecule of *sp*<sup>2</sup> type. The angular defect and the spherical curvature follow the same trend as the previous properties.

* Molecule *C*<sub>11</sub>*H*<sub>8</sub>: 7bH-cyclopenta-[cd]-indene (n°3) [6]

Composed by 2 pentagons and 1 hexagon the molecule has a *C*<sub>1</sub> symmetry:

<figure><img src="image_26.png" /><figcaption>Figure 2.11: Representation of the molecule 7bH-cyclopenta-[cd]-indene (*C*<sub>11</sub>*H*<sub>8</sub>)</figcaption></figure>

<table><caption>Table 2.6: 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*)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-424.01</td></tr><tr><td>Pyr (°)</td><td>0.39</td><td>12.11</td></tr><tr><td>Hybridization</td><td>2.00</td><td>2.30</td></tr><tr><td>Angular defect (°)</td><td>-70.76</td><td>13.12</td></tr><tr><td>Spherical curvature</td><td>0.01</td><td>0.29</td></tr></tbody></table>
<figure><img src="image_27.png" /><figcaption>Figure 2.12: Cartographies of the parameters of 7bH-cyclopenta-[cd]-indene ($C_{11}H_8$)</figcaption></figure>

Let us note here that the values of *Pyr* are the most important in the regions of junctions between geometrical patterns i.e. where the molecule tends to bend. The hybridization values remain in the same range, however we notice an increase concerning the carbon where the *Pyr* is the highest. The angular defect is once again, higher at the carbon with the maximum of *Pyr*. The spherical curvature is more important at the pentagon-pentagon junction where the curvature seems to be more significant.

* Molecule $C_{16}H_{14}$: syn-1,6,8,13-bismethano-[14]-annulene (n°4) [6]

This molecule (of symmetry $C_1$) is constituted of 3 hexagonal patterns bridged together.

<figure><img src="image_28.png" /><figcaption>Figure 2.13: Representation of the molecule syn-1,6,8,13-bismethano-[14]-annulene ($C_{16}H_{14}$)</figcaption></figure>

<table><caption>Table 2.7: Set of minimum and maximum values of parameters of the molecule syn-1,6,8,13-bismethano-[14]-annulene ($C_{16}H_{14}$, *Pyr*, *Hyb*, *AngDef*, *SphCurv*)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-618.19</td></tr><tr><td>Pyr (°)</td><td>2.43</td><td>4.78</td></tr><tr><td>Hybridization</td><td>2.01</td><td>2.04</td></tr><tr><td>Angular defect (°)</td><td>-67.57</td><td>2.07</td></tr><tr><td>Spherical curvature</td><td>0.07</td><td>0.13</td></tr></tbody></table>

We observe here that the *Pyr* are more important at the extremities and in the core of the molecule and the hybridizations are relatively uniform of sp² type. The angular defect shows 2 trends. The skeleton as a whole has relatively small values between 0 and 2, and only the 2 atoms at the bridge vertices have much smaller values. The spherical curvature approximates the *Pyr* cartography in terms of the distribution of values within the molecule.
<figure><img src="image_29.png" /><figcaption>Figure 2.14: Cartographies of the parameters of syn-1,6,8,13-bismethano-[14]-annulene ($C_{16}H_{14}$)</figcaption></figure>

* Molecule $C_{20}H_{20}$: trans-15,16-diethyldihydropyrene (n°5) [6]

This molecule has of formula $C_{20}H_{20}$ presents 2 substituents $-CH_2 - CH_3$, and has a symmetry $C_1$.

<figure><img src="image_30.png" /><figcaption>Figure 2.15: Representation of the molecule trans-15,16-diethyldihydropyrene</figcaption></figure>

All the information of the different properties are available in the cartographies:

<table><caption>Table 2.8: Set of minimum and maximum values of parameters of the molecule *trans*-15,16-diethyldihydropyrene (*Pyr*, *Hyb*, *AngDef*, *SphCurv*)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-774.28</td></tr><tr><td>Pyr (°)</td><td>0.18</td><td>4.32</td></tr><tr><td>Hybridization</td><td>2.00</td><td>2.03</td></tr><tr><td>Angular defect (°)</td><td>-74.99</td><td>1.69</td></tr><tr><td>Spherical curvature</td><td>0.004</td><td>0.11</td></tr></tbody></table>

We note that the *Pyr* are relatively low on the whole molecule. The hybridizations are close to 2.00 so of sp²-type. The angular defect shows 2 trends with values close to 1 within the hexagonal units and very negative values on the atoms bonded to substituents. The spherical curvature is relatively low.

<figure><img src="image_31.png" /><figcaption>Figure 2.16: Cartographies of the parameters of trans-15,16-diethyldihydropyrene</figcaption></figure>
* Molecule $C_7H_{10}$: bicyclo-[2.2.1]-hept-2-ene (n°6) [6]

<figure><img src="image_32.png" /><figcaption>Figure 2.17: Representation of the molecule bicyclo-[2.2.1]-hept-2-ene</figcaption></figure>

For this molecule of symmetry $C_1$ we get the following data:

<table><caption>Table 2.9: Set of minimum and maximum values of parameters of the bicyclo-[2.2.1]-hept-2-ene (*Pyr*, *Hyb*, *AngDef*, *SphCurv*)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-272.72</td></tr><tr><td>Pyr (°)</td><td>14.21</td><td>23.07</td></tr><tr><td>Hybridization</td><td>2.44</td><td>3.71</td></tr><tr><td>Angular defect (°)</td><td>-75.02</td><td>49.28</td></tr><tr><td>Spherical curvature</td><td>0.38</td><td>0.53</td></tr></tbody></table>

This molecule has 4 neighbors atoms and only 2 trivalent atoms where our parameters are computable. The 2 calculated *Pyr* have quite high values and the hybridizations show situations higher than $sp^2$ and $sp^3$. The computable angular defects are negative on all atoms except the 2 atoms which are trivalent.

<figure><img src="image_33.png" /><figcaption>Figure 2.18: Cartographies of the parameters of bicyclo-[2.2.1]-hept-2-ene</figcaption></figure>

* Molecule $C_{12}H_{16}$: Tricyclo[4.2.2.2$^{2.5}$]dodecane (n°7) [30]

The following molecule has a $D_{2h}$ geometry:

<figure><img src="image_34.png" /><figcaption>Figure 2.19: Representation of the molecule $C_{12}H_{16}$</figcaption></figure>
For all the properties calculated next, we will note that only 2 values emerge, we thus identify carbons in the center of the molecule and the 4 other carbons of the extremities.

<table><caption>Table 2.10: Set of minimum and maximum values of parameters of the molecule $C_{12}H_{16}$ (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-466.93</td></tr><tr><td>Pyr (°)</td><td>11.17</td><td>-</td></tr><tr><td>Hybridization</td><td>2.25</td><td>-</td></tr><tr><td>Angular defect (°)</td><td>-75.02</td><td>11.06</td></tr><tr><td>Spherical curvature</td><td>0.26</td><td>-</td></tr></tbody></table>

A unique value of *Pyr* and spherical curvature in this molecule 11.17° and 0.26. The hybridization indicates a single carbon type with a *sp*² tendency. The angular defect is negative on the external carbons and positive on the central atoms.

<figure><img src="image_35.png" /><figcaption>Figure 2.20: Cartographies of the parameters of Tricyclo[4.2.2.2².⁵]dodecane</figcaption></figure>

* Molecule $C_{28}H_{16}$: 9,9',10,10'-tetrahydrodianthracene (n°8) [30]

<figure><img src="image_36.png" /><figcaption>Figure 2.21: Representation of the molecule $C_{28}H_{16}$</figcaption></figure>

<table><caption>Table 2.11: Set of minimum and maximum values of parameters of 9,9',10,10'-tetrahydrodianthracene (Pyra, Hyb, AngDef, SphCurv)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-1076.74</td></tr><tr><td>Pyr (°)</td><td>0.02</td><td>16.79</td></tr><tr><td>Hybridization</td><td>2.00</td><td>2.67</td></tr><tr><td>Angular defect (°)</td><td>0.00</td><td>24.05</td></tr><tr><td>Spherical curvature</td><td>0.0004</td><td>0.38</td></tr></tbody></table>

Of $C_{2h}$ symmetry, we notice that the *Pyr* are the highest in the center of the molecule at the junctions which present the most important curvature on the 4 carbons. A similar behavior is observed for
hybridizations close to 2.00 and increase up to 2.66 for the 4 central atoms. The situation is the same for the 2 other parameters.

<figure><img src="image_37.png" /><figcaption>Figure 2.22: Cartographies of the parameters of 9,9',10,10'-tetrahydrodianthracene</figcaption></figure>

* Molecule $C_{20}H_{20}$: Sesquinorbornatriene (bridged down) (n°9) [30]

<figure><img src="image_38.png" /><figcaption>Figure 2.23: Representation of the molecule sesquinorbornatriene (bridged down)</figcaption></figure>

<table><caption>Table 2.12: Set of minimum and maximum values of parameters of sesquinorbornatriene (bridged down) (*Pyr*, *Hybr*, *AngDef*, *SphCurv*)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-773.92</td></tr><tr><td>Pyr (°)</td><td>4.04</td><td>18.12</td></tr><tr><td>Hybridization</td><td>2.03</td><td>2.82</td></tr><tr><td>Angular defect (°)</td><td>-75.21</td><td>33.19</td></tr><tr><td>Spherical curvature</td><td>0.11</td><td>0.42</td></tr></tbody></table>

The molecule is $C_1$ symmetry, the *Pyr* have higher values in the deformed zone of the molecule, the spherical curvature follows the same behavior but with values lower than 1. We observe 2 types of carbons $sp^2$ and $sp^3$. The angular defect is calculable on all carbons, the deformed zone has the highest values.

<figure><img src="image_39.png" /><figcaption>Figure 2.24: Cartographies of the parameters of sesquinorbornatriene (bridged down)</figcaption></figure>
* Molecule C$_{20}$H$_{20}$: Sesquinorbornatriene (bridged to the top) (n°10) [30]

<figure><img src="image_40.png" /><figcaption>Figure 2.25: Representation of the molecule sesquinorbornatriene (bridged to the top)</figcaption></figure>

<table><caption>Table 2.13: Set of minimum and maximum values of parameters of the sesquinorbornatriene (bridged to the top) (*Pyr*, *Hyb*, *AngDef*, *SphCurv*)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-773.93</td></tr><tr><td>Pyr (°)</td><td>0.01</td><td>27.88</td></tr><tr><td>Hybridization</td><td>2.00</td><td>5.81</td></tr><tr><td>Angular defect (°)</td><td>-74.71</td><td>71.74</td></tr><tr><td>Spherical curvature</td><td>0.0003</td><td>0.61</td></tr></tbody></table>

The *Pyr* are in the same order of importance as in the top-bridged case but slightly lower. The hybridizations and the angular defect are higher. Finally, the spherical curvature is quite low, less than 1.

<figure><img src="image_41.png" /><figcaption>Figure 2.26: Cartographies of the parameters of sesquinorbornatriene (bridged to the top)</figcaption></figure>

* Molecule C$_{8}$H$_{14}$: *Trans*-cyclooctene (n°11) [30]

Of symmetry C$_{1}$ the hybridization of the 2 carbon atoms is of type sp$^{2}$, the angular defect is negative and very weak except on the 2 trivalent atoms where it is lower than 1. The same behavior is observed with values lower than 0.5.

<figure><img src="image_42.png" /><figcaption>Figure 2.27: Representation of the molecule *Trans*-cyclooctene</figcaption></figure>
<table><caption>Table 2.14: Set of minimum and maximum values of parameters of the *Trans*-cyclooctene (*Pyr*, *Hyb*, *AngDef*, *SphCurv*)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-313.35</td></tr><tr><td>Pyr (°)</td><td>1.41</td><td>3.10</td></tr><tr><td>Hybridization</td><td>2.00</td><td>2.02</td></tr><tr><td>Angular defect (°)</td><td>-78.60</td><td>0.87</td></tr><tr><td>Spherical curvature</td><td>0.04</td><td>0.08</td></tr></tbody></table>

<figure><img src="image_43.png" /><figcaption>Figure 2.28: Cartographies of the parameters of $C_8H_{14}$: *Trans*-cyclooctene</figcaption></figure>

* Molecule $C_{24}H_{24}$ (n°12) [30]

<figure><img src="image_44.png" /><figcaption>Figure 2.29: Representation of the molecule $C_{24}H_{24}$</figcaption></figure>

<table><caption>Table 2.15: Set of minimum and maximum values of parameters of the molecule $C_{24}H_{24}$ (*Pyr*, *Hyb*, *AngDef*, *SphCurv*)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-929.13</td></tr><tr><td>Pyr (°)</td><td>7.05</td><td>-</td></tr><tr><td>Hybridization</td><td>2.09</td><td>-</td></tr><tr><td>Angular defect (°)</td><td>-76.26</td><td>4.46</td></tr><tr><td>Spherical curvature</td><td>0.17</td><td>-</td></tr></tbody></table>
<figure><img src="image_45.png" /><figcaption>Figure 2.30: Cartographies of the parameters of C<sub>24</sub>H<sub>24</sub></figcaption></figure>

The extrema are summarized for a C<sub>6h</sub> symmetry. The *Pyr* are identical throughout the molecule as the *sp*<sup>2</sup> hybridization. The angular defect is according to 2 values and the spherical curvature less than 1 uniformly.

* Molecule C<sub>20</sub>H<sub>10</sub>: Corannulene (n°13) [30]

<figure><img src="image_46.png" /><figcaption>Figure 2.31: Representation of the molecule of corannulene</figcaption></figure>

<table><caption>Table 2.16: Set of minimum and maximum values of parameters of corannulene (*Pyr*, *Hy*, *AngDef*, *SphCurv*)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-768.32</td></tr><tr><td>Pyr (°)</td><td>0.03</td><td>0.07</td></tr><tr><td>Hybridization</td><td>2.00</td><td>-</td></tr><tr><td>Angular defect (°)</td><td>0.000088</td><td>0.0004</td></tr><tr><td>Spherical curvature</td><td>0.008</td><td>0.002</td></tr></tbody></table>

<figure><img src="image_47.png" /><figcaption>Figure 2.32: Cartographies of the parameters of the corannulene</figcaption></figure>
The molecule of formula $C_{20}H_{10}$ of symmetry $C_{5h}$, includes 5 hexagonal patterns surrounding 1 pentagon. During the calculations we observe that the molecule bends progressively. The *Pyr* will be more pronounced in the center of the molecule than in its periphery. The hybridization is uniform showing a *sp*² character. The angular defect is rather reliable being weaker on the carbons in periphery than in the center. This agrees with the data of *Pyr*. The spherical curvature also correlates with the previous results with a greater spherical curvature on the atoms forming the pentagon.

### 2.3.2 Fullerenic molecules

* Molecule $C_{50}-D_3$ (n°14) [31]

This molecule is a $C_{50}$ fullerene with $D_3$ symmetry, composed by pentagons and a majority of hexagons.

<figure><img src="image_48.png" /><figcaption>Figure 2.33: Representation of the molecule fullerene $C_{50}-D_3$</figcaption></figure>

<table><caption>Table 2.17: Set of minimum and maximum values of parameters of the molecule fullerene $C_{50}-D_3$ (*Pyr*, *Hyb*, *AngDef*, *SphCurv*)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-1894.39</td></tr><tr><td>Pyr (°)</td><td>12.06</td><td>13.86</td></tr><tr><td>Hybridization</td><td>2.30</td><td>2.42</td></tr><tr><td>Angular defect (°)</td><td>12.88</td><td>16.81</td></tr><tr><td>Spherical curvature</td><td>0.30</td><td>0.34</td></tr></tbody></table>

The *Pyr* data are quite uniform with relatively close minimum and maximum values. The hybridizations are also relatively similar around 2.3. The angular defect and the spherical curvature have a uniform behavior.

<figure><img src="image_49.png" /><figcaption>Figure 2.34: Cartographies of the parameters of fullerene $C_{50}-D_3$</figcaption></figure>
* Molecule $C_{50}-D_{5h}$ (n°15) [31]

<figure><img src="image_50.png" /><figcaption>Figure 2.35: Representation of the molecule fullerene $C_{50}-D_{5h}$</figcaption></figure>

This fullerene of $D_{5h}$ symmetry consists of pentagons and a majority of hexagons. The *Pyr* are in a relatively close range of values, however we note that 2 pentagons on both sides of the molecule have lower angles around 11°. The hybridizations are uniform. The angular defect values are wider. Finally the spherical curvature remains lower than 0.5.

<table><caption>Table 2.18: Set of minimum and maximum values of parameters of the molecule fullerene $C_{50}-D_{5h}$ (*Pyr*, *Hyb*, *AngDef*, *SphCurv*)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-1894.38</td></tr><tr><td>Pyr (°)</td><td>11.12</td><td>13.59</td></tr><tr><td>Hybridization</td><td>2.25</td><td>2.40</td></tr><tr><td>Angular defect (°)</td><td>10.98</td><td>16.26</td></tr><tr><td>Spherical curvature</td><td>0.28</td><td>0.34</td></tr></tbody></table>

We note few differences between the values of parameters of the 2 isomers.

<figure><img src="image_51.png" /><figcaption>Figure 2.36: Cartographies of the parameters of fullerene $C_{50}-D_{5h}$</figcaption></figure>

* Molecule Chlorofullerene $C_{50}Cl_{10}-D_{3}$ (n°16) [34]

This chlorofullerene, constituted of a $C_{50}$ and 10 chlorine atoms on the equatorial zone, presents an original structure called "saturne". It should be noted that this fullerene exists under 2 different symmetries.
<figure><img src="image_52.png" /><figcaption>Figure 2.37: Representation of the fullerenic molecule $C_{50}Cl_{10}-D_3$</figcaption></figure>

<table><caption>Table 2.19: Set of minimum and maximum values of parameters of the fullerenic molecule $C_{50}Cl_{10}-D_3$ (*Pyr*, *Hyb*, *AngDef*, *SphCurv*)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-6475.74</td></tr><tr><td>Pyr (°)</td><td>9.66</td><td>10.92</td></tr><tr><td>Hybridization</td><td>2.18</td><td>2.24</td></tr><tr><td>Angular defect (°)</td><td>-72.60</td><td>10.54</td></tr><tr><td>Spherical curvature</td><td>0.24</td><td>0.27</td></tr></tbody></table>

<figure><img src="image_53.png" /><figcaption>Figure 2.38: Cartographies of the parameters of cholorofullerene $D_3$</figcaption></figure>

We observe 2 trends in the values of *Pyr*. The whole molecule is in the range of 10° except for the carbons where the chlorine atoms are added where the *Pyr* reaches 24°. The hybridizations are of the average of 2.2 on the quasi-totality of the molecule, however the carbons bounded with the chlorines have hybridization of almost 4. Finally the angular defect and the spherical curvature have the same behavior.

* Molecule Chlorofullerene $C_{50}Cl_{10}-D_{5h}$ (n°17) [34]

<figure><img src="image_54.png" /><figcaption>Figure 2.39: Representation of the fullerenic molecule $C_{50}Cl_{10}-D_{5h}$</figcaption></figure>
<table><caption>Table 2.20: Set of minimum and maximum values of parameters of the fullerene molecule $C_{50}Cl_{10}-D_{5h}$ (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-6475.74</td></tr><tr><td>Pyr (°)</td><td>9.67</td><td>10.92</td></tr><tr><td>Hybridization</td><td>2.18</td><td>2.24</td></tr><tr><td>Angular defect (°)</td><td>-72.60</td><td>10.54</td></tr><tr><td>Spherical curvature</td><td>0.23</td><td>0.27</td></tr></tbody></table>

<figure><img src="image_55.png" /><figcaption>Figure 2.40: Cartographies of the parameters of cholorofullerene $D_{5h}$</figcaption></figure>

This second molecule is similar but with a $D_{5h}$ symmetry. In all the cartographies we observe the same behavior as for the previous molecule with lower values on the whole molecule but higher values on the carbon atoms of the equatorial part where the chlorine atoms are added.

* Molecule of fullerene $C_{64}-C_{3v}$ (n°18) [34]

<figure><img src="image_56.png" /><figcaption>Figure 2.41: Representation of the molecule $C_{64}-C_{3v}$</figcaption></figure>

<table><caption>Table 2.21: Set of minimum and maximum values of parameters of the molecule $C_{64}-C_{3v}$ (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-2424.79</td></tr><tr><td>Pyr (°)</td><td>11.61</td><td>16.43</td></tr><tr><td>Hybridization</td><td>2.28</td><td>3.50</td></tr><tr><td>Angular defect (°)</td><td>11.94</td><td>40.17</td></tr><tr><td>Spherical curvature</td><td>0.28</td><td>0.53</td></tr></tbody></table>
<figure><img src="image_57.png" /><figcaption>Figure 2.42: Cartographies of the parameters of the fullerene $C_{64}-C_{3v}$</figcaption></figure>

This fullerene of 64 carbon atoms has a $C_{3v}$ symmetry. This fullerene is not perfectly spherical and shows a deformation at one of its poles. We notice that the *Pyr* are the highest in 4 atoms at the top which is deformed. The hybridizations are uniform of $sp^2$ type but are beyond 3 at the top. For the angular defect and the spherical curvature the same behavior is observed at the top.

* Molecule of fullerene $C_{64}-C_2$ (n°19) [34]

<figure><img src="image_58.png" /><figcaption>Figure 2.43: Representation of the molecule $C_{64}-C_2$</figcaption></figure>

<table><caption>Table 2.22: Set of minimum and maximum values of parameters of the molecule $C_{64}-C_2$ (*Pyr*, *Hyb*, *AngDef*, *SphCurv*)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-2425.03</td></tr><tr><td>Pyr (°)</td><td>7.06</td><td>22.97</td></tr><tr><td>Hybridization</td><td>2.10</td><td>3.68</td></tr><tr><td>Angular defect (°)</td><td>4.46</td><td>42.85</td></tr><tr><td>Spherical curvature</td><td>0.17</td><td>0.51</td></tr></tbody></table>

We note once again that this $C_{64}$ is not perfectly spherical, which will certainly induce variations in the properties. The *Pyr* are more pronounced in the non-spherical regions. This deformed region also shows higher hybridizations close to 4 while the rest of the molecule has $sp^2$.
<figure><img src="image_59.png" /><figcaption>Figure 2.44: Cartographies of the parameters of the fullerene $C_{64}-D_2$</figcaption></figure>

* Molecule of fullerene $C_{64}-D_2$ (n°20) [34]

<figure><img src="image_60.png" /><figcaption>Figure 2.45: Representation of the molecule $C_{64}-D_2$</figcaption></figure>

<table><caption>Table 2.23: Set of minimum and maximum values of parameters of the molecule $C_{64}-D_2$ (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-2425.05</td></tr><tr><td>Pyr (°)</td><td>7.77</td><td>22.79</td></tr><tr><td>Hybridization</td><td>2.12</td><td>3.64</td></tr><tr><td>Angular defect (°)</td><td>6.23</td><td>42.22</td></tr><tr><td>Spherical curvature</td><td>0.19</td><td>0.50</td></tr></tbody></table>

<figure><img src="image_61.png" /><figcaption>Figure 2.46: Cartographies of the parameters of the fullerene $C_{64}-D_2$</figcaption></figure>

We observe once again that the fullerene is not perfectly spherical, in particular by the pyramidal angles which are clearly higher at the poles of the molecule. We note the same situation with the
hybridizations which are close to 2 on the whole molecule and beyond 3 at the vertices. Concerning the angular defect and the spherical curvature the situations are rather similar.

* Molecule of fullerene $C_{64}-C_s$ (n°21) [34]

<figure><img src="image_62.png" /><figcaption>Figure 2.47: Representation of the molecule $C_{64}-C_s$</figcaption></figure>

<table><caption>Table 2.24: Set of minimum and maximum values of parameters of the molecule $C_{64}-C_s$ (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-2425.04</td></tr><tr><td>Pyr (°)</td><td>6.94</td><td>22.79</td></tr><tr><td>Hybridization</td><td>2.09</td><td>3.64</td></tr><tr><td>Angular defect (°)</td><td>4.31</td><td>42.26</td></tr><tr><td>Spherical curvature</td><td>0.17</td><td>0.50</td></tr></tbody></table>

Once again this $C_{64}$ is not perfectly spherical. We thus observe higher pyramidal angles at the more deformed poles of the molecule. The hybridizations are around 2.2 on the whole molecule except on the extremities.

<figure><img src="image_63.png" /><figcaption>Figure 2.48: Cartographies of the parameters of the fullerene $C_{64}-C_s$</figcaption></figure>

As for the $C_{50}Cl_{10}$ molecules of the 2008 [34] publication reports chlorofullerenes said to be "pineapple-shaped" from $C_{64}$ and 4 chlorine atoms. We were able to study the possible chlofullerenes from the different C64 studied which are not found in the "pineapple" category. The first cholorofullerene is the one constructed from $C_{3v}$ symmetry which adopts the original shape described by the publication [34].

* Molecule of fullerene $C_{64}Cl_4-C_{3v}$ (n°22) [34]

As already commented concerning the fullerene $C_{64}$ the deformation of the molecule is accentuated with the addition of chlorine atoms. We thus observe a zone where the values of the properties will be more significant.
<figure><img src="image_64.png" /><figcaption>Figure 2.49: Representation of the molecule $C_{64}Cl_{4}-C_{3v}$</figcaption></figure>

<table><caption>Table 2.25: Set of minimum and maximum values of parameters of the molecule $C_{64}Cl_{4}-C_{3v}$ (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-4257.59</td></tr><tr><td>Pyr (°)</td><td>7.43</td><td>11.92</td></tr><tr><td>Hybridization</td><td>2.11</td><td>2.29</td></tr><tr><td>Angular defect (°)</td><td>-78.45</td><td>12.59</td></tr><tr><td>Spherical curvature</td><td>0.18</td><td>0.29</td></tr></tbody></table>

<figure><img src="image_65.png" /><figcaption>Figure 2.50: Cartographies of the parameters of the fullerene $C_{64}Cl_{4}-C_{3v}$</figcaption></figure>

We note a maximum of 12° for the angles of pyramidalisation on the lower part of the chlorofullerene and smaller angles towards the top. The hybridizations follow the same behavior with a maximum hybridization of 2.3. The angular defect is calculable in each atom of the molecule, we note that it reaches 13° on the main part of the molecule until the top which shows negative values.

* Molecule of fullerene $C_{64}Cl_{4}-C_{s}$ (n°23) [34]

This second molecule has a different geometry because the chlorine atoms are not added on the same carbon atoms as the previous molecule but on both sides of the molecule as we can see on the figure:

<figure><img src="image_66.png" /><figcaption>Figure 2.51: Representation of the molecule $C_{64}Cl_{4}-C_{s}$</figcaption></figure>
<table><caption>Table 2.26: Set of minimum and maximum values of parameters of the molecule $C_{64}Cl_4-C_s$ (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-4257.52</td></tr><tr><td>Pyr (°)</td><td>6.94</td><td>13.33</td></tr><tr><td>Hybridization</td><td>2.12</td><td>2.38</td></tr><tr><td>Angular defect (°)</td><td>-70.57</td><td>15.56</td></tr><tr><td>Spherical curvature</td><td>0.19</td><td>0.32</td></tr></tbody></table>

The pyrmidalization angles range from 7° to 13° on the whole molecule. The hybridizations are uniform on the whole molecule showing that the carbons are hybridized sp². The angular defect is positive on the whole carbon skeleton except on the 4 addition sites where the values are negative.

<figure><img src="image_67.png" /><figcaption>Figure 2.52: Cartographies of the parameters of the fullerene $C_{64}Cl_4-C_s$</figcaption></figure>

* Molecule of fullerene $C_{64}Cl_4-D_2$ (n°24)[34]

This chlorofullerene is presented in the same way as the previous one with the addition of 2 chlorine atoms at each pole and presents the $D_2$ symmetry.

<figure><img src="image_68.png" /><figcaption>Figure 2.53: Representation of the molecule $C_{64}Cl_4-D_2$</figcaption></figure>

<table><caption>Table 2.27: Set of minimum and maximum values of parameters of the molecule $C_{64}Cl_4-D_2$ (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-4257.51</td></tr><tr><td>Pyr (°)</td><td>7.77</td><td>12.61</td></tr><tr><td>Hybridization</td><td>2.12</td><td>2.33</td></tr><tr><td>Angular defect (°)</td><td>-70.47</td><td>14.00</td></tr><tr><td>Spherical curvature</td><td>0.19</td><td>0.31</td></tr></tbody></table>
<figure><img src="image_69.png" /><figcaption>Figure 2.54: Cartographies of the parameters of the fullerene $C_{64}Cl_4-D_2$</figcaption></figure>

Similar to the other chlorofullerenes, $C_{64}Cl_4$ with $D_2$ symmetry shows non-uniform *Pyr* throughout the molecule. The angular defect values are positive throughout the molecule but negative here as well at sites of chlorine atom addition. Finally, the spherical curvature is not very uniform on the whole molecule.

* Molecule of fullerene $C_{64}Cl_4-C_2$ (n°25) [34]

<figure><img src="image_70.png" /><figcaption>Figure 2.55: Representation of the molecule $C_{64}Cl_4-C_2$</figcaption></figure>

<table><caption>Table 2.28: Set of minimum and maximum values of parameters of the molecule $C_{64}Cl_4-C_2$ (*Pyr*, *Hyb*, *AngDef*, *SphCurv*)</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Energy (u.a)</td><td colspan="2">-4257.51</td></tr><tr><td>Pyr (°)</td><td>6.94</td><td>13.33</td></tr><tr><td>Hybridization</td><td>2.12</td><td>2.38</td></tr><tr><td>Angular defect (°)</td><td>-70.57</td><td>15.56</td></tr><tr><td>Spherical curvature</td><td>0.20</td><td>0.50</td></tr></tbody></table>

<figure><img src="image_71.png" /><figcaption>Figure 2.56: Cartographies of the parameters of the fullerene $C_{64}Cl_4-C_2$</figcaption></figure>
We observe angles of pyramidalization between 7° and 13°. The hybridizations are uniform. In this case again the angular defect is positive on the molecule except on the 4 addition sites of the chlorine atoms.

**Overview of the pyramidalization angle and hybridization:** The two following graphs (figure 2.57) illustrate the data of the molecules according to the parameters of pyramidalization angle and hybridization, taking into account the neighboring environment and the hybridization of each carbon.

<figure><img src="image_72.png" /><figcaption>Figure 2.57: Data of the pyramidalization angles and the hybridization of each molecule considering the carbon environment</figcaption></figure>
**2.4 Application to reactivity**

**2.4.1 Presentation of the ring expansion reaction path of a Nitrile Imine**

The results presented in this section are based on data from the publication: "New rearrangements of Nitriles Imines: Ring expansion of benzonitrile imines to cycloheptatetraenes and ring closure to 3-phenyl-3H-diazirines" [32].

<figure><img src="image_73.png" /><figcaption>Figure 2.58: Stages ring expansion calculated by DFT B3LYP-6-311G method [32]</figcaption></figure>

The **context** of this study is based on **nitrile imines** which are important intermediates in 1,3-dipolar cycloaddition reactions. They are known to undergo efficient, unimolecular rearrangements to 1H-diazirines, imidoylnitrenes and carbodiimides under thermal and photochemical reaction conditions.
C. Wentrup et al. [32] have recently reported a competing rearrangement in which C-phenylnitrile imines undergo ring expansion to 1-diazenyl-1,2,4,6-cycloheptatetraenes akin to the phenylcarbene - cycloheptatetraene rearrangement. Amino-, hydroxy-, and thiol-groups in the meta positions of C-phenylnitrile imine lower the activation energies so that this rearrangement becomes competitive with cyclization to 1H-diazirines and hence rearrangement to carbodiimides. With two NMe₂ groups in the 3- and 5-positions, the activation energy for the ring expansion drops below 50 kcal/mol. The diazenylcycloheptatetraenes so formed can rearrange further to 2-diazenyl-phenylcarbenes over modest activation barriers, and these carbenes cyclize very easily to 2H- and 3H-indazoles, from which 6-methylenecyclohexadienylidene, phenylcarbene, and fulvenallene are potentially obtainable. Moreover, another new rearrangement of benzonitrile imine forms 3-phenyl-3H-diazirine, which is a precursor of phenyldiazomethane and hence phenylcarbene. This reaction is competitive with the ring expansion. The new rearrangements predicted by Wentrup et al., should be experimentally observable, e.g. under FVP or matrix-photolysis conditions.

In this **theoretical work** authors sought to **describe** and **compare** three **intramolecular rearrangement** routes of aromatic nitrile imines, as set out in figure 2.59.

**Route (i)** describes the new ring expansion of benzonitrile imine (8a) to 1-diazenylcycloheptatetraene (12a) and subsequent ring contraction to 2-diazenylphenylcarbene (16a).

**Route (ii)** is the previously investigated ring closure to 3-phenyl-1H-diazirine (18a) and further to phenylcarbodiimide (20a).

**Route (iii)** is the new ring closure to 3-phenyl-3H-diazirine (22a), which results in the easy formation of phenyldiazomethane (24) and then phenylcarbene (26).
Electronic energies of ground and transition states were studied at the DFT and CASPT2 levels of theories (figure 2.59). Compounds are designated 8a, 10a...etc for the parent compounds and transition states TS9a...etc. A designate propargylic structure of nitrile imine.

<figure><img src="image_74.png" /><figcaption>Figure 2.59: Rearrangement routes of aromatic nitrile imines [32]</figcaption></figure>

If such a study provides information on the topology of the potential surfaces this kind of approach does not inform the chemist about the evolution of the chemical character of the species present during the chemical reaction (along the reaction path). Indeed, for a stable equilibrium molecule, principal theoretical interest generally focuses on the Lewis-structural bonding pattern and associated orbitals that are optimal for its description. For a chemical reaction, however, the primary focus turns to bond shifts between reactant and product species and the corresponding dynamical transformations of orbital form and location along a reactive pathway on the potential energy surface (PES). An NBO (Natural Bond Orbital) analysis of the electron distributions during these transformations is generally developed by chemist but the theoretical level of these interpretations remains sometimes questionable especially in the presence of compounds as complex as those studied in our work because of their multi-determinantal character.

We believe that another way of approaching the problem is possible thanks to the **tools** we have presented so far in this work. A very **preliminary attempt** of the analysis of the route (i) is presented in this work as illustration. A thorough study of this work is still to be done. So, along this reaction path we have studied each of the points of the curve of route (i). For each of the points four parameters were studied: the pyramidalization angle, the angular defect, the spherical curvature and the hybridization.

*Note that all the visualizations in this section are once again made from the program "Pychemcurv" [10].*
## 2.4.2 Results

### 2.4.3 Conformation 8aA - Allenic Benzonitrile Imine

<figure><img src="image_75.png" /><figcaption>Figure 2.60: Conformation 8aA and classification of atoms according to their environment and hybridization</figcaption></figure>

<figure><img src="image_76.png" /><figcaption>Figure 2.61: Cartographies and representation of the parameters of conformation 8aA</figcaption></figure>

<table><caption>Table 2.29: Set of environment and energy of conformation 8aA</caption><tr><td>Environment C-C-C</td><td>sp<sup>2</sup></td><td>C5</td></tr><tr><td>Environment C-C-H</td><td>sp<sup>2</sup></td><td>C1 C2 C3 C4 C6</td></tr><tr><td>Environment C-C-N</td><td>sp</td><td>C7</td></tr><tr><td>E</td><td>-</td><td>-379.827126 u.a</td></tr></table>

<table><caption>Figure 2.62: Set of values of parameters of conformation 8aA (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><td>Atom</td><td>Pyr</td><td>DefAng</td><td>Sph</td><td>Hyb</td></tr></thead><tbody><tr><td>1</td><td>1.667498</td><td>0.252008</td><td>0.045012</td><td>2.005094</td></tr><tr><td>2</td><td>0.964742</td><td>0.084388</td><td>0.026089</td><td>2.001702</td></tr><tr><td>3</td><td>0.379465</td><td>0.013059</td><td>0.010298</td><td>2.000263</td></tr><tr><td>4</td><td>3.978158</td><td>1.429812</td><td>0.107605</td><td>2.029301</td></tr><tr><td>5</td><td>6.321044</td><td>3.649031</td><td>0.154860</td><td>2.075476</td></tr><tr><td>6</td><td>2.65232</td><td>0.637254</td><td>0.071664</td><td>2.012931</td></tr></tbody></table>
### Characteristics of V1

<figure><img src="image_77.png" /><figcaption>Figure 2.63: Cartographies and representation of the parameters of conformation V1</figcaption></figure>

<table><caption>Table 2.30: Set of environment and energy of conformation V1</caption><tr><td>Environment C-C-C</td><td>sp<sup>2</sup></td><td>C5</td></tr><tr><td>Environment C-C-H</td><td>sp<sup>2</sup></td><td>C1 C2 C3 C4 C6</td></tr><tr><td>Environment C-C-N</td><td>sp</td><td>C7</td></tr><tr><td>E</td><td>-</td><td>-379.816260 u.a</td></tr></table>

<table><caption>Figure 2.64: Set of values of parameters of conformation V1 (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><td>Atom</td><td>Pyr</td><td>DefAng</td><td>Sph</td><td>Hyb</td></tr></thead><tbody><tr><td>1</td><td>1.936569</td><td>0.33981180</td><td>0.052259</td><td>2.006875</td></tr><tr><td>2</td><td>1.024235</td><td>0.09511434</td><td>0.027681</td><td>2.001919</td></tr><tr><td>3</td><td>0.002085</td><td>3.942986.10<sup>-7</sup></td><td>0.000057</td><td>2.000000</td></tr><tr><td>4</td><td>4.513634</td><td>1.83891400</td><td>0.122170</td><td>2.037862</td></tr><tr><td>5</td><td>7.603411</td><td>5.34616100</td><td>0.186513</td><td>2.110867</td></tr><tr><td>6</td><td>2.681132</td><td>0.65144840</td><td>0.072442</td><td>2.013216</td></tr></tbody></table>

The "V1" state of the reaction path of figure 2.58 has been reached.
Characteristics of V2

<figure><img src="image_78.png" /><figcaption>Figure 2.65: Cartographies and representation of the parameters of conformation V2</figcaption></figure>

<table><caption>Table 2.31: Set of environment and energy of conformation V2</caption><tr><td>Environment C-C-C</td><td>sp<sup>2</sup></td><td>C5</td></tr><tr><td>Environment C-C-H</td><td>sp<sup>2</sup></td><td>C1 C2 C3 C4 C6</td></tr><tr><td>Environment C-C-N</td><td>sp</td><td>C7</td></tr><tr><td>E</td><td>-</td><td>-379.792368 u.a</td></tr></table>

<table><caption>Figure 2.66: Set of values of parameters of conformation V2 (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><td>Atom</td><td>Pyr</td><td>DefAng</td><td>Sph</td><td>Hyb</td></tr></thead><tbody><tr><td>1</td><td>1.753834</td><td>0.278776</td><td>0.047307</td><td>2.005636</td></tr><tr><td>2</td><td>1.190437</td><td>0.128510</td><td>0.032101</td><td>2.002593</td></tr><tr><td>3</td><td>1.156046</td><td>0.121208</td><td>0.031440</td><td>2.002445</td></tr><tr><td>4</td><td>2.992608</td><td>0.811263</td><td>0.080689</td><td>2.016488</td></tr><tr><td>5</td><td>10.016094</td><td>9.688596</td><td>0.246477</td><td>2.199614</td></tr><tr><td>6</td><td>1.981983</td><td>0.35674</td><td>0.053581</td><td>2.007203</td></tr></tbody></table>

This point corresponds to point "V2" of the reaction path.
### Characteristics of V3

<figure><img src="image_79.png" /><figcaption>Figure 2.67: Cartographies and representation of the parameters of conformation V3</figcaption></figure>

<table><caption>Table 2.32: Set of environment and energy of conformation V3</caption><tr><td>Environment C-C-C</td><td>sp<sup>2</sup></td><td>C5</td></tr><tr><td>Environment C-C-H</td><td>sp<sup>2</sup></td><td>C1 C2 C3 C4 C6</td></tr><tr><td>Environment C-C-N</td><td>sp</td><td>C7</td></tr><tr><td>E</td><td>-</td><td>-379.774313 u.a</td></tr></table>

<table><caption>Figure 2.68: Set of values of parameters of conformation V3 (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><td>Atom</td><td>Pyr</td><td>DefAng</td><td>Sph</td><td>Hyb</td></tr></thead><tbody><tr><td>1</td><td>1.438139</td><td>0.187526</td><td>0.038757</td><td>2.003786</td></tr><tr><td>2</td><td>1.210689</td><td>0.132956</td><td>0.032592</td><td>2.002682</td></tr><tr><td>3</td><td>1.764184</td><td>0.282108</td><td>0.047811</td><td>2.005703</td></tr><tr><td>4</td><td>1.194216</td><td>0.129531</td><td>0.031908</td><td>2.002610</td></tr><tr><td>5</td><td>11.524357</td><td>13.559333</td><td>0.284061</td><td>2.272060</td></tr><tr><td>6</td><td>1.468643</td><td>0.196024</td><td>0.039750</td><td>2.003949</td></tr></tbody></table>

State "V3" of the reaction path is reached.
Characteristics of V4

<figure><img src="image_80.png" /><figcaption>Figure 2.69: Cartographies and representation of the parameters of conformation V4</figcaption></figure>

<table><caption>Table 2.33: Set of environment and energy of conformation V4</caption><tr><td>Environment C-C-C</td><td>sp<sup>2</sup></td><td>C5</td></tr><tr><td>Environment C-C-H</td><td>sp<sup>2</sup></td><td>C1 C2 C3 C4 C6</td></tr><tr><td>Environment C-C-N</td><td>sp</td><td>C7</td></tr><tr><td>E</td><td>-</td><td>-379.768938 u.a</td></tr></table>

<table><caption>Figure 2.70: Set of values of parameters of conformation V4 (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><th>Atom</th><th>Pyr</th><th>DefAng</th><th>Sph</th><th>Hyb</th></tr></thead><tbody><tr><td>1</td><td>1.314700</td><td>0.156744</td><td>0.035407</td><td>2.003164</td></tr><tr><td>2</td><td>1.187049</td><td>0.127832</td><td>0.031937</td><td>2.002578</td></tr><tr><td>3</td><td>1.884989</td><td>0.321999</td><td>0.050948</td><td>2.006513</td></tr><tr><td>4</td><td>NaN</td><td>-21.387858</td><td>NaN</td><td>NaN</td></tr><tr><td>5</td><td>12.059293</td><td>15.280032</td><td>0.297491</td><td>2.301357</td></tr><tr><td>6</td><td>1.312094</td><td>0.156485</td><td>0.035541</td><td>2.003151</td></tr><tr><td>7</td><td>11.034023</td><td>16.893025</td><td>0.269333</td><td>2.246920</td></tr></tbody></table>

The transition state approaches the sp³ state.

The point coincides with the V4 state of the path in the figure 2.58.
### 2.4.4 Conformation 2 : transition state TS9a

<figure><img src="image_81.png" /><figcaption>Figure 2.71: Cartographies and representation of the parameters of conformation TS9a</figcaption></figure>

<table><caption>Table 2.34: Set of environment and energy of conformation TS9a</caption><tr><td>Environment C-C-C</td><td>sp<sup>2</sup></td><td>C5</td></tr><tr><td>Environment C-C-C-H</td><td>sp<sup>3</sup></td><td>C4</td></tr><tr><td>Environment C-C-H</td><td>sp<sup>2</sup></td><td>C1 C2 C3 C6</td></tr><tr><td>Environment C-C-N</td><td>sp</td><td>C7</td></tr><tr><td>E</td><td>-</td><td>-379.765890 u.a</td></tr></table>

<table><caption>Figure 2.72: Set of values of parameters of conformation TS9a (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><td>Atom</td><td>Pyr</td><td>DefAng</td><td>Sph</td><td>Hyb</td></tr></thead><tbody><tr><td>1</td><td>1.191238</td><td>0.128711</td><td>0.032057</td><td>2.002597</td></tr><tr><td>2</td><td>1.154461</td><td>0.120929</td><td>0.031043</td><td>2.002439</td></tr><tr><td>3</td><td>1.958262</td><td>0.347467</td><td>0.052759</td><td>2.007031</td></tr><tr><td>4</td><td>NaN</td><td>-24.73652</td><td>NaN</td><td>NaN</td></tr><tr><td>5</td><td>12.544951</td><td>17.073513</td><td>0.309727</td><td>2.329739</td></tr><tr><td>6</td><td>1.188212</td><td>0.128351</td><td>0.032213</td><td>2.002583</td></tr><tr><td>7</td><td>9.677130</td><td>12.801086</td><td>0.236120</td><td>2.185240</td></tr></tbody></table>

This point corresponds to the "TS9a" state of the path, so the conformation change takes place between the point "V4" and "V6" of the path of figure 2.58.
Characteristics of V6

<figure><img src="image_82.png" /><figcaption>Figure 2.73: Cartographies and representation of the parameters of conformation V6</figcaption></figure>

<table><caption>Table 2.35: Set of environment and energy of conformation V6</caption><tr><td>Environment C-C-C</td><td>sp<sup>2</sup></td><td>C5</td></tr><tr><td>Environment C-C-C-H</td><td>sp<sup>3</sup></td><td>C4</td></tr><tr><td>Environment C-C-H</td><td>sp<sup>2</sup></td><td>C1 C2 C3 C6</td></tr><tr><td>Environment C-C-N</td><td>sp</td><td>C7</td></tr><tr><td>E</td><td>-</td><td>-379.765726 u.a</td></tr></table>

<table><caption>Figure 2.74: Set of values of parameters of conformation V6 (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><th>Atom</th><th>Pyr</th><th>DefAng</th><th>Sph</th><th>Hyb</th></tr></thead><tbody><tr><td>1</td><td>0.805861</td><td>0.058935</td><td>0.021642</td><td>2.001188</td></tr><tr><td>2</td><td>1.109085</td><td>0.111680</td><td>0.029776</td><td>2.002250</td></tr><tr><td>3</td><td>2.154805</td><td>0.420597</td><td>0.057685</td><td>2.008518</td></tr><tr><td>4</td><td>NaN</td><td>-30.916412</td><td>NaN</td><td>NaN</td></tr><tr><td>5</td><td>13.349291</td><td>21.076803</td><td>0.329821</td><td>2.380743</td></tr><tr><td>6</td><td>1.022567</td><td>0.095154</td><td>0.027767</td><td>2.001913</td></tr><tr><td>7</td><td>6.611699</td><td>5.875616</td><td>0.162842</td><td>2.082838</td></tr></tbody></table>

We are at point "V6" of the path (figure 2.58).
### 2.4.5 Conformation 3 - 10a - bicyclo[4.1.0]hepta-2,4,7-triene

<figure><img src="image_83.png" /><figcaption>Figure 2.75: Cartographies and representation of the parameters of conformation 10a</figcaption></figure>

<table><caption>Table 2.36: Set of environment and energy of conformation 10a</caption><tr><td>Environment C-C-C</td><td>sp<sup>2</sup></td><td>C5</td></tr><tr><td>Environment C-C-C-H</td><td>sp<sup>3</sup></td><td>C4</td></tr><tr><td>Environment C-C-H</td><td>sp<sup>2</sup></td><td>C1 C2 C3 C6</td></tr><tr><td>Environment C-C-N</td><td>sp</td><td>C7</td></tr><tr><td>E</td><td>-</td><td>-379.766669 u.a</td></tr></table>

<table><caption>Figure 2.76: Set of values of parameters of conformation 10a (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><th>Atom</th><th>Pyr</th><th>DefAng</th><th>Sph</th><th>Hyb</th></tr></thead><tbody><tr><td>1</td><td>1.090407</td><td>0.108527</td><td>0.029665</td><td>2.002175</td></tr><tr><td>2</td><td>0.009218</td><td>0.000008</td><td>0.000247</td><td>2</td></tr><tr><td>3</td><td>2.200591</td><td>0.439445</td><td>0.059013</td><td>2.008886</td></tr><tr><td>4</td><td>1.927923</td><td>0.336771</td><td>0.051559</td><td>2.006814</td></tr><tr><td>5</td><td>NaN</td><td>-34.932909</td><td>NaN</td><td>NaN</td></tr><tr><td>6</td><td>12.532256</td><td>22.126278</td><td>0.311692</td><td>2.328974</td></tr><tr><td>7</td><td>3.974531</td><td>1.977641</td><td>0.099852</td><td>2.029247</td></tr></tbody></table>

This is the "10a" point of the reaction path (figure 2.58).
Characteristics of TS11a

<figure><img src="image_84.png" /><figcaption>Figure 2.77: Cartographies and representation of the parameters of conformation TS11a</figcaption></figure>

<table><caption>Table 2.37: Set of environment and energy of conformation TS11a</caption><tr><td>Environment C-C-C</td><td>sp<sup>2</sup></td><td>C5</td></tr><tr><td>Environment C-C-C-H</td><td>sp<sup>3</sup></td><td>C4</td></tr><tr><td>Environment C-C-H</td><td>sp<sup>2</sup></td><td>C1 C2 C3 C6</td></tr><tr><td>Environment C-C-N</td><td>sp</td><td>C7</td></tr><tr><td>E</td><td>-</td><td>-379.762883 u.a</td></tr></table>

<table><caption>Figure 2.78: Set of values of parameters of conformation TS11a (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><th>Atom</th><th>Pyr</th><th>DefAng</th><th>Sph</th><th>Hyb</th></tr></thead><tbody><tr><td>1</td><td>0.089150</td><td>0.000725</td><td>0.002434</td><td>2.000015</td></tr><tr><td>2</td><td>1.418131</td><td>0.182615</td><td>0.038093</td><td>2.003682</td></tr><tr><td>3</td><td>3.829610</td><td>1.328587</td><td>0.102723</td><td>2.027128</td></tr><tr><td>4</td><td>3.008701</td><td>0.821332</td><td>0.080555</td><td>2.016667</td></tr><tr><td>5</td><td>NaN</td><td>-28.082354</td><td>NaN</td><td>NaN</td></tr><tr><td>6</td><td>12.727553</td><td>31.058654</td><td>0.325574</td><td>2.340869</td></tr><tr><td>7</td><td>3.030169</td><td>1.006072</td><td>0.076200</td><td>2.016908</td></tr></tbody></table>

This point represents the TS11a state of the path.
### 2.4.6 Conformation 4 - 12a cycloheptatetraene

<figure><img src="image_85.png" /><figcaption>Figure 2.79: Cartographies and representation of the parameters of conformation 12a</figcaption></figure>

<table><caption>Table 2.38: Set of environment and energy of conformation 12a</caption><tr><td>Environment C-C</td><td>sp</td><td>C6</td></tr><tr><td>Environment C-C-H</td><td>sp<sup>2</sup></td><td>C1 C2 C3 C4 C5</td></tr><tr><td>Environment C-C-N</td><td>sp<sup>2</sup></td><td>C7</td></tr><tr><td>E</td><td>-</td><td>-379.788760 u.a</td></tr></table>

<table><caption>Figure 2.80: Set of values of parameters of conformation 12a (Pyr, Hyb, AngDef, SphCurv)</caption><thead><tr><th>Atom</th><th>Pyr</th><th>DefAng</th><th>Sph</th><th>Hyb</th></tr></thead><tbody><tr><td>1</td><td>0.661212</td><td>0.039825</td><td>0.018010</td><td>2.000799</td></tr><tr><td>2</td><td>1.237321</td><td>0.138851</td><td>0.033134</td><td>2.002802</td></tr><tr><td>3</td><td>2.797924</td><td>0.715678</td><td>0.074597</td><td>2.014400</td></tr><tr><td>4</td><td>0.175846</td><td>0.002846</td><td>0.004692</td><td>2.000057</td></tr><tr><td>5</td><td>2.058030</td><td>0.383770</td><td>0.055007</td><td>2.007768</td></tr><tr><td>6</td><td>NaN</td><td>NaN</td><td>NaN</td><td>NaN</td></tr><tr><td>7</td><td>2.927266</td><td>0.783591</td><td>0.072709</td><td>2.015771</td></tr></tbody></table>

The last point of the path is reached point "12a".
2.4.7 Comparative graphs

<figure><img src="image_86.png" /><figcaption>Figure 2.81: Graph displaying the angle of pyramidalization as a function of carbon atoms</figcaption></figure>

<figure><img src="image_87.png" /><figcaption>Figure 2.82: Graph displaying the angle of pyramidalization as a function of carbon atoms (taking into account the configurations)</figcaption></figure>
* Hybridization

<figure><img src="image_88.png" /><figcaption>Figure 2.83: Graph displaying the hybridization as a function of carbon atoms</figcaption></figure>

<figure><img src="image_89.png" /><figcaption>Figure 2.84: Graph showing the hybridization and the energies as a function of transition states</figcaption></figure>
<figure><img src="image_90.png" /><figcaption>Figure 2.85: Graph showing the angular defect as a function of carbon atoms</figcaption></figure>

<figure><img src="image_91.png" /><figcaption>Figure 2.86: Graph representing the spherical curvature as a function of carbon atoms</figcaption></figure>
These analyses allow us today to propose a **physical representation** of the reaction path (see figure 2.87) of the route (i) directly connected to the classical expected description of the chemist in terms of Lewis representation.

<figure><img src="image_92.png" /><figcaption>Figure 2.87: Representation of the reaction path of route (i) 2.58</figcaption></figure>

## 2.5 Conclusion

In this chapter we studied a large range of examples following tools giving access to all the information necessary to characterize the deformation and the reactivity of the systems. 3 major categories of molecules are studied: **non-fullerenic**, **fullerenic** and **nitrile imines**.

In the case of the **non-fullerenic molecules** we observed that:

* the *Pyr* are more important at the vertices, at the junctions of the patterns and at the bridged zones,

* the **hybridization** is generally uniform around 2.00 which means $sp^2$ atoms hybridization. The case n°13 corannulene is totally uniform with all atoms hybridized $sp^2$. But in some cases the hybridization > 3 in bridged molecules,

* the **spherical curvature** is still < 1.

In general, we observe limitations of the methods for atoms whose degree is > 3. The **angular defect** provides a way to go beyond this limit but presents negative values. On the set of these molecules, the deformation is more significant on the junction zones which seem then more susceptible to be **reactive**.

For the **fullerenic molecules** we note that:

* the *Pyr* is quite uniform on this type of spherical molecule but the *Pyr* is never > 6 but is higher when the fullerene is a few distorted at the poles,

* the **hybridization** is around 2 but is still < 4 on the poles atoms,

* the **angular defect** is calculable on all the atoms but negatives values on the atoms of degree > 4,
• the **spherical curvature** is still < 1.

The same general limitations are observed with the atoms of degree > 3.

In a general way, fullerenic molecules seem to be more adapted to the use of the set of tools and are potentially **more reactive** on more deformed molecular regions.

We note an obvious limitation of our tools with the angular defect, since even if the angular defect is computable in all cases, the negative values observed have no physical meaning. This is an important point which requires additional research beyond this thesis work.

Concerning the **2 figures (2.57)** (for reminder the non-fullerenic molecules are of number 1 to 13 and the fullerenic molecules from 14 to 25):

For the *Pyr*, the 2 groups are clearly observed: the non-fullerenic with rather low pyr values between 0 and 10° for an environment with *sp*²-hybridized carbon-carbon-hydrogen type atoms and the fullerenic with higher *Pyr* between 8 and 15° for an environment predominantly carbon *sp*² hybridized environment. This observation is logical since the non-fullerenic exhibit less curvature.

For **hybridization** we also have 2 groups for the same environment as *Pyr*:

Non-fullerenes molecules have a hybridization between 2 and 2.5. For the fullerenes the hybridization is also between 2.05 and 2.6 but we observe much higher and less classic values with *sp*⁴-type hybridizations.

For the **nitrile imines** molecules:

We show that the parameters are useful in the case of study of **reaction path**, and allow to follow the evolution of the bonds of a system. By the different graphs, we observe that a complete study of the system can be carried out.

We show by this specific example that the use of the tools described in this work is simple and gives access to useful geometrical and chemical information along reaction path.
Chapter 3

Beyond the π-Orbital Axis Vector: the
POAV2 theory

In this chapter we look beyond the theories presented in the 2 previous chapters. We expose the POAV2
theory revised, initially developed by R. C. Haddon in 1986 [6].

We give a complete derivation of the POAV2 theory with complete proofs according to different param-
eters:

• the σ and π-orbitals,

• the existence and uniqueness of POAV2,

• the orthogonality,

• and the hybridization numbers.

Then, we make a partial comparison between POAV1 and POAV2 theories and an exploration with
possible applications.

We used results from the work of R. C. Haddon [6] and our preprint work on POAV2 [36].

**3.1 Study of POAV2 theory**

As proven in the previous chapters, the **POAV1** theory gives tools to identify the **local geometry** of a molecule by emancipating from the bond lengths, angles between bonds of neighboring atoms and laborious quantum computations.

If we consider the situation of POAV1, it is a very particular case, typically the $C_{3v}$ symmetry. The use of POAV1 is perfectly valid for the description of local properties but for a more global description we have to generalize the notion. It is judicious to take into consideration more information such as the length and angles between bonds. This is what R. C. Haddon proves in his 1986 article [6]. The **POAV1** is a **pure geometric object** and its construction depends only on the **local geometry** for a vertex and therefore for a considered atom. The relation with the electronic structure depends on the chosen basis of molecular orbitals. By the concept **POAV2**, R. C. Haddon describes the characteristics in a more **physical** way.

According to R. C. Haddon, the description of the σ-system is possible by choosing a set of orbitals
in the direction of the bonds. Then, an orbital is added to define the π-system. The family of orbitals
has an orthogonality with a certain vector v. The validity of the existence of v depends on the angles
between bonds.
Thus, R. C. Haddon introduced two notions to study non-planar molecules and their electronic structure: the POAV1 for π-orthogonal average vector theory and then, the POAV2 theory aiming to solve some particular problems related to the POAV1 theory.

### 3.1.1 σ and π-orbitals

We consider a **trivalent molecule** as we have defined it in section 0.1 as a set of atoms $A_i$ (with $i = 1, \dots, n$) and 3 atoms $B_i$ (with $i = 1, 2, 3$), and $\theta_{i,j}$ the angle between the bonds $AB_i$ and $AB_j$ (with $i$ and $j \in \{1, 2, 3\}$).

We set **σ-orbitals** as previously in section 1.2.1, $h_i$ such as:

$$h_i = c_i s + \lambda_i u_i \qquad (3.1)$$

with $i = 1, 2, 3$ and where $u_i$ is the unitary vector along the internuclear axes between A and $B_i$ in a trivalent case, $u_i \in \mathbb{R}^3$ and $\lambda_i > 0$. $AB_i = \lambda_i u_i$ with $\lambda_i = \|AB_i\|$ and $\|u_i\| = 1$.

Then, the **π-orbital** is given as:

$$h_{\pi} = c_{\pi} s + \lambda_{\pi} u_{\pi} \qquad (3.2)$$

where $u_\pi$ is the unitary vector.

The $u_\pi$ vector is, when it exists, an **unitary vector** of $\mathbb{R}^2$ such that the family of orbitals $\{h_\pi, h_1, h_2, h_3\}$ is orthogonal, meaning that $\langle h_i, h_j \rangle = 0$ and $\langle h_i, h_\pi \rangle = 0$ for $i$ and $j \in \{1, 2, 3\}$ and $\langle h_i, h_\pi \rangle = 0$ for $i = 1, 2, 3$, where $\langle \cdot, \cdot \rangle$ is the classical scalar product on $\mathbb{R}^3$.

We know that such a vector exists when the molecule possesses a $C_{3v}$ symmetry and is given by the POAV1 vector. However, the existence of such a vector in the general case or its uniqueness is **not trivial**.

### 3.1.2 Existence, uniqueness of POAV2 and orthogonality conditions

Contrary to the POAV1, the **existence and uniqueness** of POAV2 does not follow pure geometric considerations. It shows by the fact that the 2 theories have very **different basis**.

In fact, the POAV1 vector satisfies only a partial orthogonality condition. Precisely, orthogonality is only required between the orbitals $h_\pi$ and $h_i$ with $i = 1, 2, 3$. We recover a complete orthogonality only in the $sp^3$ case.

The POAV2 vector exists and is unique up to **orientation**.

The orientation of the POAV2 vector can be fixed as: if the molecule determine an orientable surface then one can defined a coherent family of normal in each atom. We then, choose the orientation of a $u_\pi$ vector in accordance with this orientation.

We then define the **POAV2** more precisely as:

The family of orbitals used $\{h_\pi, h_1, h_2, h_3\}$ is orthogonal to $u_\pi$ the unitary vector of $\mathbb{R}^2$ when it exists. This means that $\langle h_i, h_j \rangle = 0$ and $\langle h_i, h_\pi \rangle = 0$ for $i$ and $j \in \{1, 2, 3\}$.

The conditions on **orthogonality** are given as:

$$H_{i,\pi} : c_i c_{\pi} + \lambda_i \lambda_{\pi} \cos(\theta_{i,\pi}) = 0 \qquad (3.3)$$

with $i = 1, 2, 3$.
$$ H_{i,j} : c_i c_j + \lambda_i \lambda_j \cos(\theta_{i,j}) \qquad (3.4) $$

with $i \neq j \in \{1, 2, 3\}$.

By the relations given for the conditions of orthogonality, we can notice a strong relation between the angles of the bonds and the angles that $u_π$ does with the bonds.

Proofs of these relations are in Appendix 2 section 11.

Then, we get the relations between the angles $\theta_{i,j}$ and $i, j \in \{1, 2, 3, \pi\}$:

$$ \cos(\theta_{j,k}) \cos(\theta_{i,\pi}) = \cos(\theta_{j,\pi}) \cos(\theta_{i,k}) \qquad (3.5) $$

with $i, j \in \{1, 2, 3\}$ and $i \neq k, j$.

We find this previous relation in the work of R. C. Haddon [6] (eq. 15) and the complete proofs are given in Appendix 2 section 12.

We note that if a molecule has a $C_{3v}$ symmetry then the angles between the bonds and the $u_π$ vector are equal. This is the only case where POAV1 and POAV2 are **identical**.

Proofs of this statement is given in Appendix 2 section 13.

The components of the vector $u_π = (x_π, y_π, z_π)$ (in an orthogonal reference system) are the solutions of a linear system whose coefficients will depend on the unit vectors $u_i (x_i, y_i, z_i)$ along the bonds and the angles between bonds (with $i = 1, 2, 3$).

The POAV2 is a solution of the **linear system** $M.u_π = 0$, with $u_π$ of coordinates $(x_π, y_π, z_π)$:

$$ M = \begin{pmatrix} x_3 \cos \theta_{1,2} - x_2 \cos \theta_{3,1} & y_3 \cos \theta_{1,2} - y_2 \cos \theta_{3,1} & z_3 \cos \theta_{1,2} - z_2 \cos \theta_{3,1} \\ x_1 \cos \theta_{2,3} - x_3 \cos \theta_{1,2} & y_1 \cos \theta_{2,3} - y_3 \cos \theta_{1,2} & z_1 \cos \theta_{2,3} - z_3 \cos \theta_{1,2} \\ x_2 \cos \theta_{3,1} - x_1 \cos \theta_{2,3} & y_2 \cos \theta_{3,1} - y_1 \cos \theta_{2,3} & z_2 \cos \theta_{3,1} - z_1 \cos \theta_{2,3} \end{pmatrix} \qquad (3.6) $$

The kernel of the matrix is of dimension 1 in the case of a non-planar molecule meaning that the bonds $u_1, u_2, u_3$ define a 3-dimensional vector space. By consequence, there is $u_π \in ker(M)$ as $ker(M) = 1$ there exists a unitary vector $u$ such that $ker(M) = \langle u \rangle$ unique up to orientation.

This system is given by R. C. Haddon in [6] (eq. 16) and the complete proofs are in Appendix 2 sections 14 and 15.

Concerning the orientation of POAV2, the molecule defines an **orientable** surface thus we can define a family of normals to each atom. The orientation of POAV2 will depend on the orientation of $u_π$.

### 3.1.3 Hybridization numbers

As mentioned in chapter 1, the **hybridization numbers** are related to the $σ$- and $π$-orbitals which were described at the beginning of this section (eq. 3.1 and 3.2).

We can determine the hybridization number $n_i$ associated to the $σ$-orbitals. We find in the publication about POAV2 of R. C. Haddon [6] the connection with hybridization under the label $n_1, n_2$ and $n_3$. Moreover R. C. Haddon refers to $sp^n$ hybridization as the $p$-part of the $σ$-orbitals and to $s^mp$ as the
s-part of the π-orbitals.

Here, we show that in the case of $h_i$ the number of hybridization $n$.

The idea is precisely to measure in some sense the weight of the orbital $p$ with respect to $s$ in $h_i$ seen as $s c_i^2 p^{\lambda_i^2}$ where $c_i^2$ and $\lambda_i^2$ are the weight of the orbitals $s$ and $p$ in the sense proposed by R. C. Haddon [6] putting all the weight on $p$ as $sp^{\frac{\lambda_i^2}{c_i^2}} = sp^{n_i}$.

We have a hybridization of the $sp_i^n$-type as:

$$n_i = \frac{\lambda_i^2}{c_i^2} \qquad (3.7)$$

Then, we note for $n_i$ with $i = 1, 2, 3$:

$$ \begin{aligned} n_1 &= -\frac{\cos \theta_{2,3}}{\cos \theta_{1,2} \cos \theta_{1,3}} \\ n_2 &= -\frac{\cos \theta_{3,1}}{\cos \theta_{2,3} \cos \theta_{2,1}} \\ n_3 &= -\frac{\cos \theta_{1,2}}{\cos \theta_{3,1} \cos \theta_{3,2}} \end{aligned} \qquad (3.8) $$

These formula coincides with the one given by R. C. Haddon [6] (eq. 10).

We give the proofs of these formula in Appendix 2 section 16.

In the case of $h_π$, the hybridization is of type $s^mp$ and the number of hybridization associated is $m$ as:

$$m = \frac{c_{\pi}^{2}}{\lambda_{\pi}^{2}} \qquad (3.9)$$

Once again, this definition is different to the one given by R. C. Haddon [30] (eq. 3) but follows the same idea.

The number $m$ can be understood as a measure of the $s$ orbital with respect to the $p$ content in the $h_π$ orbital, i.e. considering $h_π$ as $s c_π^2 p^{\lambda_π^2}$ and finally $s^{\lambda_π^2} p$ which corresponds to $s^mp$.

Imposing a **normalization** of the $s$ content in the three hybrids, i.e. as:

$$\sum_{i=1}^{3} \frac{c_i^2}{c_i^2 + \lambda_i^2} + \frac{c_\pi^2}{c_\pi^2 + \lambda_\pi^2} = 1 \qquad (3.10)$$

this is called by R. C. Haddon the $sp^3$ normalization in section 1.2.1.

We obtain then, the expression of the quantity $W_σ$ for the π-hybridization number:

$$W_{\sigma} = \sum_{i=1}^{3} \frac{1}{1 + n_i} \qquad (3.11)$$
Then, we set under the sp³ normalization condition, the π-hybridization number is given following m such that:

$$m = \frac{1}{W_{\sigma}} - 1 \qquad (3.12)$$

The proof is given in Appendix 2 section 17, the formula looks like the one given by R. C. Haddon in [30] (eq. 14).

## 3.2 Comparison of the POAV1 and POAV2

We know that the **POAV theory** allows to predict the maximum direction of the reactivity by the local study of the π-system. As we have presented, 2 approaches exist to determine the π-system: by quantum calculations and by the POAV theory.

The choice of the POAV method is obvious since it is simpler. The selection between POAV1 and POAV2 is relatively trivial since POAV2 involves more parameters and this is what we try to show in this section.

The work on **POAV2 theory** has not yet been explored sufficiently at this time of the thesis to provide complete conclusions. We present here **partial** results. This part of the work has been possible thanks to our collaboration with Dr Germain Salvato-Vallverdu who realized computations and python programs in order to carry out this section.

### 3.2.1 Relative variation of POAV2 versus POAV1

As we know the POAV theory provides the study of the **planarity** of molecules and here we are trying to know if the POAV2 is more adapted than the POAV1. It should be noted that the POAV1 theory is more widely used (as examples some recent works using POAV1 [37][38][39]).

The POAV2 theory takes into account additional parameters compared to POAV1. We therefore wanted to know if there was a real observable difference between the POAV1 and POAV2 theories. The first idea was to make a **comparison** between the 2 vectors.

Our first exploration was on the molecule n°3 (from section 2.3.1) from the work of R. C. Haddon [6]:

<figure><img src="image_93.png" /><figcaption>Figure 3.1: Representation of the molecule C₁₁H₈ with the POAV1 (in green) and POAV2 (in red) vectors</figcaption></figure>

*The molecule being symmetrical we have represented the vectors on only a part of the molecule to make it clearer.*

On this figure we observe that the 2 vectors are very **close** in fact the maximum variation is of 4°. The difference is observed on the atom n°10 which is finally the most interesting since it is at the junction between 2 pentagons and according to our detailed study (section 2.3.1) it is where *Pyr* is the highest.
<figure><img src="image_94.png" /><figcaption>Figure 3.2: Representation of the molecule $C_{11}H_8$ with the POAV1 (in pink) and POAV2 (in grey) vectors and the HOMO</figcaption></figure>

The figure represents the Highest Occupied Molecular Orbital (HOMO) of the molecule n°3. We observe the **isosurfaces** for values ranging from 0.02 (in red) to 0.09 (in magenta). We note that around the atom n°10 it is quite isotropic.

These images were carried out by DFT calculations and by extracting data from the generated `.cube` files. These calculations have been done with fine grids at a distance of 1 Å and with angles between 2 points of 1°.

The main difficulty consists in the size of the grid used on which the wave function is computed, it must be fine enough to make comparison possible and is not necessarily adaptable in Gaussview software. On this illustration we note a difference between the 2 vectors but it remains **minor**. Based only on this example we cannot decide if the POAV2 is more appropriate than the POAV1.

### 3.2.2 Quality of POAV1 versus POAV2 to predict the direction of the π-system

To investigate the comparisons, we used a simpler system: ethylene.

The first step was to calculate by classical DFT quantum methods, the π-system of the molecule, what we call the "real" π-system.

By taking into account again the HOMO of the molecule (with an isovalue of ±0.03Å, in blue the negative part and in red the positive part) and the **nodal surface** (the white grid) of the system, the following figure obtained with VMD software [40] illustrates the molecule in 3D:

<figure><img src="image_95.png" /><figcaption>Figure 3.3: Representation of the HOMO and the nodal surface of the ethylene molecule in 3D</figcaption></figure>

To determine and carry out each of the constructions (POAV1, POAV2, π-system), we investigated different methods by **linear interpolation**.

The first approach was to perform a linear regression to obtain a fitted line on the intersection between
the nodal surface and a chosen plane.

The other possibility was to look for the normal vector (the "real" $\pi$-system) to the mean plane fitted on the points of the nodal surface around a given atom.

The goal is to see if the normal vector to the nodal surface is in the plane defined by POAV1 and POAV2. As a result, it is essential to have a very fine grid to obtain precise results.

By linear interpolation and by the second approach, we made figures (figure 3.4) in the plane. The circle points represent the atoms: *black* for carbon and *grey* for hydrogen. In the left figure, we observe the HOMO in the plane in *red* and *blue*. The isosurface is represented by the curve in *grey* (at $\pm 0.03$ Å). Finally, the *black* curve is the intersection between the plane and the nodal surface, which represents the plane defined by POAV1 and POAV2.

<figure><img src="image_96.png" /><figcaption>Figure 3.4: Representation of the linear interpolation for the ethylene molecule</figcaption></figure>

Using a fine grid (in the right figure), i.e. a distance between 2 points of $0.005$ Å, the grid is located around an atom of interest and the dotted curve represents the nodal surface.

<figure><img src="image_97.png" /><figcaption>Figure 3.5: Representation by linear interpolation of POAV1 and POAV2 (in blue the normal, in green the POAV1 and in red the POAV2)</figcaption></figure>
To clarify the construction of this last figure: we consider the closest point on the nodal surface, i.e. the orthogonal projection of the atom on the nodal curve. This means that we take the tangent to the curve and the vector we are interested in is the one that is normal to this tangent. Once this vector is determined, we relate it to the atom and compare its direction with POAV1 and POAV2.

We note in this case studied more precisely, that there is a difference between POAV1 and POAV2. This difference has not been quantified but merits to be quantified. We notice that the POAV2 is closer to the "real" π-system than the POAV1. Thus, by comparison the **POAV2** seems **more accurate** for the considered atom.

### 3.3 To reach further...

By the tools presented since the beginning of this manuscript, we can consider using them to **characterize** the geometries and topologies of different kind of systems as **Covalent Organic Frameworks (COFs) and Metal-organic frameworks (MOFs) for their reactivity**. This is in line with a strategy of decarbonization and energy transition. The systems could be integrated into hydrogen or methane storage or CO₂ capture processes. The purpose of these processes is not the same. Hydrogen and methane are intended to be stored for use primarily as an energy source. CO₂ capture is aimed at reducing anthropogenic emissions directly at the source.

We will make some reminders about MOFs and COFs. These complexes are crystalline solids said to be porous. Their structure makes them promising complexes for gas storage by physisorption. These materials are listed in two main databases: from Cambridge University [41, 42] and Northwestern University [43], both listing about 255,000 structures. These materials with high sorption properties can be used in surface chemistry and their reactivity can be controlled generating new electronic and chemical properties. The high number of existing structures, leads to a problem of identification of structures for a specified application. Therefore, with our tools we could propose a **topological** and **geometrical** identification according to a given property.

We could provide a selection process for COFs and MOFs based on tools whose scientific development is clear in order to organize the COFs and MOFs bases in the best possible way.

The steps that could be implemented would be based: 1 - on the development of a tool analyzing the reactivity properties of the COFs and MOFs of all the databases, 2 - then, the extraction of specific indicators such as reactivity and 3 - then, validate these tools by experimentation.

Steps 1 and 2 can be completed by our POAV tools to connect the geometrical and topological structure of the molecules with the electronic and chemical properties. In fact, POAV vectors intrinsically include all the quantum and orbital information that allow to study the chemical reactivity of the atoms of the material. The POAV vectors are computable from the local geometrical structure without using quantum calculations. The simplicity of use leads to a pretty large scale database, as we have done previously for our first article on the fullerene database [4]. In a simple way, it allows to detect on the materials the most reactive sites and to cartography them (as we have already achieved). Thus, we suggest the use of POAVs to cartography the structures of COFs and MOFs.

Following these cartographies, it will be necessary to obtain different indicators which will reflect the reactivity and selectivity of COFs and MOFs for a given target. Statistical by nature, these indicators will show the properties of the POAVs, in particular for the density of the highly reactive sites and for the average reactivity of the structure.

Finally, the last step, step 3, will validate and complete the theoretical data.
### 3.4 Conclusion

In this chapter, we take up the POAV1 theory with the POAV2 which completes in a **physical way** the POAV1 which is more **local**.

We have proven as precisely as possible mathematically the work of R. C. Haddon on the POAV2, on the basis of $\sigma$ and $\pi$-orbitals but also on the conditions of orthogonality and based on the tools previously developed as the hybridization numbers.

The comparison of the work initiated here between the 2 tools shows from a more practical point of view and from the molecular point of view that the consequences are **not significant** between POAV1 and POAV2. In fact for our example, there is only 4° difference between the 2 vectors, it is then, complicated to conclude only on this observation.

However, when we compare the difference between POAV1 and the $\pi$-system and between POAV2 and the $\pi$-system it is **more significant**. We can clearly see that POAV2 is closer to the $\pi$-system. From this point of view the POAV2 theory is more interesting since it is more representative of the system. This is in accordance with the general observations related to molecules with large bond angles, POAV2 is quite different from POAV1 but this is logical since the bond parameters are taken into account.

The observations from these comparisons are **clearly not conclusive** and the development of the proofs related to these tools having required a long work it seems interesting to focus on it in the future.
Conclusion of the part II

This part II concerns the **POAV theories**. It gives a complete study of the tools and methods avail-
able to the chemistry community to understand and determine the information of molecular systems
related to **deformation, hybridization** and **reactivity**.

Chapter 1 and 2 are complementary since the notions and tools (pyramidalization angle, POAV1, hybridizations...) are used in the wide range of examples in chapter 2 and are clearly studied, generalized and proven. Chapter 2 gives a concrete idea of the use of these tools from a computational and illustrative point of view with meaningful cartographies on a large panel of molecules. We show that the evolution of the transformations of the bonds (formation, weakening...) can be followed in simple ways and thus allow the understanding of the reactivity.

Chapter 3 is an **extension** of POAV1 theory with the introduction of POAV2 theory which seems theoretically more complete but does not provide clearly any practical benefit for the moment.
This remains to be investigated since the study of this last chapter is not completed.

In a general way, this part answers the initial issues generated by R. C. Haddon’s work about plar-
narity and reactivity, and it is important to mention that all the methods and tools presented in this
manuscript are useful and easily implementable by chemists.
# Part III

## Aromaticity, helical states and applications
In this third part, we are interested both in the aromaticity criteria, and in the curiosity of helical electronic states.

The aromaticity criteria are studied :

• in a general way by the Hückel rules,

• by the 2 visions of Hückel and Möbius in the case of annulenes.

We review these criteria from a mathematical point of view with Hückel and Möbius by distinguishing the cases of linear molecules and cyclic molecules.

Then, we go beyond these aromaticity criteria with the study of Möbius systems according to Roald Hoffmann and show the criteria and the existence of helicoidal states.

Finally, applications and visualizations of helical states are presented.

The main results of this part come from the work of R. Hoffmann's research team [9] and from our preprint work [44].
# Introduction

This third part gives the opportunity to go through different aspects, both **theoretical** and **applicative**. The study of concepts related to molecular deformation and tools for understanding the reactivity are associated with the aromaticity topics.

Chapter 4 leads us back to the theory with the **aromaticity criteria** (of cyclic molecules among others) according to the rules stated by **Hückel**. These famous Hückel's rules are widely used in chemistry and in particular by organic chemists.

We have relied on Hückel's rules to highlight the comparison between the so-called **Hückel and Möbius model systems**. First, we prove the geometrical construction of these systems, and then their **optimal and equivalent** structures. To improve the comprehension and to put in evidence more details, we needed to review some mathematical tools. In particular, concerning the **Hückel matrices** according to different cases. In fact, we have chosen to work according to different situations, namely in **cyclic** situations (to get closer to an annulenic system) or in **linear** situations (as cumulenic systems). After these different steps, and a brief review of the Möbius molecules, we established the **energy equations** according to the two model systems. To complete the study, we took into account both in the Möbius and Hückel case the **number of electrons** of the system: even or odd.

In the Möbius systems, we observe a **rotation** of the orbitals along the cycle, and by consequence different **angle distributions**. We have therefore decided to study these distributions and the influence of the $C_2$ symmetry on them.

To be more precise and applicative, we conclude with some simple examples to prove the calculation of the distribution of angles.

Chapter 5, presents how Möbius systems have found interest in recent years from research teams, in particular the Nobel prize laureate Roald Hoffmann.

It is due to the attention given to the work of R. Hoffmann that we are particularly interested in the **linear systems** called **cumulenes** and the curiosities that they provide with the **helical molecular orbitals**. We then details associated to the concepts concerning the **helical states**. We prove mathematically in a precise way the angle distributions in the general case and in the cumulene cases. The rest of the chapter presents the **criteria of existence of helices** from a mathematical and then by a chemical way.

Then, the last chapter provides a link with the chapter 5 by different **applications**. The chapter presents the applications and visualizations that we carried out on cumulenes and on more complex molecules. All these investigations are very complete since we have performed DFT and multideterminantal CASSCF(CASPT2 single points) calculations both in fundamental and excited states, to be confident that the helices observed by DFT were not dependent on the method used.
Chapter 4

Criteria of aromaticity - theory

In this chapter we discuss :

• the criteria of aromaticity,

• reminders of mathematical tools for Hückel matrices,

• and, the determination of energy equations according to the Hückel and Möbius systems.

For the criteria of aromaticity we expose:

– Hückel’s rules,

– the 2 visions of Hückel and Möbius concerning the annulenic systems.

Concerning the mathematical tools necessary to understand the Hückel matrices we review:

– Hückel matrices for cyclic ($S_N^1(\phi)$) and linear ($L_N(\phi)$) molecules.

Finally, for the determination of the energy equations of Möbius and Hückel systems, we expose:

– reminders about Möbius molecules,

– results concerning the cases of cyclic molecules with zero distribution ($S_N^1(0)$) and with angle distribution ($S_N^1(\phi)$),

– the energetic comparison of the 2 cases and their stability,

– the study of the molecule with angle distribution following a $C_2$ symmetry invariance,

– the results for linear molecules with null distribution ($L_N(0)$) and angle distribution ($L_N(\phi)$),

– the study of optimal distribution for cyclic molecules ($S_N^1(\phi)$),

– the study of the distribution of linear molecules ($L_N(\phi)$) equivalent to cyclic molecules with angle distribution ($S_N^1(\phi)$),

– the asymptotic study of the angle distribution on a linear chain.

All the results of this chapter are based on R. Hoffmann’s work [9] and from our preprint dedicated to the study of the optimal structures and equivalences of some molecules [45]. This chapter has required a lot of work and time.