**4.1 Hückel aromaticity**

We have presented the Hückel method in part I, but here we are interested in another aspect: the **aromaticity** according to Hückel [46, 47].

Different approaches to Hückel's rules exist both on criteria based on the **number of electrons**, but also based on the **geometrical construction** related to the distribution of orbitals within the system.

**4.1.1 Hückel rules**

From the point of view of organic chemists, **Hückel's rules** are well-known. It is in 1931 that Erich Hückel proposes the theory of determination of the **aromatic compounds** in the case of plane cyclic molecules (also called annulenes). These **aromaticity criteria** are based on several points: the molecule must be **cyclic**, must be **plane**, must be **conjugated** and must **satisfy** the relation 4N+2.

*For more details, in a simple way, the conjugation is the alternation of single (σ) and double bonds (π) and the relation is equal to the number of π-electrons of the molecule.*

The aromaticity reflects the **stability** and consequently simplifies certain reactions such as substitutions.

In the fundamental state for a cyclic molecule, if the electronic system has 4N + 2 electrons then, the system is said to be **aromatic** and **stable**. On the contrary, if the electronic system has 4N electrons, the system is called **anti-aromatic**.

The 4N + 2 rule applies when there is an **even** number of electrons. These systems are then called **Hückel systems**.

To illustrate this rule, we can take the simple example of **benzene** which is the well-known conjugated cyclic molecule. Containing 6 π-electrons, there are 2 electrons on the first lowest energy orbital and then the remaining 4 electrons are distributed on the following orbitals, as shown in the following diagram:

<figure><img src="image_98.png" /><figcaption>Figure 4.1: Simplified energetic diagram of the benzene</figcaption></figure>

In this simple example, 4N + 2 = 6 then, N = 1, we obtain a positive integer, the rule is respected.
The Hückel rule is used to determine the aromaticity thus, it only considers the π-electrons, i.e. the
electrons involved in the π-bonds. The atoms implied are therefore sp²-hybridized.

For these cyclic systems, a simple mnemonic circle method proposed by Frost and Musulin [48] in 1953
allows access to the energies of the molecular orbitals.

In 1966, on the basis of the method of Frost and Musulin [48], Zimmerman [8] proposes according to
the 4N + 2 rule of Hückel, that the systems **satisfying** the rule will be called of **Hückel** but that
those satisfying 4N electrons will be called systems of **Möbius** [8].
### 4.1.2 Two visions of annulenes : Hückel and Möbius

#### Geometrical construction

As we have already mentioned in part I with a **geometrical** point of view, we can study different cases of **arrangement** of atoms.

The atoms of a molecule can be disposed following two possibilities:

* a **line** segment ($L_N$) which corresponds to a **linear chain** (as allene molecule)

* a **circle** ($S_N^1$) which corresponds to **cyclic** molecule (as annulene molecule)

We are interested in **two distributions** depending on the arrangement of the atoms. These distributions have a **fundamental role**.

A **distribution** is defined along a molecule $\gamma$ constituted by N atoms noted $s$ (a preliminary definition is given in the reminder 0.2.4). The distribution is given in $\mathbb{R}^3$ as a family of vectors $v$ $\{v_1, ..., v_N\}$ in the planes $\{P_{s_1}, ..., P_{s_N}\}$.

<figure><img src="image_99.png" /><figcaption>Figure 4.2: Schematic representation of the distribution along a molecule $\gamma$</figcaption></figure>

In fact if we have: $\gamma: s \in [a,b] \rightarrow \gamma(s) \in \mathbb{R}^3$ a regular curve of $\mathbb{R}^3$ on which are uniformly disposed N atoms. We note $s_i \in [a,b]$ the position $\gamma(s_i)$ of the atom $C_i$ (with $i = 1, ..., N$). We assume that $s_0 = a$ and $s_N = b$:

$$s_i = a + (i - 1) \frac{b - a}{N - 1} \qquad (4.1)$$

Set a family of vectors $u_i$ in $\gamma(s_i)$ contained in the plane normal to the tangent vector $\gamma'(s_i)$ to the $\gamma$ curve in $s_i$. These vectors will represent in our case, particular orbitals in each atom. For each $i = 1, ..., N$, let us note $D_i$ the segment described by:

$$t \in [-1, 1] \rightarrow \gamma(s_i) + t \frac{u_i}{\|u_i\|} \in \mathbb{R}^3 \qquad (4.2)$$

We note that $D_i(0) = \gamma(s_i)$ and that the line segment is symmetrical about this point.

A **homotopy** is carried out between each $D_i$ and $D_{i+1}$ (with $i = 1, ..., N - 1$) in the following way:

we denote $D(s)$ the continuous family of segments defined for $s \in [a, b]$ by:

$$D(s) = \left\{ t \in [-1, 1] \rightarrow D(s)(t) = \gamma(s) + t \left[ \frac{s_{i+1} - s}{s_{i+1} - s_i} \frac{u_i}{\|u_i\|} + \frac{s - s_i}{s_{i+1} - s_i} \frac{u_{i+1}}{\|u_{i+1}\|} \right] \in \mathbb{R}^3 \right\} \quad (4.3)$$

with $s \in [s_i, s_{i+1}]$.

Note that $D(s_i) = D_i$ for $i = 1, ..., N$. We thus generate a connected parameter surface of $\mathbb{R}^3$ associated with the data of $\gamma$ and the family of vectors $(u_1, ..., u_N)$ noted $\gamma_N(u_1, ..., u_N)$ such as:
$$ \gamma_N(u_1, \dots, u_N) = s \in [a, b] \rightarrow D(s)(t) \qquad (4.4) $$

In the situation where the $\gamma$ curve is **closed**, i.e. $\gamma(a) = \gamma(b)$, the surface described is not necessarily continuous. To be continuous, it is necessary to have $u_N = \pm u_1$ so that $D(a) = D(b)$ and thus, that there is recollection of the segments, when the curve is **crossed**.

In the case where the vectors $u_i$ are associated with a given $\phi = (\phi_1, \dots, \phi_N)$ distribution such as:

$$ u_{i+1} = e^{j\phi_i} u_i \qquad (4.5) $$

with $j^2 = -1$, $i = 1, \dots, N$.

In the case where $\phi = 0$ it is enough to take $u_i = u_1$ for $i = 2, \dots, N$. We obtain then in the case where $\gamma$ is the segment $\gamma(s) = A + sB$ with A and B $\in \mathbb{R}^3$, $B \neq A$, $s \in [0, 1]$ the parameter surface generated by the transfer of the segment:

$$ D(s) = \left\{ t \in [-1, 1] \rightarrow D(s)(t) = \gamma(s) + t \frac{u_1}{\| u_1 \|} \in \mathbb{R}^3 \right\} \qquad (4.6) $$

along the $\gamma$ segment which represents a strip.

If we assume that: $L_N$ is a linear chain with N atoms and $S_N^1$ is a cyclic molecule with N uniformly distributed atoms.

We consider the following geometries:

* the **zero distribution**, noted $\phi = (0, \dots, 0)$ or $\phi = 0$ is represented by a **ribbon** or a strip which is called **"Hückel"-type distribution**.

* the **"Möbius"-type distribution** $\phi = (\pi/N, \dots, \pi/N)$ which is a Möbius ribbon i.e. a **twisted strip** of angle $\pi$ [47].

<figure><img src="image_100.png" /><figcaption>Figure 4.3: Illustration of the 2 distributions: Hückel and Möbius-types [49]</figcaption></figure>

The terminology ("Hückel" or "Möbius") comes from the geometric construction.

In the case of a **Möbius distribution** we have:

$$ u_{i+1} = e^{j\frac{\pi}{N}} u_i \qquad (4.7) $$

with $j^2 = -1$ and $i = 1, \dots, N$.

The $D(s)$ segment is given for $s \in [0, 1]$ by:

$$ D(s) = \left\{ t \in [-1, 1] \rightarrow D(s)(t) = \gamma(s) + t e^{js \frac{\pi}{N}} \frac{u_1}{\| u_1 \|} \in \mathbb{R}^3 \right\} \qquad (4.8) $$
which represents a **180° twisted strip**.

This same construction in the case where $\gamma$ is a circle gives respectively a ribbon and the famous **Möbius ribbon** which is a continuous surface since in this case $u_N = -u_1$.

## Optimal structures

For a given $S_N^1(\phi)$ structure, several questions arise: intuitively, we expect the zero distribution to be the most energetically stable, i.e. it carries out the minimum of the molecular energy $L_N(\phi)$ or $S_N^1(\phi)$ over all possible distributions. However, it can be proven that for some values of N, this **minimum** is realized by a **Möbius-type distribution**. Moreover, the calculations comparing these two particular distributions do not answer the question.

By definition, for a given S and N, we characterize the set of $\phi$ distributions such as $S_N(\phi)$ satisfies the **minimum energy**. In this manuscript we study the problem in all generality for $S = S^1$ and $S = L$.

We specify the previous problem by writing explicitly the optimization problem for a given N and $\phi$ a given distribution. We note $E(S_N^1(\phi))$ the energy of the $S_N^1(\phi)$ molecule.

We notice $E_{min}$ the quantity:

$$E_{min} = \min_{\phi} E(S_N^1(\phi)) \qquad (4.9)$$

In fact, we are looking for the set of distributions such as : $E(S_N^1(\phi)) = E_{min}$.

The **strategy** to solve the question, is to determine the structure of the orbitals for a molecule with N atoms on the circle with a given distribution $\phi$. The approximation of the MOs are obtained using the Hückel method. The energy of the molecule is then calculated on the set of distributions. In a simpler way one can **compare** for a given geometry and type of atom, the **energy stability** for two fixed distributions.

We can easily **compare** for a given geometry and type of atom, the **geometrical stability** for two fixed distributions. We explicitly note a comparison between $S_N^1(0)$ and $S_N^1(\phi_M)$ in the case of carbon.

We specify here $\phi_M$, it describes the Möbius-type distribution.

## Equivalent structures

From a point of view of **equivalent representations**, we can represent the geometry of the $\pi$-orbitals of the molecules of the allene type (called cumulenes) either by a linear chain or a circle.

*Note that a cumulene, which will be mainly discussed in the continuation of the manuscript, are carbon molecules with only double bonds.*

For a given cumulene, how to describe precisely the equivalent linear and circular representations? In particular, how to specify the distribution associated with each of the representations?

To specify the previous question it is necessary to take in consideration the **symmetries** of the molecule. The problem of **equivalent representations** is expressed by the research of a **geometry adapted** to a minor symmetry than the initial molecule.

The computations to obtain the structure of the orbitals in the case of the circle can be used to determine the distribution of a linear chain whose structure of the orbitals is imposed by the decomposition obtained for the configuration of the circle in a given distribution $\phi$. We then obtain for any $\phi$ on the circle, a distribution $L_N(\phi)$ on a chain.
## 4.2 Reviews of mathematical tools with Hückel matrices

### 4.2.1 Hückel matrices for molecules as $S_N^1(\phi)$, $L_N(\phi)$

To better understand the systems $S_N^1(\phi)$ and $L_N(\phi)$, in the following, we express the **Hückel matrices** of the 2 situations.

#### Molecules $S_N^1(\phi)$

As presented in part I, the atoms $A_i$ (with $i = 1, ..., N$) are positioned on a circle and to simplify, we have the notations:

$$A_{N-1} = A_N \qquad (4.10)$$

$$A_{N+1} = A_1 \qquad (4.11)$$

To simplify the construction of the **Hückel matrix**, for a given atom, only **two neighboring atoms** on the curve have a significant overlap, thus, a given atom $A_i$ is bonded to the atom $A_{i-1}$ and $A_{i+1}$ for $i = 1, ..., N$.

We obtain a **symmetrical matrix** $\mathbf{M}[S_N^1(\phi)](E)$ as :

$$ \begin{pmatrix} \alpha - E & \cos(\phi_1)\beta & 0 & \dots & \dots & 0 & \cos(\phi_N)\beta \\ \cos(\phi_1)\beta & \alpha - E & \cos(\phi_2)\beta & 0 & \dots & \dots & 0 \\ 0 & \cos(\phi_2)\beta & \alpha - E & \cos(\phi_3)\beta & 0 & \dots & 0 \\ \vdots & & & & & & & \\ 0 & \dots & \dots & 0 & \cos(\phi_{N-2})\beta & \alpha - E & \cos(\phi_{N-1})\beta \\ \cos(\phi_N)\beta & 0 & \dots & \dots & 0 & \cos(\phi_{N-1})\beta & \alpha - E \end{pmatrix} \qquad (4.12) $$

By posing $x = \frac{\alpha - E}{\beta}$, there is the **reduced matrix** $\mathbf{U}[S_N^1(\phi)](x)$ defined by :

$$ \begin{pmatrix} x & \cos(\phi_1) & 0 & \dots & \dots & 0 & \cos(\phi_N) \\ \cos(\phi_1) & x & \cos(\phi_2) & 0 & \dots & \dots & 0 \\ 0 & \cos(\phi_2) & x & \cos(\phi_3) & 0 & \dots & 0 \\ \vdots & & & & & & & \\ 0 & \dots & \dots & 0 & \cos(\phi_{N-2}) & x & \cos(\phi_{N-1}) \\ \cos(\phi_N) & 0 & \dots & \dots & 0 & \cos(\phi_{N-1}) & x \end{pmatrix} \qquad (4.13) $$

The two matrices can be linked as:

$$ \mathbf{M}[S_N^1(\phi)](E) = \beta \mathbf{U}[S_N^1(\phi)](x) \qquad (4.14) $$

To simplify, we have the family of so-called **Hückel matrices** of type $S^1$ and parameter a ($a = (a_1, ..., a_N) \in \mathbb{R}^N$):

$$ \mathbf{U}_a = \begin{pmatrix} x & a_1 & 0 & \dots & \dots & 0 & a_N \\ a_1 & x & a_2 & 0 & \dots & \dots & 0 \\ 0 & a_2 & x & a_3 & 0 & \dots & 0 \\ \vdots & & & & & & & \\ 0 & \dots & \dots & 0 & a_{N-2} & x & a_{N-1} \\ a_N & 0 & \dots & \dots & 0 & a_{N-1} & x \end{pmatrix} \qquad (4.15) $$

The matrix $\mathbf{U}_a$ is the **matrix associated** with the simple **Hückel matrix** of type $S^1$ where $a_1 = ... = a_N = a$ with the parameter $a \in \mathbb{R}$.
## Molecules $L_N(\phi)$

In this configuration, the atoms are arranged on a **line** and the following simplifying hypothesis is admitted: for a given atom, only two neighboring atoms on the curve have a significant overlap, i.e. a given atom $A_i$ is bounded to atom $A_{i-1}$ and $A_{i+1}$ for $i = 2, ..., N-1$. The atom $A_1$ is bounded to $A_2$ and the atom $A_N$ to $A_{N-1}$.

By this hypothesis, the **Hückel matrix** $M[L_N(\phi)]$ ($E$) is purely tridiagonal:

$$ \begin{pmatrix}
\alpha - E & \cos(\phi_1)\beta & 0 & \dots & \dots & 0 & 0 \\
\cos(\phi_1)\beta & \alpha - E & \cos(\phi_2)\beta & 0 & \dots & \dots & 0 \\
0 & \cos(\phi_2)\beta & \alpha - E & \cos(\phi_3)\beta & 0 & \dots & 0 \\
\vdots & & & & & & & \\
0 & \dots & \dots & 0 & \cos(\phi_{N-2})\beta & \alpha - E & \cos(\phi_{N-1})\beta \\
0 & 0 & \dots & \dots & 0 & \cos(\phi_{N-1})\beta & \alpha - E
\end{pmatrix} \quad (4.16) $$

The matrix reduction leads to $U[S_N^1(\phi)](x)$:

$$ \begin{pmatrix}
x & \cos(\phi_1) & 0 & \dots & \dots & 0 & 0 \\
\cos(\phi_1) & x & \cos(\phi_2) & 0 & \dots & \dots & 0 \\
0 & \cos(\phi_2) & x & \cos(\phi_3) & 0 & \dots & 0 \\
\vdots & & & & & & & \\
0 & \dots & \dots & 0 & \cos(\phi_{N-2})\beta & x & \cos(\phi_{N-1}) \\
0 & 0 & \dots & \dots & 0 & \cos(\phi_{N-1}) & x
\end{pmatrix} \quad (4.17) $$

As in the previous case $S_N^1(\phi)$, we have the **Hückel matrix** of type $L^1$, of parameter a (with $a = (a_1, ..., a_{N-1})$) and of dimension $N \times N$ such that:

$$ M_a = \begin{pmatrix}
0 & a_1 & 0 & \dots & \dots & 0 & 0 \\
a_1 & 0 & a_2 & 0 & \dots & \dots & 0 \\
0 & a_2 & 0 & a & 0 & \dots & 0 \\
\vdots & & & & & & & \\
0 & \dots & \dots & 0 & a_{N-2} & 0 & a_{N-1} \\
0 & 0 & \dots & \dots & 0 & a_{N-1} & 0
\end{pmatrix} \quad (4.18) $$

Here also, we have $a_1 = \dots = a_{N-1}$, the Hückel matrix $L^1$ is called simple.

# 4.3 Determination of energy equations Möbius - Hückel

## 4.3.1 Reviews of Möbius molecules

The so-called **Möbius molecules** have their origins in the first works published by A. F. Möbius and J. B. Listing about projective planes and one-sided surfaces in 1858 [50, 51]. Probable conflicts of publications between the 2 researchers are at the origin of the name of "Möbius" ribbon and not "Listing".

Most of the mathematical objects are two-dimensional and we are familiar with, i.e. they have an inside and an outside surface. But the Möbius ribbon is an exception since it is a **closed strip** with an **odd number of unilateral 180° twists** and an **even number of bilateral 180° twists**. This means that the Möbius ribbon is a **non-orientable** system [47].
<figure><img src="image_101.png" /><figcaption>Figure 4.4: Illustrative representation of the Möbius ribbons [47]</figcaption></figure>

Note that the works of Möbius and Listing have inspired many artists in terms of non-orientable surfaces.

From a chemical point of view, to define a **twisted plane** it is necessary to have a $\pi$-system. In this type of twisted $\pi$-system, the unilateral surface is defined along the nodal plane of the $\pi$-system. This type of molecule are **Möbius annulenes**. We note that the so-called **Hückel annulenes** have $p$-orbitals which are associated in order to create interactions. In the case of Möbius there will be at least one sign inversion of the system which will result in a 180° twist.

As a reminder, annulenes are conjugated cyclic molecules. The simplest example is the benzene which is aromatic or the cyclobutadiene which is anti-aromatic. These 2 examples are perfectly plane.

For an ideal **Möbius ribbon**, the twist is uniformly distributed, therefore, the inversion point is not precisely defined. If we assimilate the Möbius ribbon to an origami type model, the sign inversion will be observed in the most twisted region. For an example of an origami Möbius annulene, the system is not planar and shows the shape of a 8 along a $C_2$ axis of symmetry. Being non-planar, the $\pi$ and $\sigma$-systems are not orthogonal.

Möbius $\pi$-systems have been particularly studied by Heilbronner in 1964 [7] who highlights the aromaticity according to Hückel which we have already discussed in this manuscript (section 4.1.1).

<figure><img src="image_102.png" /><figcaption>Figure 4.5: Scheme representing a Hückel and Möbius type annulene [47]</figcaption></figure>

As shown in the Herges review [47] and in figure 4.5, for the Hückel case, the $\pi$-plane has two sides, thus, there are two $\pi$ clouds on either side of the annulene. In the Möbius case, there is only one $\pi$-system. The presence of the torsion generates a sign inversion so a rotation of the $p$-orbitals.

Note that Heilbronner gives a rather good precision to the stability of the large Möbius annulenes [7].

Finally, the properties of **aromaticity** and **anti-aromaticity** are largely proven by the work of Zimmerman and Frost-Musilin that we have mentioned previously (section 4.1.1).

There are **different** types of Möbius ribbons: we can mention the cyclacenes (purely theoretical, the synthesis being complicated). They are composed exclusively of benzene units which makes them highly unstable but remain very interesting by their electronic structures.
<figure><img src="image_103.png" /><figcaption>Figure 4.6: Representation of a polyacene, a cyclacene and a Möbius cyclacene [47]</figcaption></figure>

Simpler than cyclacenes, polycyclic aromatic hydrocarbons are more interesting. But they present more isomers and the ways to introduce the torsions are numerous. This type of system is therefore more complex. For example, the smallest system is the Möbius coronene.

<figure><img src="image_104.png" /><figcaption>Figure 4.7: Representation of a coronene and a Möbius coronene [47]</figcaption></figure>

Concerning the synthesis, it was necessary to wait 40 years between their discovery in 1964 and the first attempts of synthesis [52, 53]. The main difficulty consists in the synthesis agents which are annulenes which are stable in their non-twisted conformations. The stabilization of Möbius annulenes is based on the rigidification of the molecular frame. The synthesized systems have large $\pi$-systems and some have $C_2$ symmetries.

Möbius annulenes have two types of aromaticity: "normal" and "in the plane". Respectively, either the atoms are $sp^2$-hybridized in a planar trigonal configuration, or $sp^2$-hybridized atoms pyramidal with a certain $sp^3$ character.

<figure><img src="image_105.png" /><figcaption>Figure 4.8: Representation of Möbius annulene [47]</figcaption></figure>

Note that we have to distinguish the **topology** and the **aromaticity criteria**.

From a **topological** point of view: we differentiate the so-called "normal" case, also called the Hückel case in the literature and here the ribbon case, from the Möbius case.

In terms of **aromaticity**, whether it is the ribbon or the Möbius system, it is a Hückel system, i.e. only one p-orbital participates per carbon atom. Indeed it is according to the closure of the ring that will give the meaning to the aromaticity (i.e. the value of the sign of the off-diagonal elements of the Hückel matrix) [9].

To finish, there are other topologies, as Möbius annulenes with double twists. Different examples exist: heterooctaphyrins or the theoretically predicted [14]-annulene.
### 4.3.2 Cases of the molecules $S_N^1(0)$ and $S_N^1(\phi_M)$

We have determined the **electronic structures** of the molecules $S_N^1(0)$ and $S_N^1(\phi_M)$ for all N. However, first, we were interested in the computation of the **energies** according to the configurations.

#### Trigonometry reminder

First of all, we remind some useful information of trigonometries, by detailed proofs.

For all $\theta \in \mathbb{R}$ and $m \in \mathbb{N}$:

$$ \sum_{k=0}^{m} \sin(k\theta) = \sin\left(m\frac{\theta}{2}\right) \frac{\sin\left((m+1)\frac{\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} \quad (4.19) $$

Here is the proof which seems to be the fastest:

$$ \sum_{k=0}^{m} \sin(k\theta) = Im \left[ \sum_{k=0}^{m} e^{jk\theta} \right] \quad (4.20) $$

where $j^2 = -1$ and $Im[z]$ refers to the imaginary part of a complex number $z \in \mathbb{C}$.

We then study, the complex geometric serie $\sum_{k=0}^{m} (e^{j\theta})^k$, the classical result gives :

$$ \sum_{k=0}^{m} z^k = \frac{1 - z^{m+1}}{1 - z} \quad (4.21) $$

For this proof we get:

$$ \sum_{k=0}^{m} (e^{j\theta})^k = \frac{1 - e^{j\theta(m+1)}}{1 - e^{j\theta}} \quad (4.22) $$

It is possible to simplify the expression by using the arc moieties such as :

$$ 1 - e^{j\theta} = e^{j\frac{\theta}{2}}(e^{-j\frac{\theta}{2}} - e^{j\frac{\theta}{2}}) = e^{j\frac{\theta}{2}}2j \sin\left(\frac{\theta}{2}\right) \quad (4.23) $$

Then,

$$ \frac{1 - e^{j\theta(m+1)}}{1 - e^{j\theta}} = \frac{e^{j(m+1)\frac{\theta}{2}} \sin\left((m+1)\frac{\theta}{2}\right)}{e^{j\frac{\theta}{2}} \sin\left(\frac{\theta}{2}\right)} \quad (4.24) $$

as,

$$ \frac{e^{j(m+1)\frac{\theta}{2}}}{e^{j\frac{\theta}{2}}} = e^{jm\frac{\theta}{2}} \quad (4.25) $$

we get:

$$ \sum_{k=0}^{m} (e^{j\theta})^k = e^{jm\frac{\theta}{2}} \frac{\sin\left((m+1)\frac{\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} \quad (4.26) $$

By considering the imaginary part of this expression, we obtain the expected equality:

$$ \sum_{k=0}^{m} \sin(k\theta) = \sin\left(m\frac{\theta}{2}\right) \frac{\sin\left((m+1)\frac{\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} \quad (4.27) $$
This concludes the proof.

The result of this trigonometry relations is used in the following of the manuscript.

## Energy proof

For all $\theta \in \mathbb{R}$ and $m \in \mathbb{N}$:

$$ \sum_{k=0}^{m} \sin(2k+1)\theta = \frac{1}{\sin\theta}[\sin((m+1)\theta)]^2 \qquad (4.28) $$

We use the trigonometry reminders,

$$ \mathrm{Im} \left[ \sum_{k=0}^{m} e^{j(2k+1)\theta} \right] = e^{j\theta} \sum_{k=0}^{m} e^{2jk\theta} \qquad (4.29) $$

By using the relation 4.26:

$$ \sum_{k=0}^{m} e^{2jk\theta} = e^{j(m+1)\theta} \frac{\sin((m+1)\theta)}{\sin\theta} \qquad (4.30) $$

To conclude the proof, by taking the imaginary part:

$$ \sum_{k=0}^{m} \sin((2k + 1)\theta) = \frac{1}{\sin\theta}[\sin((m + 1)\theta)]^2 \qquad (4.31) $$

## Energy level structure and conditions on N

The two configurations $S_N^1(0)$ and $S_N^1(\phi_M)$ are different in the structure of the energy levels of the molecule.

In the **Möbius case** $S_N^1(\phi_M)$, we get energy levels which are given by pairs. We fill several levels when $\pi(S_N^1(\phi_M))$ ($\pi(S)$: number of electrons, see the part I to complete explanations) is a multiple of 4.

In the **ribbon case** $S_N^1(0)$, we have a minimal energy level followed by energy levels in pairs. In the same way, we saturate several energy levels if $\pi(S_N^1(0)) - 2$ is a multiple of 4.

This is resumed by what is called **electronic saturation** where $x \equiv y$ [mod.4] meaning that $x$ is congruent to $y$ modulo 4 ie $x = y + 4m, m \in \mathbb{N}$.

To remind, we use here the terms of "Möbius" and "ribbon" applied for the study of aromaticity criteria.

The **electronic saturation** is defined as: the energy levels are saturated for $S_N^1(\phi_M)$ if $\pi(S_N^1) \equiv 0$ [mod.4] and for $S_N^1(0)$ if $\pi(S_N^1) - 2 \equiv 0$ [mod.4].

We have studied different cases according to the number of electrons, the case called "even" $\pi(S_N^1) \equiv 0$ [mod.4] and the "odd" case $\pi(S_N^1) \neq 0$ [mod.4].

**Even case** $\pi(S_N^1) \equiv 0$ [mod.4]: In this paragraph, the energy levels of a structure S are noted k and the energy associated to this energy level for an electron is $\epsilon_k(S)$.
**In the Möbius case:** From C.A. Coulson, B. O'Leary and R. B. Mallion calculation techniques [54], the eigenvalues of the Hückel matrix are given in this case by [54]:

$$ \lambda_k = x + 2a \cos \left( (2k + 1)\pi \frac{k}{N} \right) \qquad (4.32) $$

with $k = 0, ..., N-1$.

In order to determine the structure of the energy levels, we study the degeneracies such as:

we have $\lambda_i = \lambda_j$ if and only if N is even and $j = N - i - 1$ for $i = 0, ..., N-1$.

The proof of this statement is given in Appendix 2 section 18.

Thus, for N even degenerated levels set by 2, $\lambda_i = \lambda_{N-i-1}$ for $i = 0, ..., N-1$.

The situation is described in the diagram :

<figure><img src="image_106.png" /><figcaption>Figure 4.9: Distribution of the energy levels of $S_N^1(\phi_M)$ in the case of $\pi(S_N^1) \equiv 0 \pmod{4}$</figcaption></figure>

In this case here, it is an **electronic saturation**. All levels from 0 to $\frac{\pi(S_N^1)}{4} - 1$ are **occupied**.

The energy $\epsilon_k(S_N^1(\phi_M))$ of an electron on the energy level k verifies:

$$ \epsilon_k(S_N^1(\phi_M)) = \alpha + 2\beta \cos((2k + 1)\phi) \qquad (4.33) $$

where $\phi = \frac{\pi}{N}$.

The Möbius energy $E(S_N^1(\phi_M))$ associated to the molecule $S_N^1(\phi_M)$ is given as:

$$ E(S_N^1(\phi_M)) = \alpha\pi(S_N^1) + 8\beta \cos \frac{\pi}{N} \sum_{k=0}^{\frac{\pi(S_N^1)}{4}-1} \cos \left( (2k+1)\frac{\pi}{N} \right) \qquad (4.34) $$

To prove this result, based on symmetry considerations where all the levels are occupied, we have:

$$ E(S_N^1(\phi_M)) = \sum_{k=0}^{\frac{\pi(S_N^1)}{4}-1} 2\epsilon_k(S_N^1(\phi_M)) + \sum_{k=N-\frac{\pi(S_N^1)}{4}}^{N-1} 2\epsilon_k(S_N^1(\phi_M)) \qquad (4.35) $$
$$ \sum_{k=0}^{\frac{\pi(S_N^1)}{4}-1} \epsilon_k(S_N^1(\phi_M)) = \sum_{k=N-\frac{\pi(S_N^1)}{4}}^{N-1} \epsilon_k(S_N^1(\phi_M)) \quad (4.36) $$

Then,

$$ E(S_N^1(\phi_M)) = 4 \sum_{k=0}^{\frac{\pi(S_N^1)}{4}-1} \epsilon_k(S_N^1(\phi_M)) \quad (4.37) $$

We can therefore substitute $\epsilon_k(S_N^1(\phi_M))$ by its expression in eq. 4.34.

**In the ribbon case:** In the ribbon case $S_N^1(0)$, the structure of the energy levels is different.

As a reminder, the eigenvalues of the Hückel matrix are given by :

$$ \lambda_k = x + 2a \cos \left( 2\pi \frac{k}{N} \right) \quad (4.38) $$

with $k = 0, ..., N - 1$, notice that the numeration of the eigenvalues from $k = 1$ to $N$ is equivalent to $k = 0$ to $N - 1$.

As in the previous case, we study the **structure of the energy levels** such as, $\lambda_i = \lambda_j$ if and only if $j = N - i$ and $i = 1, ..., N - 1$.

The proof is given Appendix 2 section 19.

We get the energy level corresponding to $\lambda_0$ and degenerate levels grouped by 2, $\lambda_i = \lambda_{N-i}$ for $i = 1, ..., N-1$.

We represent this situation by the diagram:

<figure><img src="image_107.png" /><figcaption>Figure 4.10: Distribution of the energy levels of $S_N^1(0)$ in the case of $\pi(S_N^1) \equiv 0 \pmod{4}$</figcaption></figure>

As expected, this is a **non-saturated** case with two available electrons on the last two energy levels.

We know that the $\epsilon_k(S_N^1(0))$ **energy** of an electron on the k-energy level of $S_N^1(0)$ is given by:

$$ \epsilon_k(S_N^1(0)) = \alpha + 2\beta \cos \left( \frac{2k\pi}{N} \right) \quad (4.39) $$
The energy $E(S_N^1(0))$ of the molecule $S_N^1(0)$ is given by:

$$E(S_N^1(0)) = -2\epsilon_0(S_N^1(0)) + 2\epsilon_{\frac{\pi(S_N^1)}{4}}(S_N^1(0)) + \alpha\pi(S_N^1) + 8\beta \sum_{k=0}^{\frac{\pi(S_N^1)}{4}-1} \cos\left(\frac{2k\pi}{N}\right) \quad (4.40)$$

with $\epsilon_0(S_N^1(0)) = \alpha + 2\beta$ and $\epsilon_{\frac{\pi(S_N^1)}{4}}(S_N^1(0)) = \alpha + 2\beta \cos(\pi(S_N^1)\frac{\pi}{2N})$.

To prove this relation, we know that:

$$E(S_N^1(0)) = 2\epsilon_0(S_N^1(0)) + \sum_{k=1}^{\frac{\pi(S_N^1)}{4}-1} 2\epsilon_k S_N^1(0) + \sum_{N-\frac{\pi(S_N^1)}{4}+1}^{N-1} 2\epsilon_k(S_N^1(0)) + \epsilon_{\frac{\pi(S_N^1)}{4}}(S_N^1(0)) + \epsilon_{N-\frac{\pi(S_N^1)}{4}}(S_N^1(0)) \quad (4.41)$$

We get the relations:

$$\epsilon_{\frac{\pi(S_N^1)}{4}}(S_N^1(0)) = \epsilon_{N-\pi(S_N^1)}(4(S_N^1(0))) \quad (4.42)$$

and,

$$\sum_{k=1}^{\frac{\pi(S_N^1)}{4}-1} 2\epsilon_k(S_N^1(0)) = \sum_{k=N-\frac{\pi(S_N^1)}{4}+1}^{N-1} 2\epsilon_k(S_N^1(0)) \quad (4.43)$$

Thus,

$$E(S_N^1(0)) = 2\epsilon_0(S_N^1(0)) + 4 \sum_{k=1}^{\frac{\pi(S_N^1)}{4}-1} \epsilon_k(S_N^1(0)) + 2\epsilon_{\frac{\pi(S_N^1)}{4}}(S_N^1(0)) \quad (4.44)$$

If the sum begins at $k=0$, we have:

$$E(S_N^1(0)) = -2\epsilon_0(S_N^1(0)) + 4 \sum_{k=0}^{\frac{\pi(S_N^1)}{4}-1} \epsilon_k(S_N^1(0)) + 2\epsilon_{\frac{\pi(S_N^1)}{4}}(S_N^1(0)) \quad (4.45)$$

If we substitute $\epsilon_k(S_N^1(0))$ by its expression we obtain the relation eq. 4.40.

**Odd case** $\pi(S_N^1) \neq 0$ [mod.4]: The situation of this paragraph is **more complicated** because it considers cases where there is **saturation** of the electronic levels in the **ribbon** case. Moreover, $\pi(S_N^1) \neq 0$ [mod.4] indicates that there exists $\delta \in \{1, 2, 3\}$ such as $\pi(S_N^1) = 4k + \delta$ (with k the energy levels).

**In the Möbius case** : Set $\delta \in \{1, 2, 3\}$, we get $m = \pi(S_N^1(\phi_M)) - \delta$. By hypothesis, we have $m \equiv 0$ [mod.4].

We are in a **non-saturated** situation, for example in the case $\delta = 3$ we have the representation as:
<figure><img src="image_108.png" /><figcaption>Figure 4.11: Distribution of the energy levels of $S_N^1(\phi_M)$ in the case of $\pi(S_N^1) \neq 0 \pmod{4}$</figcaption></figure>

The energy of this Möbius situation $E(S_N^1(\phi_M))$ of the molecule $S_N^1(\phi_M)$, with $\delta \in 1, 2, 3$ as $\pi(S_N^1) - \delta \equiv 0 \pmod{4}$ is given by:

$$E(S_N^1(\phi_M)) = \alpha(\pi(S_N^1) - \delta) + 8\beta \cos\left(\frac{\pi}{N}\right) \sum_{k=0}^{\frac{\pi(S_N^1)-\delta}{4}-1} \cos\left((2k+1)\frac{\pi}{N}\right) \delta\epsilon\frac{\pi(S_N^1)-\delta}{4} (S_N^1(\phi_M)) \quad (4.46)$$

For proof of this expression, we are interested in the first contribution of the energy given by the saturated energy levels associated with the $\pi(S_N^1) - \delta$ first electrons. To calculate this contribution, we have to use the result of the $\pi(S_N^1) \equiv 0 \pmod{4}$ case (eq. 4.34) by substituting $\pi(S_N^1)$ by $\pi(S_N^1) - \delta$.

We obtain a term such as:

$$\alpha(\pi(S_N^1) - \delta) + 8\beta \cos\left(\frac{\pi}{N}\right) \sum_{k=0}^{\frac{\pi(S_N^1)-\delta}{4}-1} \cos\left((2k+1)\frac{\pi}{N}\right) \quad (4.47)$$

Then, it remains $\delta$ electrons on the $\frac{\pi(S_N^1)-\delta}{4}$ level. These electrons contribute to the energy given by:

$$\delta\epsilon\frac{\pi(S_N^1)-\delta}{4} (S_N^1(\phi_M)) \quad (4.48)$$

We then deduce by summing the different terms:

$$E(S_N^1(\phi_M)) = \alpha(\pi(S_N^1) - \delta) + 8\beta \cos\left(\frac{\pi}{N}\right) \sum_{k=0}^{\frac{\pi(S_N^1)-\delta}{4}-1} \cos\left((2k+1)\frac{\pi}{N}\right) \delta\epsilon\frac{\pi(S_N^1)-\delta}{4} (S_N^1(\phi_M)) \quad (4.49)$$

**In the ribbon case:** This is the more complex case because three situations can happen according to the value of $\delta \in 1, 2, 3$ defined by $\pi(S_N^1) - \delta \equiv 0 \pmod{4}$.

Section 4.3.2, the electronic saturation gives a particular status to the case $\delta = 2$ since in this case (and only this one) the energy levels of the ribbon are saturated.

We will therefore see in the energy formula the implementation of $\delta$ in relation to $\delta = 2$. For this, we indicate $\mu \in \{-1, 0, 1\}$ and we put $\delta = 2+\mu$. The case $\mu = 0$ corresponds then to the saturated situation.
Let's take for example, the case $\mu = 1$ corresponding to $\delta = 3$, an electron is alone on the level of energy $\frac{\pi(S_N^1)-\delta}{4} + 1$, the levels of energy prior to this value being saturated we obtain the following diagram:

<figure><img src="image_109.png" /><figcaption>Figure 4.12: Distribution of the energy levels of $S_N^1(0)$ in the case of $\pi(S_N^1) - 3 \equiv 0 \pmod{4}$</figcaption></figure>

The energy $E(S_N^1(0))$ of the molecule $S_N^1(0)$ is given by:

$$ E(S_N^1(0)) = -2\epsilon_0(S_N^1(0)) + 4\epsilon\frac{\pi(S_N^1)-\delta}{4}(S_N^1(0)) + \alpha(\pi(S_N^1)-\delta) + 8\beta \sum_{k=0}^{\frac{\pi(S_N^1)-\delta}{4}-1} \cos\left(\frac{2k\pi}{N}\right) \\ + \begin{cases} -\epsilon\frac{\pi(S_N^1)-\delta}{4}(S_N^1(0)), & \text{if } \mu = -1, \\ 0, & \text{if } \mu = 0, \\ +\epsilon\frac{\pi(S_N^1)-\delta}{4}+1(S_N^1(0)), & \text{if } \mu = 1 \end{cases} \tag{4.50} $$

where $\delta \in \{1, 2, 3\}$ is defined by $\pi(S_N^1) - \delta \equiv 0 \pmod{4}$ and $\mu \in \{-1, 0, 1\}$ is given by $\mu = \delta - 2$.

For proof, the number $\pi(S_N^1) - \delta$ is a multiple of 4 for which we already calculated the energy. This one is given in eq. 4.40 by replacing in the formulas $\pi(S_N^1)$ by $\pi(S_N^1) - \delta$ gives a contribution to the total energy as:

$$ -2\epsilon_0(S_N^1(0)) + 2\epsilon\frac{\pi(S_N^1)-\delta}{4}(S_N^1(0)) + \alpha(\pi(S_N^1)-\delta) + 8\beta \sum_{k=0}^{\frac{\pi(S_N^1)-\delta}{4}-1} \cos\left(\frac{2k\pi}{N}\right) \tag{4.51} $$

A corrective term has to be applied which is dependent on $\delta$.

If $\delta = 1$ or 2, the energy level $\frac{\pi(S_N^1)-\delta}{4}$ is completed.

If $\delta = 3$ we have to add an electron on the energy level $\frac{\pi(S_N^1)-\delta}{4} + 1$.

A manner of coding the totality of these operations is to take the saturated situation as reference. The energy of the saturated case corresponding to $\frac{\pi(S_N^1)-\delta}{4} + 1$ is obtained using the previous formula by adding 2 electrons on the $\frac{\pi(S_N^1)-\delta}{4}$ level.

We get an energy given by:
$$ -2\epsilon_0(S_N^1(0)) + 4\epsilon\frac{\pi(S_N^1)-\delta}{4}(S_N^1(0)) + \alpha(\pi(S_N^1)-\delta) + 8\beta\sum_{k=0}^{\frac{\pi(S_N^1)-\delta}{4}-1} \cos\left(\frac{2k\pi}{N}\right) \quad (4.52) $$

According to the value of $\mu$ we can add or subtract the contribution of the energy levels. More precisely, we have a corrective term to the energy value of the saturated case with $\frac{\pi(S_N^1)-\delta}{4} + 2$ electrons defined as follows:

$$ \left\{ \begin{array}{ll} -\epsilon\frac{\pi(S_N^1)-\delta}{4}(S_N^1(0)), & \text{if } \mu = -1, \\ 0, & \text{if } \mu = 0, \\ +\epsilon\frac{\pi(S_N^1)-\delta}{4}+1(S_N^1(0)), & \text{if } \mu = 1 \end{array} \right. \quad (4.53) $$

If we combine these contributions we obtain the expected formula eq. 4.50.

### 4.3.3 Energetic comparison of structures $S_N^1(0)$ and $S_N^1(\phi_M)$ - stability

Using the calculations of the previous section, we can determine which of the structures $S_N^1(0)$ and $S_N^1(\phi_M)$ is the **most energetically favorable**. We note that the number of electrons $(\pi(S_N^1))$ available in the structure takes on a primordial role.

**The $\pi(S_N^1) \equiv 0$ [mod.4] case**

We obtain the next result which is quite surprising, if we assume that $\pi(S_N^1) \equiv 0$ [mod.4], then:

$$ E(S_N^1(\phi_M)) - E(S_N^1(0)) = 0 \quad (4.54) $$

In this electronic configuration, from an energetic point of view, the **Möbius case is equivalent to the ribbon structure** which is the natural situation to be expectable.

For proof we use the previously proven formulas (section 4.3.2) we have:

$$ E(S_N^1(\phi_M)) - E(S_N^1(0)) = 8\beta \sum_{k=0}^{\frac{\pi(S_N^1)}{4}-1} \left[ \cos\left(\frac{\pi}{N}\right) \cos\left((2k+1)\frac{\pi}{N}\right) - \cos\left(\frac{2k\pi}{N}\right) \right] \\ + 2 \left( \epsilon_0(S_N^1(0)) - \epsilon_{\frac{S_N^1(0)}{4}}(S_N^1(0)) \right) \quad (4.55) $$

As a trigonometric reminder we know that: $\cos(a)\cos(b) = \frac{1}{2}(\cos(a-b) + \cos(a+b))$.

By this trigonometric reminder we can rewrite the equality such as:

$$ E(S_N^1(\phi_M)) - E(S_N^1(0)) = 4\beta \sum_{k=0}^{\frac{\pi(S_N^1)}{4}-1} \left[ \cos\left((2k+2)\frac{\pi}{N}\right) - \cos\left(\frac{2k\pi}{N}\right) \right] \\ + 2 \left( \epsilon_0(S_N^1(0)) - \epsilon_{\frac{\pi(S_N^1)}{4}}(S_N^1(0)) \right) \quad (4.56) $$

If we consider once again the trigonometric formula, we know that for all $a, b \in \mathbb{R}$, $2\sin(a)\sin(b) = \cos(a-b) - \cos(a+b)$.
By setting: $a = (2k + 1)\frac{\pi}{N}$ and $b = \frac{\pi}{N}$, we have: $-2\sin(\frac{2k\pi}{N})\sin(\frac{\pi}{N}) = \cos((2k + 2)\frac{\pi}{N}) - \cos(\frac{2k\pi}{N})$.

By substituting the previous equality in the expression of $E(S_N^1(\phi_M)) - E(S_N^1(0))$, we obtain:

$$E(S_N^1(\phi_M)) - E(S_N^1(0)) = -4\beta \sin\left(\frac{\pi}{N}\right) \sum_{k=0}^{\frac{\pi(S_N^1)}{4}-1} \sin\left((2k+1)\frac{\pi}{N}\right) + 2\left(\epsilon_0(S_N^1(0)) - \epsilon_{\frac{\pi(S_N^1)}{4}}(S_N^1(0))\right) \quad (4.57)$$

We finally get:

$$E(S_N^1(\phi_M)) - E(S_N^1(0)) = -4\beta \left(\pi(S_N^1)\frac{\pi}{4N}\right)^2 + 2\left(\epsilon_0(S_N^1(0)) - \epsilon_{\frac{\pi(S_N^1)}{4}}(S_N^1(0))\right) \quad (4.58)$$

In addition,

$$\epsilon_0(S_N^1(0)) - \epsilon_{\frac{\pi(S_N^1)}{4}}(S_N^1(0)) = 2\beta \left(1 - \cos\left(\pi(S_N^1)\frac{\pi}{2N}\right)\right) \quad (4.59)$$

and,

$$E(S_N^1(\phi_M)) - E(S_N^1(0)) = -4\beta \left(\sin(\pi(S_N^1)\frac{\pi}{4N})\right)^2 + \cos\left(\pi(S_N^1)\frac{\pi}{2N}\right) + 4\beta \quad (4.60)$$

Then, as $\pi(S_N^1) = N$, we get:

$$E(S_N^1(\phi_M)) - E(S_N^1(0)) = 0 \quad (4.61)$$

**The $\pi(S_N^1) \neq 0$ [mod.4] case**

This is the **most complicated situation**, because each energy equation in the Möbius and ribbon case is composed of a part corresponding to the case presented earlier and of corrective terms. The energy difference in the $\pi(S_N^1) \neq 0$ [mod.4] case is equal to zero, thus, the comparison of the two energies is equivalent to the **comparison** of the **corrective terms**.

If we assume that $\pi(S_N^1) \neq 0$ [mod.4] then:

$$E(S_N^1(\phi_M)) - E(S_N^1(0)) = \begin{cases} 2\beta\delta\sin\left(\frac{\pi}{N}\right)\sin\left(\left(\frac{\pi(S_N^1)-\delta}{2}+1\right)\frac{\pi}{N}\right) & \text{if } \delta = 1 \text{ or } 2, \\ 4\beta\sin\left(\frac{\pi}{N}\right)\sin\left(\left(\frac{\pi(S_N^1)-\delta}{2}+1\right)\frac{\pi}{N}\right) + \\ -\alpha - 2\beta\cos\left(\left(\frac{\pi(S_N^1)-\delta}{2}+2\right)\frac{\pi}{N}\right) & \text{if } \delta = 3 \end{cases} \quad (4.62)$$

These expressions indicate that in this case, the sign of the energy difference is not fixed and **depends** explicitly on $N$. We are therefore, in the situation where the Möbius configuration is the most stable compared to the ribbon case.

For proof, by using the previous results, we can obtain the energy difference as:

$$E(S_N^1(\phi_M)) - E(S_N^1(0)) = \delta\epsilon \frac{\pi(S_N^1)-\delta}{4} (S_N^1(\phi_M)) - \begin{cases} \delta\epsilon \frac{\pi(S_N^1)-\delta}{4} (S_N^1(0)), & \text{if } \delta = 1 \text{ or } 2, \\ 2\epsilon \frac{\pi(S_N^1)-\delta}{4} (S_N^1(0)) + \epsilon \frac{\pi(S_N^1)-\delta}{4} (S_N^1(0)), & \text{if } \delta = 3 \end{cases} \quad (4.63)$$
For $\delta = 1$ or 2, the expression is now:

$$E(S_N^1(\phi_M)) - E(S_N^1(0)) = \delta \left( \epsilon_{\frac{\pi(S_N^1)-\delta}{4}}(S_N^1(\phi_M)) - \epsilon_{\frac{\pi(S_N^1)-\delta}{4}}(S_N^1(0)) \right) \quad (4.64)$$

By using the expression of the energy levels:

$$\epsilon_{\frac{\pi(S_N^1)-\delta}{4}}(S_N^1(\phi_M)) - \epsilon_{\frac{\pi(S_N^1)-\delta}{4}}(S_N^1(0)) = 2\beta \left[ \cos\left(\frac{\pi}{N}\right) \cos\left(\left(\frac{\pi(S_N^1)-\delta}{2} + 1\right)\frac{\pi}{N}\right) - \cos\left(\frac{\pi(S_N^1)-\delta}{2}\frac{\pi}{N}\right) \right] \quad (4.65)$$

If we use once again the trigonometry equality, $(2 \cos(a) \cos(b) = \cos(a+b) + \cos(a-b))$, with $a = \left(\frac{\pi(S_N^1)-\delta}{2} + 1\right) \frac{\pi}{N}$ and $b = \left(\frac{\pi(S_N^1)-\delta}{2}\right) \frac{\pi}{N}$, we can rewrite this quantity such as :

$$\cos\left(\frac{\pi}{N} \cos\left(\frac{\pi(S_N^1) - \delta}{2} + 1\right) \frac{\pi}{N}\right) - \cos\left(\frac{\pi(S_N^1) - \delta}{2} \frac{\pi}{N}\right) = \frac{1}{2} \left[ \cos\left(\frac{\pi(S_N^1) - \delta}{2} + 2\right) \frac{\pi}{N} - \cos\left(\frac{\pi(S_N^1) - \delta}{2}\right) \frac{\pi}{N} \right] \quad (4.66)$$

This expression can be simplified by using the relation, for all $a, b \in \mathbb{R}$, $2 \sin(a) \sin(b) = \cos(a-b) - \cos(a+b)$.

By taking, $a = \left(\frac{\pi(S_N^1)-\delta}{2} + 1\right) \frac{\pi}{N}$ and $b = \frac{\pi}{N}$, we have:

$$\cos\left(\left(\frac{\pi(S_N^1) - \delta}{2} + 2\right) \frac{\pi}{N}\right) - \cos\left(\frac{\pi(S_N^1) - \delta}{2} \frac{\pi}{N}\right) = \sin\frac{\pi}{N} \sin\left(\left(\frac{\pi(S_N^1) - \delta}{2} + 1\right) \frac{\pi}{N}\right) \quad (4.67)$$

Thus, we get a term as:

$$E(S_N^1(\phi_M)) - E(S_N^1(0)) = 2\beta\delta \sin\frac{\pi}{N} \sin\left(\left(\frac{\pi(S_N^1) - \delta}{2} + 1\right)\frac{\pi}{N}\right) \quad (4.68)$$

$$2\epsilon_{\frac{\pi(S_N^1)-\delta}{4}}(S_N^1(0)) + \epsilon_{\frac{\pi(S_N^1)-\delta}{4}+1}(S_N^1(0)), \text{ if } \delta = 3. \quad (4.69)$$

### 4.3.4 Molecule $S_N^1$ twisted by a distribution and a $C_2$ invariance

#### Geometry of a twisted molecule $S_N^1$

The geometry of a molecule $S_N^1$ twisted by a $\phi$ distribution ($\phi = (\phi_1, ..., \phi_N)$) is characterized by two families of angles. Firstly, a family of $\theta_i$ angles related to the position of the atoms on the circle, and the $\phi_i$ angles providing the orientation of the orbitals in relation to each other in each atom taking as reference the first one. We give some precisions in the rest of the manuscript.

**Parametrization of the circle and $\theta_i$ angles:** We consider a reference frame $(x, y, z)$ of $\mathbb{R}^3$ and a circle in the $(y, z)$ plane centered in $O = (0, 0, 0)$ of radius $R > 0$ parametrized by an angle $\theta_i \in [0, 2\pi]$ as:

$$C_R = \{(0, R \sin(\theta_i), -R \cos(\theta_i))\} \quad (4.70)$$

with $\theta_i \in [0, 2\pi[$.
Note that $x$ is normal to the plane containing the circle.

We get a $\phi = (\phi_1, ..., \phi_N)$ distribution. We dispose $N$ atoms $C_i$ along the circle and we note $\theta_i$ (with $i = 1, ..., N$) the angles corresponding to this distribution. The $C_i$ atoms have the coordinates $(0, R \sin(\theta_i), -R \cos(\theta_j))$ with $j^2 = -1$.

**$\phi$ distribution and representation:** Let $n_i$ be the tangent vector to the circle at point $C_i$ i.e. $n_i = (0, R \cos(\theta_i), R \sin(\theta_i))$. Let $P_i$ be the normal plane to $n_i$ at point $C_i$. Let $u_i$ be a family of vectors in the normal plane to $n_i$. A basis of $\mathbb{R}^3$ at the point $C_i$ consists of $e_1 = (1, 0, 0)$, $n_i$ and the vector $OC_i$. The vectors $e_1$ and $OC_i$ form a basis of $P_i$. A vector $u_i \in P_i$ is therefore represented by a vector $U_i$ of $\mathbb{R}^2$ whose coordinates are given by the coordinates of $u_i$ in the basis $\langle e_1, OC_i \rangle$.

<figure><img src="image_110.png" /><figcaption>Figure 4.13: Evolution of the orbitals along the circle - Möbius case</figcaption></figure>

Using these notations, the $\phi$ distribution is related to the vectors $u_i$ by identifying the planes $P_i$ by the relation:

$$ \widehat{U_i, U_{i+1}} = \phi_i \qquad (4.71) $$

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

With the vector $u_1$ fixed, we get the vector $u_k$ by positioning the vector in the $P_k$ plane:

$$ U_k = e^{-j(\phi_1 + \cdots + \phi_{k-1})} U_1 \qquad (4.72) $$

At this step of the manuscript, we have not imposed any particular symmetry, we will then impose the $C_2$ invariant molecule.

Note that in many publications, the direction of the circle is clockwise, which is contrary to the usual trigonometric direction used in mathematics. This is what explains the presence of the "-" in the rotation that expresses $U_k$ as a function of $U_1$. It is also what explains the particular form of the parameterization of the circle.

### $C_2$ invariance of a twisted molecule

We are interested in identifying the conditions imposed by the $C_2$ invariance condition on the data of the twisted molecule.

**Axis positioning:** The first evident but important result indicates the potential axis around which the molecule $S_N^1(\phi)$ can be eventually $C_2$ invariant.

We can define if a molecule $S_N^1(\phi)$ is $C_2$ invariant then, the axis of symmetry depends:
* either to the plane of the molecule, i.e. a line passing through the center of the circle

* or, is perpendicular to the plane of the molecule passing through the center of the circle

For proof, the molecule being $S_N^1(\phi)$, the set of atoms corresponds to the plane containing the circle. The image of this molecule by a $C_2$ symmetry whose the axis is outside the plane does not belong to the molecule unless the axis is perpendicular to the plane and through the center of the circle. If we are not in this situation, we consider an axis in the plane containing the circle. In the same way, the circle is invariant by a reflection around an axis if and only if this axis passes through the center of the circle.

The two situations are represented in the following figures:

<figure><img src="image_111.png" /><figcaption>Figure 4.14: Position of the $C_2$ axis - transverse to the plane (on the left) and in the plane (on the right)</figcaption></figure>

We can thus assume that the axis of $C_2$ symmetry is carried by the z-axis and that the molecule is contained in the $x = 0$ plane.

**Restrictions on the positions of the atoms:** We suppose that the invariant $C_2$ molecule is subject to the action of a **rotation** of angle $\pi$ around the axis carried by $z$. We note this rotation: $r_z(\pi)$ and it is given by:

$$r_z(\pi) = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \qquad (4.73)$$

The first family of **constraints** imposed by the $C_2$ invariance is on the **position** and **number** of atoms.

We define the geometric constraints for a $C_2$ invariance. Let $S_N^1(\phi)$ be a $C_2$ molecule invariant by a rotation of angle $\pi$ carried by the $x = y = 0$ axis.

Then, we have the two following properties:

* let $k$ be the number of atoms on the $x = y = 0$ axis. We have $k = 1$ or 2. Then, there exists $n \in \mathbb{N}$ such that the number of atoms $N$ is given by $N = 2n + k$.

* let $C_i$ be an atom whose position is marked by the angle $\theta_i$ then its image by $r_z(\pi)$ is indicated by the angle $2\pi - \theta_i$.

For proof, these properties are relatively trivial. The molecule is by definition symmetric to the axis $x = y = 0$ in the plane $x = 0$. If we get $n$ atoms outside the axis in the $\theta_i \in ]0, \pi[$ part, then we have the images of these atoms by symmetry in the $\theta_i \in ]0, 2\pi[$ part. Therefore, we have $2n$ atoms by symmetry for the molecule out of the axis, which justifies our result.

Moreover, $r_z(\pi)[(0, R \sin(\theta_i), -R \cos(\theta_i))] = (0, -R \sin(\theta_i), -R \cos(\theta_i))$ corresponds to an angle $\tilde{\theta}$ between $[0, 2\pi[$ equals to $\tilde{\theta}_i = 2\pi - \theta_i$.
**Orbital constraints and orbitals distribution:** Regarding the **constraints** on the atomic orbitals carried by the atoms and associated to $u_i$ (with $i = 1, ..., N$) vectors data, the first obvious constraint is : let $S_N^1$ be a $C_2$ invariant molecule where the axis is carried by $z$. If an atom $C_i$ is part of the rotation axis, then its $u_C$ vector is of the form $(0, 0, a)$ for a certain $a \in \mathbb{R}$.

For proof, the molecule being $C_2$ invariant, $r_z(\pi)[C_i]$ has to belong to the molecule. As $C_i$ is carried by the axis, it is invariant by $r_z(\pi)[C_i] = C_i$. The image of the orbital $u_C = (\alpha, \beta, \gamma)$ has to verify the invariance condition, which is i.e. $r_z(\pi)[u_C]$. This is only possible if $u_C$ is of the form $(0, 0, a)$ for a given $a \in \mathbb{R}$. In fact, $r_z(\pi)[u_C] = (-\alpha, -\beta, \gamma)$ thus, $\alpha = \beta = 0$.

The constraint implies the following corollary:

let $S_N^1$ be a $C_2$ invariant molecule where the axis is carried by $z$. If a $C_i$ atom has a $u_C$ orbital which is not carried by the axis of symmetry, then the considered atom cannot be on the axis of symmetry.

**Structure of molecules $S_N^1$ - $C_2$ invariant:** In most of the Möbius type molecules, we consider the first atom with a $u_1 = (1, 0, 0)$ orbital along the $x$ axis. This atom has to be positioned out of the symmetry axis, since we assume that the $C_2$ symmetry is carried by the $z$ axis. We thus have to consider the first atom with a given $\theta_i \neq 0$ coordinate.

We consider the following situation:

let the molecule $S_N^1$ made of N atoms whose first $C_1$ is denoted by an angle $\theta \neq 0$ and carrying a vector $u_1 = (1, 0, 0)$. There are $n$ atoms with $\theta_i \in ]0, \pi[$ coordinates and thus $n$ atoms by symmetry with $\theta_i \in ]\pi, 2\pi[$ coordinates.

There are **two** possible situations:

* either there is **an atom on the axis** which in this case has an orbital belonging to the axis. We then have $N = 2n + 1$, and the distribution has to satisfy $\phi_1 + ... + \phi_{n+1} = \frac{\pi}{2}$ in order to guarantee that the image of $u_1 = (1, 0, 0)$ by the successive rotations of angles $\phi_i$ gives a vector belonging to the $z$ axis.

* either there are **no atoms** on the axis and $N = 2n$. A global constraint on the distribution comes from the fact that the image of $u_1 = (1, 0, 0)$ by $r_z(\pi)$ corresponds to the orbital of the atom N, i.e. $u_N = (-1, 0, 0)$ which imposes the distribution as $\phi_1 + ... + \phi_{N-1} = \pi$. Moreover, the $C_2$ symmetry refers the $C_i$ atom to the $C_{N-i+1}$ atom, thus, we have $\theta_{N-i+1} = 2\pi - \theta_i$.

We can summarize these results as follows:

let a molecule $S_N^1(\phi)$, $C_2$ **invariant** by a symmetry carried by the $z$-axis. We suppose $u_1 = (1, 0, 0)$, then we have :

* the $\phi$ distribution has to satisfy $\phi_1 + ... + \phi_{N-1} = \pi$,

* the $C_1$ atom is not on the axis,

* if the number of atoms is of the form $N = 2n$, then there is no atom on the symmetry axis,

* if the number of atoms is as $N = 2n + 1$, then $n + 1$ is on the axis and its orbital is carried by the axis. We have then $\phi_1 + ... + \phi_n = \frac{\pi}{2}$,

* and, the positions of the atoms are related by the relation $\theta_{N-i+1} = 2\pi - \theta_i$.
The hypothesis that a molecule $S_N^1(\phi)$ is $C_2$ symmetric, imposes a **Möbius condition** if the vector $u_1 = (1, 0, 0)$.

The $C_2$ symmetry imposes **other constraints** as well:

Let $S_N^1(\phi)$ be a $C_2$ molecule invariant by a rotation of angle $\pi$ carried by the axis $x = y = 0$. Then, we have the property of the **torsion distribution** such that:

$$ \phi_{N-i+1} = \phi_i \qquad (4.74) $$

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

The simple proof of this relation is given in Appendix 2 section 20.

At this stage of the manuscript we can describe some distributions widely found in the literature. The **standard Möbius case** is obtained by imposing a **constant angle** and the $C_2$ symmetry of axis carried by $z$.

A molecule $S_N^1(\phi)$ is a standard Möbius if it is invariant by $C_2$ symmetry of axis carried by $z$ and, if $N = 2n$ and $\phi_1 = ... = \phi_n$.

To show the standard Möbius, we know that by **invariance** we get $\phi_1+...+\phi_{2n} = \pi$ and $\phi_1+...+\phi_n = \frac{\pi}{2}$.

The symmetry relation on the angles implies $\phi_{n+1} + ... + \phi_{2n} = \phi_1 + ... + \phi_n$.

As $\phi_1 = ... = \phi_n = \phi$, we obtain $\frac{N-1}{2}\phi = \frac{\pi}{2}$ or $\phi = \frac{\pi}{N-1}$.

**Molecule $S_N^1(\phi)$ constituted of one type of atoms and $C_2$ invariance:** All the definitions presented previously do not assume **any specificity** on the atoms positioned on the circle. However, the fact of considering a molecule with only one type of atom or with several type of atoms does not have the same consequences.

Let us suppose that the molecules are constituted of only **one type** of atom. Then, the **length of the bonds** between the successive atoms should *a priori* be **constant** (or approximately constant). This imposes the distribution of $\theta_i$ angles involving the atoms, such as:

$$ \theta_{i+1} - \theta_i = \theta_2 - \theta_1 \qquad (4.75) $$

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

We have used the $\theta_{N+1} = \theta_1$ convention. In other words, each atom is **uniformly** distributed on the circle, which implies:

$$ \theta_{i+1} - \theta_i = \frac{2\pi}{N} \qquad (4.76) $$

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

Concerning the positioning of the $C_1$ atom: if we still consider an $C_2$-invariant $S_N^1(\phi)$ molecule composed of only one type of atoms, we have, $\theta_{i+1} - \theta_i = \frac{2\pi}{N}$ (with $i = 1, ..., N$) and $\theta_1 = \frac{\pi}{N}$.

To prove this situation, we consider a single atom which is in relation by symmetry with its predecessor is $C_1$ which is referred to $C_N$. Now, the angle between $C_N$ and $C_1$ is $2\theta_1$ and we then obtain: $\theta_{N+1} - \theta_i = 2\theta_1 = \frac{2\pi}{N}$ or $\theta_1 = \frac{\pi}{N}$.
### 4.3.5 Molecules $L_N(0)$ and $L_N(\phi)$

#### Geometry of the molecules $L_N(\phi)$

We get N atoms on a **linear** segment and a $\phi$ **distribution** for the **orbitals** along the molecular chain. We consider a reference frame $(x, y, z)$ of $\mathbb{R}^3$ such as the atoms are arranged along the z-axis with the $C_1$ atom at $z = 0$. We then assume $u_1 = (1, 0, 0)$.

*Note that the above hypothesis is not restrictive since the data of $u_1$ simply depends on the reference frame $(x, y, z)$ to localize the first orbital.*

We can ask, why studying these linear molecules?

As we have seen in this manuscript, $S_N^1(\phi)$ molecules of Möbius-type cannot be $C_2$ invariant.

A Möbius-type $S_N^1(\phi)$ molecule verifies the $C_2$ invariance conditions globally at the geometric level but partially at the orbital level. Only the $C_1$ atom placed on the axis obstructs the global symmetry. One idea is therefore to look for a geometry that gets around this difficulty. To do so, it is necessary to divide an $S_N^1(\phi)$ molecule of Möbius type at the level of the $C_1$ atom which causes the problem. We then obtain a linear invariant $C_2$ molecule whose axis of symmetry passes through the middle of the chain and is orthogonal to it.

In the continuation of the manuscript, we study in all generality the characterization of $C_2$ invariant $L_N(\phi)$ molecules.

#### Twisted $C_2$ invariant $L_N(\phi)$ molecules

We look for distributions such as $L_N(\phi)$ is $C_2$ invariant.

Using the previous definitions of $C_2$ invariance, we adapt it to $L_N(\phi)$ molecules.

By definition, if $L_N(\phi)$ is $C_2$ invariant then, its symmetry axis is orthogonal to the molecule.

By proof, if the axis coincides neither with the chain or with the orthogonal direction, a given atom of $L_N(\phi)$ is moved by the action of the symmetry out of $L_N(\phi)$. It is therefore impossible.

On the other hand, if the axis coincides with the chain, then as $C_1$ is returned to $C_1$, the vector $u_1$ has to be returned to itself. However, by the action of $C_2$ it is returned to $-u_1$ which implies $u_1 = 0$. However, we have assumed $u_1 = (1, 0, 0)$ and we have a contradiction. The only possible case is when the axis of symmetry is orthogonal to the chain.

Let $L_N(\phi)$ a $C_2$ invariant linear chain. Thus, we have the following properties:

* if $N = 2n$, we get $n$ atoms **symmetrically distributed** around the axis passing through the middle of the chain.

* if $N = 2n + 1$, then the axis passes through the atom $n + 1$ which is positioned in the **middle** of the chain with $u_{n+1}$ carried by the axis. The $2n$ remaining atoms are symmetrically distributed on the chain around this axis.

To prove this, for reasons of symmetry, if the axis does not pass through the center of the chain, we necessarily have atoms whose reflection by the symmetry is outside the chain. We therefore assume that the axis passes through the center of the chain. If we have $n$ atoms on one side, then their images also belong to the chain and so we have $2n$ atoms on the chain. Two cases are to be studied: the one where there is an atom on the middle of the chain and the one where there is not.

In the case where there are no atoms in the center of the chain, we necessarily have an even number
of atoms $N = 2n$. If an atom $C_i$ is placed in the middle, then the previous proposition imposes that the associated vector $u_c$ is situated along the axis.

We specify more precisely the action of the $C_2$ symmetry of axis carried by $y$ passing through the center of the chain. For that, we note $l = Nd$ the length of the chain (where $d$ is the distance between two consecutive atoms). The atoms $C_i$ are represented in the reference frame $(x, y, z)$ of $\mathbb{R}^3$ whose origin is situated in $C_1$ and such that $x = 0, y = 0$ carries the chain. The coordinates of $C_i$ are of the form: $(0, 0, z_i)$, $z_i = (i - 1)d, i = 1, ..., N$.

The vectors $u_i$ are of the form $(x_i, y_i, z_i)$. By symmetry, the atom $C_i$ is projected on the atom $C_{N-i+1}$ and the vector $u_i$ on $u_{N-i+1}$. But, the image of $u_i$ is given by $(-x_i, y_i, z_{N-i+1})$. We note $U_i$ the vectors of $\mathbb{R}^2$ given by $(x_i, y_i)$. The angle $\phi_i$ is the angle between the vector $U_i$ and $U_{i+1}$.

Let $L_N(\phi)$ be a $C_2$ invariant linear chain, we have :

$$ \bullet \phi_1 + ... + \phi_{N-1} = \pi $$

$$ \bullet \phi_i = \phi_{N-i} \ (with \ i = 1, ..., N-1). $$

In other words, the existence of a $C_2$ symmetry for the molecule means that we are in a **Möbius-type** configuration.

By proof, the rotation of axis $y$ passing through $M = (0, 0, \frac{l}{2})$ puts $C_i$ on $C_{N-i+1}$. As by hypothesis $u_1 = (1, 0, 0)$, its image is the vector $(-1, 0, 0)$. Thus, we have a total rotation of angle $\pi$ between the atom $C_1$ and $C_N$. As the vector $u_n$ is obtained from $u_1$ by a rotation of angle $\phi_1 + ... + \phi_{N-1}$, we get $\phi_1 + ... + \phi_{N-1} = \pi$. We are then in a Möbius-type configuration.

The $C_2$ symmetry being an isometry, it preserves the geometrical angles (i.e. not oriented) between the vectors.

Then, we have :

$$ \phi_i = \widehat{U_i, U_{i+1}} = \widehat{U_{N-i+1}, U_{N-i}} = \phi_{N-i} \qquad (4.77) $$

We can specify according to the number of atoms the configuration if $N = 2n + 1$ then the molecule $L_N(\phi)$ is invariant if $\phi_1 + ... + \phi_n = \frac{\phi}{2}$.

To prove this, it is sufficient to use the previous proposition by noticing that $\phi_1 = \phi_{N-1}, ..., \phi_n = \phi_{n+1}$, or $\phi_1 + ... + \phi_N = 2(\phi_1 + ... + \phi_n) = \pi$ and $\phi_1 + ... + \phi_n = \frac{\pi}{2}$. The standard linear Möbius is obtained by assuming that a chain has $N = 2n + 1$ atoms, is of constant distribution and $C_2$ invariant.

If $L_N(\phi)$ is a molecule of constant distribution $\phi$ and $C_2$ invariant then $\phi = \frac{\pi}{N-1}$. If $N = 2n + 1$ then $\phi = \frac{\pi}{2n}$ and the central orbital is carried by $y$ i.e. $\phi_1 + ... + \phi_n = \frac{\pi}{2}$.

As we can see, $C_2$ invariance in this type of geometry **practically imposes** a Möbius configuration.

### 4.3.6 Optimal distribution of $S_N^1(\phi)$ molecules

In the following, we study whether there is only an **optimal distribution** or several **distributions** for $S_N^1(\phi)$ molecules in the $C_2$ invariant case. We distinguish here: the **standard Möbius case** (symmetrical) and the **Möbius case** (with a variable distribution). Against all expectations, the standard Möbius distribution which is at the origin of the study of structures of this type, is not the optimal distribution.

#### Optimal distributions $C_2$ invariant

We interested in two cases $N = 3$ and $N = 4$.
N = 3 **case:** We consider an invariant $S_3^1(\phi) C_2$ molecule. We get $\phi_1 = \phi_2$ and $\phi_1 + \phi_2 = \pi$ thus, $\phi_1 = \phi_2 = \frac{\pi}{2}$.

Let relax this condition and just assume that we have the $\phi_1 + \phi_2 = \pi$ relation providing that we have a Möbius type molecule.

A $\phi = (\phi_1, \phi_2)$ distribution, gives the **Hückel matrix** of the form:

$$ \begin{pmatrix} x & \cos(\phi_1) & -\cos(\phi_1) \\ \cos(\phi_1) & x & \cos(\phi_2) \\ -\cos(\phi_1) & \cos(\phi_2) & x \end{pmatrix} \qquad (4.78) $$

We need to have $\phi_1 + \phi_2 = \pi$, and $\cos(\phi_2) = -\cos(\phi_1)$. Note that $a = \cos(\phi_1)$, the Hückel matrix is of the form:

$$ \begin{pmatrix} x & a & -a \\ a & x & -a \\ -a & -a & x \end{pmatrix} \qquad (4.79) $$

We obtain the **eigenvalues**: $\lambda_1 = \lambda_2 = a$ and $\lambda_3 = -2a$

The **energy** is then given by: $E = 2\lambda_1 + \lambda_2 = 3a$ if $a > 0$ and $E = 2\lambda_3 + \lambda_1 = -3a$ if $a < 0$

By plotting the function of $\phi_1$ obtained, we see that the minimum is reached for $\phi_1 = 0$. We then obtain: among the $S_3^1(\phi)$ molecules such as $\phi_1 + \phi_2 = \pi$, the one that carries out the **minimum** of energy corresponds to the $\phi = 0$ and $\phi_2 = \pi$ distribution.

In other words, the **optimal distribution is not the Möbius distribution**.

N = 4 **case:** We consider the $S_4^1(\phi)$ molecule as:

$$ \phi_1 = \phi_3 \qquad (4.80) $$

and,

$$ \phi_1 + \phi_2 + \phi_3 = \pi \qquad (4.81) $$

The **Hückel matrix** is given by:

$$ \begin{pmatrix} x & \cos(\phi_1) & 0 & -\cos(\phi_1) \\ \cos(\phi_1) & x & \cos(\phi_2) & 0 \\ 0 & \cos(\phi_2) & x & \cos(\phi_3) \\ -\cos(\phi_1) & 0 & \cos(\phi_3) & x \end{pmatrix} \qquad (4.82) $$

Symmetry **constraints** gives:

$$ \cos(\phi_3) = \cos(\phi_1) \text{ and } \cos(\phi_2) = \cos(\pi - 2\phi_1) = -\cos(2\phi_1) = 1 - 2(\cos(\phi_1))^2. $$

By posing $a = \cos(\phi_1)$, we get the simplified Hückel matrix:

$$ \begin{pmatrix} x & a & 0 & -a \\ a & x & 1 - 2a^2 & 0 \\ 0 & 1 - 2a^2 & x & a \\ -a & 0 & a & x \end{pmatrix} \qquad (4.83) $$
Note $\delta(a) = 4a^4 - 4a^3 + a^2 + 2a + 1$.

Then, the **eigenvalues** are given as:

$$ \begin{aligned} \lambda_{1,\pm} &= \frac{1}{2} (2a^2 + a - 1 \pm \sqrt{\Delta(a)}) \\ \lambda_{2,\pm} &= -\frac{1}{2} (2a^2 + a - 1 \pm \sqrt{\Delta(a)}) \end{aligned} \qquad (4.84) $$

We have the relations $\lambda_{2,+} = -\lambda_{1,-}$ and $\lambda_{2,-} = -\lambda_{1,+}$. It is sufficient to trace $\lambda_{1,\pm}$ to have an idea of the energies:

<figure><img src="image_112.png" /><figcaption>Figure 4.15: Representations of the $\lambda_{1,+}$ and $\lambda_{1,-}$ energy solutions</figcaption></figure>

The **energy** of the molecule is given by:

$$ E(a) = 2\lambda_{2,-} + 2\lambda_{1,-} = -2\sqrt{\Delta(a)} \qquad (4.85) $$

<figure><img src="image_113.png" /><figcaption>Figure 4.16: Representations of the energy of a Möbius-$C_2$ invariant</figcaption></figure>

By minimization, we obtain: for $N = 4$, the **optimal distribution** in the case of a $C_2$ invariant molecule is given by $\phi_1 = \phi_3 = 0$ and $\phi_2 = \pi$.

For proof, the graph of the function shows that the minimum is reached for $a = 1$ which corresponds to $\phi_1 = 0$. The symmetry conditions then imply $\phi_3 = 0$ and then $\phi_2 = \pi$. This is very different from the Möbius type configurations. But, the energy of such a configuration is $-4$ which corresponds to the **standard Möbius**.

## Optimal Möbius and symmetric Möbius distributions

The results of **optimality** with more or less constraints in the class of Möbius-type distributions are as follows:
a distribution $(\phi_1, ..., \phi_{2n+2})$ is considered as Möbius if:

$$ \phi_1 + \dots + \phi_{n+1} = \frac{\pi}{2} \qquad (4.86) $$

$$ \phi_{n+2} + \dots + \phi_{2n+2} = \frac{\pi}{2} \qquad (4.87) $$

A Möbius distribution is defined as **symmetric** if we have:

$$ \phi_i = \phi_{2n+2-i+1} \qquad (4.88) $$

with $i = 1, ..., n+1$.

We remind the schematic representation of a **symmetric Möbius** (standard) configuration:

<figure><img src="image_114.png" /><figcaption>Figure 4.17: Representation of a symmetric Möbius configuration</figcaption></figure>

**Symmetrical Möbius N = 4:** In this case, we have: $\phi_1 + \phi_2 = \frac{\pi}{2}$ and $\phi_3 = \phi_2$.

As $\cos(\frac{\pi}{2} - \phi_1) = \sin(\phi_1)$, the **Hückel matrix** is only expressed in terms of $\phi_1$:

$$ \begin{pmatrix} x & \cos(\phi_1) & 0 & -\cos(\phi_1) \\ \cos(\phi_1) & x & \sin(\phi_1) & 0 \\ 0 & \sin(\phi_1) & x & \sin(\phi_1) \\ -\cos(\phi_1) & 0 & \sin(\phi_1) & x \end{pmatrix} \qquad (4.89) $$

As $\phi_1 \in [0, \frac{\pi}{2}]$, we get $\sin(\phi_1) = \sqrt{1 - \cos^2(\phi_1)}$.

By posing $a = \cos(\phi_1) \in [0, 1]$, we have the matrix:

$$ \begin{pmatrix} x & a & 0 & -a \\ a & x & \sqrt{1-a^2} & 0 \\ 0 & \sqrt{1-a^2} & x & \sqrt{1-a^2} \\ -a & 0 & \sqrt{1-a^2} & x \end{pmatrix} \qquad (4.90) $$

The **eigenvalues** are given as:

$$ \lambda_{\pm} = \pm\sqrt{2a} \qquad (4.91) $$

$$ \mu_{\pm} = \pm\sqrt{2\sqrt{1-a^2}} \qquad (4.92) $$

In terms of **degeneracy**, for $N = 4$, a Möbius system is degenerate if and only if it is a standard Möbius with $\phi_4^{\frac{\pi}{4}}$.

For proof, it is necessary to solve the equation $a = \sqrt{1 - a^2}$ for $a \in [0, 1]$. We obtain $a = \frac{1}{\sqrt{2}}$ which corresponds to $\phi_1 = \frac{\pi}{4}$.
The energy of the molecule is:

$$E(a) = -2\sqrt{2}a - 2\sqrt{2}\sqrt{1 - a^2} = -2\sqrt{2}(\cos(\phi_1) + \sin(\phi_1)) \quad (4.93)$$

As $E'(\phi_1) = -2\sqrt{2}(-\sin(\phi_1) + \cos(\phi_1))$, the extremums are situated on $\cos(\phi_1) = \sin(\phi_1)$, then, $\phi_1 = \frac{\pi}{4}$ because $\phi_1 \in [0, \frac{\pi}{2}]$.

It is a minimum because $E''(\phi_1) = -E(\phi_1)$ then, $E''(\frac{\pi}{4}) = 4 > 0$.

Thus, we have: for $N = 4$, the **optimal distribution** among the symmetrical distributions of Möbius type is given by the **standard Möbius**.

**Möbius case N = 4:** In the Möbius case, we have the **Hückel matrix** as:

$$\begin{pmatrix}
x & \cos(\phi_1) & 0 & -\cos(\phi_1) \\
\cos(\phi_1) & x & \sin(\phi_1) & 0 \\
0 & \sin(\phi_1) & x & \cos(\phi_3) \\
-\cos(\phi_1) & 0 & \cos(\phi_3) & x
\end{pmatrix}
\quad (4.94)$$

By posing $a = \cos(\phi_1)$ and $b = \cos(\phi_3)$, the matrix is:

$$\begin{pmatrix}
x & a & 0 & -a \\
a & x & \sqrt{1-a^2} & 0 \\
0 & \sqrt{1-a^2} & x & b \\
-a & 0 & b & x
\end{pmatrix}
\quad (4.95)$$

The **eigenvalues** are given by:

$$\lambda_{\pm} = -\frac{\sqrt{2}}{2}\sqrt{1+a^2+b^2 \pm \sqrt{\Delta}}, \quad \mu_{\pm} = \frac{\sqrt{2}}{2}\sqrt{1+a^2+b^2 \pm \sqrt{\Delta}} \quad (4.96)$$

where,

$$\Delta = 5a^4 - 2a^2b^2 - 8a^2b\sqrt{1-a^2} - 2a^2 + (1+b^2)^2 \quad (4.97)$$

The molecule **energy** is:

$$E(a, b) = -\sqrt{2} \left[ \sqrt{1+a^2+b^2+\sqrt{\Delta}} + \sqrt{1+a^2+b^2-\sqrt{\Delta}} \right] \quad (4.98)$$

If we set $b = 1$, we get the energy $E(a, 1)$:

<figure><img src="image_115.png" /><figcaption>Figure 4.18: Representation of the energy $E(a, 1)$ for a Möbius case</figcaption></figure>
We note that on the section in $b = 1$ the energy increases, but in the direction of $b$ the energy is always decreasing as we can see on the following figure:

<figure><img src="image_116.png" /><figcaption>Figure 4.19: Representation of the energy $E(a, b)$ for a Möbius case</figcaption></figure>

We therefore expect to obtain a minimum of the energy for a value of $b = 1$ and a value of $a$ close but different from 1. This is what the research of the minimum proves.

For $N = 4$, the Möbius-type optimal distribution is given by $\phi_1 = 0.3831$ rad, $\phi_2 = \frac{\pi}{2} - \phi_1$, $\phi_3 = 0$ and $\phi_4 = \frac{\pi}{2}$.

In fact for $E(a, b)$, $a$ and $b \in [0, 1]$ the minimum in $a = 0.9275$ and $b = 1$ which is equal to $\phi_1 = 0.3831$ rad and $\phi_3 = 0$ rad, thus we understand the result.

The angle $\phi_1$ is about $21.951^\circ$. The energy of the optimal Möbius distribution is $-4.6513$ u.a which is quite below the energy of the symmetric Möbius which is $-4$ u.a.

*Note that unlike the symmetric Möbius case, the optimal Möbius case is non-degenerate.*

### 4.3.7 Distribution of an $L_N(\phi)$ molecule equivalent to a $S_N^1(\phi_M)$ molecule

In this section, we study the distribution of an $L_N(\phi)$ molecule equivalent to a standard Möbius-type $S_N^1(\phi_M)$ molecule. In other words, we consider the **effect of imposing the $C_2$ invariance** condition on the description of the molecule. In this manuscript, obtaining a $C_2$ invariant representation complicates the shape of the distribution. From a uniform distribution for the $S_N^1(\phi_M)$ Möbius case, we progress to an irregular distribution for the equivalent $L_N(\phi)$.

A precise description of this distribution is given for any $N$. We also study the asymptotics of these distributions, which corresponds to the structure of an infinite linear chain.

#### Notion of equivalent molecule

The problem of **equivalence** is not easy because the notion requires a formal definition which is not a priori available in the literature. Before giving the definition that we will use, we will take **classical examples** as the conjugated polyenes.

In the case of a **planar molecule**, we take into account the separation between the $\pi$ and $\sigma$-systems which is possible. The $\pi$-system is constituted by the set of MOs said to be **antisymmetric** with respect to the plane of the molecule and each carbon atom participates in its formation via a single $p_z$-orbital (if $x$ and $y$ constitute the molecular plane) (chap. 9 [11]).
**Equivalence of polygonal $H_n$ and cyclic $C_nH_n$:** For $n \ge 3$, the **linear** molecules $H_n$ and **linear** $C_nH_{n+2}$ or **cyclic** $C_nH_n$ are equivalent in the following meaning:

* they have the same symmetry group,

* the coefficients of the MOs decomposition of $H_n$ according to the $1s_H$ orbitals of each hydrogen atom are the same as the coefficients of the MO decomposition of $C_nH_n$ in the $\pi$-system.

* the relative energy of the $\pi$-MOs of $C_nH_n$ is equal to the model system $H_n$.

In the linear case $H_n$, the notion of equivalence changes slightly.

**(Weak) equivalence of linear $H_n$ and linear $C_nH_{n+2}$:** For $n \ge 3$, the **linear** $H_n$ and **linear** $C_nH_{n+2}$ molecules are equivalent as:

* the coefficients of the MOs decomposition of $H_n$ according to the $1s_H$ orbitals of each hydrogen atom are the same as the coefficients of the MOs decomposition of $C_nH_n$ in the $\pi$ system.

* the relative energy of the $\pi$-MOs of $C_nH_n$ is equal to the $H_n$ model system.

This weak equivalence is therefore about the fact that the molecule has not conserved the symmetry group of the initial molecule. Nevertheless, the MOs of a "hypothetical" linear version of $C_nH_{n+2}$ differ slightly from those of $C_nH_{n+2}$. We can then, reason on an "ideal" linear version of $C_nH_{n+2}$ and thus, establish the "usual" equivalence of the molecules.

We can now specify the notion of **equivalence** that we will use in the rest of the manuscript.

**Equivalence of a molecule $M_1$ and a molecule $M_2$:** Let $M_1$ be a molecule of symmetry group $G_1$ and $M_2$ a molecule of symmetry group $G_2$. Let $G$ be a mutual subgroup of $G_1$ and $G_2$. We impose that $M_1$ is equivalent to $M_2$ modulo $G$ if:

* the coefficients of the MOs decomposition of $M_1$ according to the orbitals of each atom of $M_1$ are the same as the coefficients of the MOs decomposition of the molecule $M_2$ in the $\pi$-system.

* the relative energy of the $\pi$-MOs of $M_2$ is equal to that of its model system $M_1$.

In the continuation, we investigate equivalent molecules and study what changes in the deformation of the adopted geometry.

### 4.3.8 Equivalence between $S_N^1(\phi_M)$ and $L_N(\phi)$

The issue that we study here, is to identify a linear chain $L_N(\phi)$ that is equivalent to a Möbius-type $S_N^1(\phi_M)$ molecule. As we have already indicated, it is sufficient to consider the coefficients obtained in the decomposition of the $S_N^1(\phi)$ molecular orbitals and to assume that they are also obtained in the $L_N(\phi)$ case. The question is then to know how the $\phi$ distribution will diverge from the standard Möbius distribution on $L_N(\phi)$.

**Properties of the coefficients $a_n$:**

Let a **linear chain** of atoms numbered from 1 to $N + 1$. We identify an atom of the chain by an integer $n$ between 1 and $N + 1$. The following calculations depend on an integer $k$. We adopt the convention: a quantity $q$ associated to the atom $n$, naturally depends on $k$ and $N$ and the complete notation is then: $q_n(N, k)$.
We define the family of coefficients {$a_n$}$_{n=1,...,N}$ and {$b_n$}$_{n=1,...,N}$ given as:

$$a_n = \sqrt{\frac{2}{N+1}} \sin \left( n \frac{k\pi}{N+1} \right) \qquad (4.99)$$

and

$$b_{n+1} = a_n \qquad (4.100)$$

$n = 1, ..., N$ and $b_1 = 0$.

These coefficients have symmetry properties that explain the behavior of the angles for the orbitals along the molecular chain.

We get for all $l = 1, ..., N$, the symmetry relation:

$$a_{N-(l-1)} = (-1)^{k+1} a_l \qquad (4.101)$$

and

$$b_{N+1-(l-1)} = (-1)^{k+1} b_{l+1} = (-1)^{k+1} a_l \qquad (4.102)$$

For proof, we have:

$$\begin{align}
a_{N-(l-1)} &= \sqrt{\frac{2}{N+1}} \sin \left( (N+1-l) \frac{k\pi}{N+1} \right) \nonumber \\
&= \sqrt{\frac{2}{N+1}} \left[ \sin(k\pi) \cos \left( l \frac{k\pi}{N+1} \right) - \cos(k\pi) \sin \left( l \frac{k\pi}{N+1} \right) \right] \tag{4.103} \\
&= \sqrt{\frac{2}{N+1}} (-1)^{k+1} \sin \left( l \frac{k\pi}{N+1} \right) = (-1)^{k+1} a_l \nonumber
\end{align}$$

We have the following relations for $N = 4$:

* if $k$ is **odd**, then $a_1 = a_4$ and $a_2 = a_3$ and of course $a_5 = 0$

* if $k$ is **even**, then $a_1 = -a_4$ and $a_2 = -a_3$ and always $a_5 = 0$

In the case of a chain of **infinite length**, the previous results are expanded by periodicity in the structure, i.e. the symmetry is found at a multiple of $N + 1$ in the coefficients.

### Torsion angle and properties:

The **vector** associated to the atom numbered $n$ is defined by:

$$u_n = \frac{1}{\sqrt{2}} \begin{pmatrix} b_n \\ a_n \end{pmatrix} \qquad (4.104)$$

Let us remark for more details, that the expression of $u_n$ represents the decomposition of the associated MO according to the local basis used in each atom. Moreover, we consider only the contribution carried by the $p_x$ and $p_y$ orbitals (along the chosen axes), which corresponds to a part located only in the plane.

We consider as reference vector $u_1$ given by $\sqrt{2}u_1 = (0, a_1)$, and we observe the **angle** formed by the vector $u_n$ with $u_1$ noted $\phi_{1,n}$. A simple calculation gives the expression of $\phi_{1,n}$.
The angle $\phi_{1,n}$ has for expression:

$$ \phi_{1,n} = \arccos \left( \epsilon(a_1) \epsilon(a_n) \frac{|\alpha_n|}{\sqrt{1 + \alpha_n^2}} \right) \qquad (4.105) $$

where $\epsilon(a_1)$ and $\epsilon(a_n)$ give the sign of $a_1$ and $a_n$ respectively and $\alpha_n$ is given by:

$$ \alpha_n = \frac{a_n}{b_n} \qquad (4.106) $$

The proof of the expression of the angle $\phi_{1,n}$ is given in Appendix 2 section 21.
We can note some points about the signs of $a_1$ and $a_n$.

The sign of $a_1$ is positive if:

$$ (N + 1)2l \le k \le (N + 1)(1 + 2l) \qquad (4.107) $$

for any integer $l \in \mathbb{N}$.

We notice by eq. 4.107, when $N$ tends to infinity, the condition will always be verified and the sign of $a_1$ will therefore not intervene. This will also be true for $0 \le k \le (N + 1)$ which includes for example the case $k = 3$ when $N = 4$ studied by the research team of R. Hoffmann [9].

The situation is more complicated for $a_n$. The positivity condition which depends explicitly on $k$ is:

$$ (N + 1)2l \le nk \le (N + 1)(1 + 2l) \qquad (4.108) $$

We can say that the observable angles are obtained from the **family of angles**:

$$ \phi_n = \arccos \left( \frac{|\alpha_n|}{\sqrt{1 + \alpha_n^2}} \right) \qquad (4.109) $$

The following formula (eq. 4.110) shows that the angles can be obtained from the $\phi_n$ family, optionally adding the family of conjugate angles $\pi - \phi_n$:

$$ \arccos(-x) = \pi - \arccos(x) \qquad (4.110) $$

In order to analyze how the angles are distributed, we prove some properties of the vectors $u_n$.

For $l = 1, \dots, N+1$ we have:

$$ \|u_{N+1-(l-1)}\| = \|u_l\| \qquad (4.111) $$

The proof of this relation is based on the fact that (except for the sign), we have:

$$ a_{N+1-(l-1)} = (-1)^{k+1} a_{l-1} = (-1)^{k+1} b_l \qquad (4.112) $$

$$ b_{N+1-(l-1)} = (-1)^{k+1} b_{l+1} = (-1)^{k+1} a_l \qquad (4.113) $$

and then the vector $u_{N+1-(l-1)}$ is obtained from $u_l$ by axial symmetry $y = x$ and multiplication by $(-1)^{k+1}$ gives:

$$ u_{N+1-(l-1)} = (-1)^{k+1} \frac{1}{\sqrt{2}} \begin{pmatrix} a_l \\ b_l \end{pmatrix} \qquad (4.114) $$

From the previous result we deduce symmetries on the distribution of angles between $\phi_1$ and $\phi_{N+1}$.

Precisely, noting by $\phi_{l,l+1}$ the angle between $u_l$ and $u_{l+1}$ we get for $l = 1, \dots, N$:

$$ \phi_{l,l+1} = \phi_{N+1-l,N+1-(l-1)} \qquad (4.115) $$

The proof of this expression relies on the symmetry property of the coefficients given in Appendix 2 section 22.
**Some examples with N = 4 and k = 1, 2, 3, 4**

In the following, we show how the **parity** on $k$ affects the **arrangement of the orbitals** along the chain. We consider the **simplest** cases: $N = 4$ and $k = 1, k = 2, k = 3$ and $k = 4$.

**Case N = 4 and k = 1:** In this situation, we have: $a_1 = 0.37$ and $a_2 = 0.6$ and by the relations: $a_3 = a_2$ and $a_4 = a_1$.

The angle between the vector $u_1 = (0, a_1)$ and $u_5 = (b_5, 0) = (a_4, 0) = (a_1, 0)$ is 90°.

We find via the symmetry relations on the angles: $\phi_{1,3} = \frac{\phi_{1,5}}{2} = 45^\circ$ which is the analog of the symmetry of the **even** case showing the angle of 135°.

By calculations: $\phi_{1,2} = 31.7^\circ$, then, by using the symmetrical relations: $\phi_{1,4} = 2\phi_{1,3} - \phi_{1,2} = 58.3^\circ$.

**Case N = 4 and k = 2:** In this case, we have: $a_1 = 0.60$ and $a_2 = 0.37$ which gives: $a_3 = -a_2$ and $a_4 = -a_1$.

The angle between the vector $u_1 = (0, a_1)$ and $u_5 = (a_4, 0) = (-a_1, 0)$ is 270°.

The previous relations imply that: $\phi_{1,2} = \phi_{4,5}$ and $\phi_{2,3} = \phi_{3,4}$.

As $\phi_{1,5} = \phi_{1,2} + \phi_{2,3} + \phi_{3,4} + \phi_{4,5} = 2(\phi_{1,2} + \phi_{2,3})$, then, $\phi_{1,2} + \phi_{2,3} = \phi_{1,3} = 135^\circ$.

It is necessary to calculate one of the $\phi_{1,2}$ or $\phi_{2,3}$ angles to obtain all the information.

We have $\phi_{1,2} = 58^\circ$, thus: $\phi_{1,4} = \phi_{1,3} + \phi_{3,4} = \phi_{1,3} + \phi_{2,3} = 2\phi_{1,3} - \phi_{1,2} = 270 - 58 = 212^\circ$.

**Case N = 4 and k = 3:** By setting $k = 3$, then, $a_1 = 0.60$ and $a_2 = 0.37$.

We still have: $a_4 = a_1$ and $a_3 = a_2$.

The angle $\phi_{1,5}$ is 90° but we need to understand this angle as: $360 + 90$ because following the vectors $u_1$ to $u_5$ on the trigonometric circle, we carry out a complete rotation and it is a significant difference.

In particular, when we write via the symmetrical relations: $\phi_{1,3} = \frac{\phi_{1,2}+\phi_{2,3}+\phi_{3,4}+\phi_{4,5}}{2}$, that we previously used: $\phi_{1,2}+\phi_{2,3}+\phi_{3,4}+\phi_{4,5} = \phi_{1,5}$ which is true but modulo $2\pi$, i.e. to a certain integer number of turns.

Here: $\phi_{1,3} = \frac{360+90}{2} = 225^\circ$. The angle $\phi_{1,2}$ gives by calculations, $\phi_{1,2} = 122^\circ$.

We deduce, $\phi_{2,3} = \phi_{1,3} - \phi_{1,2} = 225 - 122 = 103^\circ$, so, $\phi_{1,4} = \phi_{1,3} + \phi_{3,4} = \phi_{1,3} + \phi_{2,3} = 225 + 103 = 328^\circ$.

We note that, $\phi_{4,5} = 360 + 90 - 328 = 122^\circ = \phi_{1,2}$ as expected.

This situation corresponds to the case $k = 1$ but for vectors rotated by $-90^\circ$.

**Case N = 4 and k = 4:** Now, let $k = 4$ then $a_1 = 0.37$ ; $a_2 = -0.60$ ; $a_4 = -a_1$, and $a_3 = -a_2$.

The angle $\phi_{1,5}$ is still 270°. Once again by following the vectors $u_1$ to $u_5$ we perform a turn of the circle and the angle $\phi_{1,3} = 315^\circ$.
The angle $\phi_{1,2}$ is given by calculations at 148°. We then obtain: $\phi_{1,4} = \phi_{1,3} + \phi_{3,4} - 360 = 120°$.

We verify again that: $\phi_{4,5} = 270 - 122 = 148°$ is equal to $\phi_{1,2}$.

**To sum up:** To resume the 4 previous examples:

<table><caption>Table 4.1: Overview of the angles (in degrees) of the 4 examples</caption><thead><tr><th>k</th><th>φ<sub>1</sub></th><th>φ<sub>2</sub></th><th>φ<sub>3</sub></th><th>φ<sub>4</sub></th><th>φ<sub>5</sub></th></tr></thead><tbody><tr><td>1</td><td>0</td><td>32</td><td>45</td><td>58</td><td>90</td></tr><tr><td>2</td><td>0</td><td>58</td><td>135</td><td>212</td><td>270</td></tr><tr><td>3</td><td>0</td><td>122</td><td>225</td><td>328</td><td>90</td></tr><tr><td>4</td><td>0</td><td>148</td><td>315</td><td>122</td><td>270</td></tr></tbody></table>

<figure><img src="image_117.png" /><figcaption>Figure 4.20: Orbital distributions for $N = 4$ and $k = 1, 2, 3, 4$</figcaption></figure>

For a general remark both on the cases $k$ even or odd. We note an asymmetry between the even and odd cases $k$ which is always present for any value of $N$.

### 4.3.9 Asymptotic distribution of angles along a chain

#### Orbital cases of the form $(b_n, a_n)$

The angle $\phi_{1,n}$ converges for fixed $n$ and $k$ when $N$ tends to infinity to the angle:

$$ \phi_{\infty,n} = \arccos \left( \frac{n}{\sqrt{n^2 + (n-1)^2}} \right) \qquad (4.116) $$

The proof is given in Appendix 2 section 23.

We note that this angle corresponds to the situation of an **infinite chain** and that it is independent of $k$. We obtain the following values for the atoms of the chain (with $n = 1,...,7$):
<table><caption>Table 4.2: Examples of angles values (in degrees) along a molecular chain</caption><thead><tr><td>n</td><td>φ<sub>1,n</sub></td></tr></thead><tbody><tr><td>1</td><td>0</td></tr><tr><td>2</td><td>26.57</td></tr><tr><td>3</td><td>33.69</td></tr><tr><td>4</td><td>36.87</td></tr><tr><td>5</td><td>38.66</td></tr><tr><td>6</td><td>39.81</td></tr><tr><td>7</td><td>40.6</td></tr></tbody></table>

<figure><img src="image_118.png" /><figcaption>Figure 4.21: Representation of asymptotic angles</figcaption></figure>

These symptomatic angles only have meaning in the case of an **infinite chain** which is a **purely abstract object**. We note a progressive stabilization around an angle of 45° which is easily proved as: the angles are accumulated when $n$ becomes large around $\frac{\pi}{4}$.

The proof is direct,

$$ \lim_{n \to \infty} \frac{n}{\sqrt{n^2 + (n-1)^2}} = \frac{1}{\sqrt{2}} \quad (4.117) $$

then,

$$ \lim_{n \to \infty} \phi_{\infty,n} = \arccos \frac{1}{\sqrt{2}} = \frac{\pi}{4} \quad (4.118) $$

### Case of orbitals of the form $(b_n, -a_n)$

As in the previous case, we can introduce the family of angles which considers the fact that the sign of an is replaced by "-" in the expression:

$$ \phi_n = \arccos -\frac{|\alpha_n|}{\sqrt{1 + \alpha_n^2}} \quad (4.119) $$

## 4.4 Conclusion

This chapter is very dense in information. Our interest was focused on the different aspects of **aromaticity** since it is related to the reactivity of the molecule. The criteria of aromaticity are linked to the rules stated by Hückel and associated with the systems that we distinguish as "Möbius" and "Hückel".
Generally, we have either **linear** $L_N$ or **cyclic** $S_N^1$ chains, where each of these representations will have different **orbital distributions**: either **null** (0) (called ribbon or Hückel), or with **angle distribution** ($\phi$) (called twisted ribbon or Möbius).

All these distributions are detailed and proven as generally as possible in order to determine the **equiv-**alent and most **optimal** structure depending on the **energy** parameter.

Using mathematical details on the **Hückel matrices** in each of the cyclic and linear cases, we showed how the energies of each system were determined and compared. In fact, the **energy difference** between the **cyclic** case with **null** distribution and the **Möbius** case with **electronic saturation** is **null**, which shows that the two cases are **equivalent**.

On the contrary, with the case of **electronic unsaturation**, it is more complex. The comparison takes into account $\delta$ **corrective terms**. It results that the comparison depends mainly on the **parameter** $N$ (i.e. the number of atoms).

To be precise in the understanding of the twisted cyclic systems $S_N^1(\phi)$, we took into account the $C_2$ **symmetry** with a set of parametrizations, showing that the **optimal distribution** is **not** the Möbius distribution (which contradicts what is intuitively stated in the work of Hoffmann et al. [55]).

Analogously, for the linear cases $L_N$, the same steps are followed (determination of the energy and taking into account the symmetry). Among other things, the $C_2$ invariance symmetry **imposes** a configuration with **Möbius distribution**.

The notion of **equivalence** between linear and cyclic molecules is shown by different classical examples and clearly detailed to identify how the divergence between the standard Möbius distribution $S_N^1(\phi)$ and the linear case $L_N(\phi)$ is presented.

Finally, we define the properties and expression of the $\phi_n$ **torsion angles** for all systems as well as for infinite linear chains.
Chapter 5

Beyond aromaticity criteria - Helical
states

In this chapter we go beyond the aromaticity criteria and focus on helical states. In a first approach we discuss the following points:

* the Möbius systems following R. Hoffmann,

• the linear cases with cumulenes,

• helical molecular orbitals,

* molecular orbitals in even and odd cases (in terms of number of double bonds).

Then, we expose:

* the properties of helical states,

• the criteria for the existence of helices.

All these results come from the work of R. Hoffmann [9], Solomon et al. [56], Jin et al. [57] and our preprint [44].

5.1 Möbius systems according to the study of Roald Hoffmann

In this section we refer to recent 2018 work of the Nobel prize laureate research team Roald Hoffmann which we considered as particularly interesting [9].

As we saw in the previous chapter, section 4.3.5, **Möbius systems** are based on p-orbitals where there is a **continuous overlap** (as shown figure 4.5) which forms a continuous ribbon that closes upon sign inversion. Following the discovery of these original systems in 1964 [7], other researchers such as Fisher and Kollmar in 1968 [58] and Zimmerman in 1971 [46], have shown with the theory of Hückel, that the topology of Möbius systems have a similarity of behavior with the orbitals of allene type molecules. In parallel, in 1968, Buenker, showed with *ab initio* studies of MOs, the rotation barrier of the allenes [59]. Then, in 1976 it is semi-empirical calculations which are carried out to demonstrate the optical properties of rotation of allenes [60].

As a reminder, allenic molecules are linear hydrocarbons containing only consecutive double bonds. From a reactivity point of view this type of system is more reactive than those composed by an alternation of single and double bonds. Moreover, note that the allenes display an axial chirality.

The following figure illustrates the parallel between the **fictive Möbius cyclic systems** and the **linear allene systems**. We observe the **rotation** and the change of sign of the orbitals within the systems.
<figure><img src="image_119.png" /><figcaption>Figure 5.1: Comparison between the behavior of orbitals in Möbius ribbons and allenes (here a cyclobutadiene and a [2]-cumulene) [9]</figcaption></figure>

On the left-hand side of the figure, we observe the **rotation** of the p-type orbitals along a Möbius cyclobutadiene. Analogously, on the right-hand side, the rotation of the orbitals of the allene, which can be described as a so-called coarctate system. The term coarctate is proposed by Herges to describe two fused π-systems (as related to the breaking and creation of bonds on one or more atoms simultaneously), an action that takes place in pericyclic reactions (alternating cyclic sequence processes) [61].

### 5.1.1 [n]-cumulenes : linear case

To be more precise, here, when we refer to allenic molecules, the most appropriate term is **cumulene**. As explained in the previous paragraph, the cumulenes are **linear systems** constituted exclusively of **double bonds** between the carbons and can be substituted at the end of the chain.

To detail the following figure, the molecule is composed of *N* carbon atoms sp-hybridized with *C*₀ and *C*ₙ are sp²-hybridized since they are at the end of the chain. To be precise, the chain of atoms is oriented along the *z*-axis. We note that *L*₁, *L*₂ and *R*₁ and *R*₂ are the left and right substituents of the chain ends, which can simply be hydrogens but can also be more complex as we will discuss in the continuation of the manuscript.

<figure><img src="image_120.png" /><figcaption>Figure 5.2: Representation of a [n]-cumulene</figcaption></figure>

To complete, in terms of nomenclature we note [n]-cumulene, [n] corresponds to the number of double bonds and *N* the number of carbons involved in the molecule. In our study we have studied **even and odd** cumulenes i.e. composed by an **even or odd number of atoms**.

We simulated the π-system with a basis set composed by *p*ₓ and *p*ᵧ-orbitals for each atom of the chain. Then, the *p*ₓ-orbitals are directed along the direction of the chain and is included in the σ-system. An illustration of this hybrid system is given in the following figure 5.3. We observe in the case of a [4]-cumulene for the HOMO-1, HOMO and LUMO, the mixing of the *p*ₓ and *p*ᵧ-systems which are
at the origin of the formation of the curiosity of helical orbitals whose the detailed study is in the
continuation of the manuscript.

<figure><img src="image_121.png" /><figcaption>Figure 5.3: Representation of the hybrid systems at the origin of the formation of orbitals in helix (lower part : $p_x$ and $p_y$-systems are represented schematically, the green and red lobes indicate the sign - higher part: the representation of the orbitals by DFT computations)</figcaption></figure>

Concerning the **synthesis** of this type of molecule: we note that the literature is **rich** in terms of synthesis of allenes since they are useful complex molecular targets (for example: as precursor in cycloadditions, or as redox reagent) in many fields such as pharmaceuticals, nanomaterials or electronics. Among their advantages, they present fundamental properties that are interesting both for structural electronic properties and in terms of reactivity [62].

There is a great variety of [n]-cumulenes. In terms of even cumulenes it has been possible to synthesize from [4]-cumulene to metallo-[6]-cumulene. For odd cumulenes, we note synthesis up to [9]-cumulene. Substituted cumulenes have also generated a lot of interest (such as S-α,ω-dimethyl-[n]-cumulene). We quote here only some examples since we do not focus on the synthesis work [63, 64, 65, 66, 67].

It should be noted, however, by studying the cumulene literature, Hendon et al. [63] proves a difference between the terms of allenes and cumulenes. According to them, allenes have an odd number of carbon atoms with orthogonal chain ends showing an electrohelicity (a term whose meaning will be explained in the rest of this work). On the other hand, cumulenes with an even number of carbons are planar and without orbital helicity. By misuse of language we will use the term cumulene in a generic way and will demonstrate the presence of orbital helicity in the continuation of the manuscript.

5.1.2 Helical molecular orbitals

The major curiosity for **cumulenes** is the presence of **helical frontier orbitals**. The important point is that these orbitals are **not due** to a helical molecular structure.

According to the literature, in particular the study carried out by Hoffmann et al. [9], the research team shows that cumulenes have **two perpendicular π-systems** along their molecular axis. Thus, for **unsubstituted [n]-cumulenes** (with even [n]), the **π-systems** are **equivalents** and the ends chain are **perpendicular** to each other providing a $D_{2d}$ symmetry and then, **π-MOs all degenerated**.

However, when a **substitution** is applied at the ends of the chain we observe a relaxation of the
degeneracy reducing the symmetry to $C_2$ resulting in a **chiral** molecule.

Notice that the same behavior is observed when a **mechanical constraint** is imposed, for example a torsion of the terminal carbons. The mechanical constraint such as setting the two ends perpendicular to each other resulting in the appearance of helical orbitals has been highlighted recently in 2013 by Hendon and Walsh research team [63]. Indeed, the forced torsion of the terminal groups lowers the symmetry of $D_{2d}$ to $D_2$, which will generate **helical orbitals**.

It is particularly on the aspect of **mechanical constraint** that we have focused. We tried to obtain all possible configurations in order to achieve **helical MOs**. The final aim being to obtain **new properties** of materials since the helical MOs provide a **delocalization** of $\pi$-electrons all along the twisted chain.

It should be noted that the **electrohelicity** phenomenon shown by Hendon [63] can be discussed according to different methods, in a classical way following the Hückel theory [9] that we have seen previously (section 0.2.1) or in a more complex method from a physical point of view by the Hamiltonian [68].

*Note that the phenomenon of electrohelicity is said for a system that presents perpendicular end groups and that the system shows helical frontier orbitals.*

We are interested in the helical MOs specific to the **linear carbon chain** with no specific terminal substituents.

Finally, it should be noted that there are contradictions in the literature since many articles specify that the presence of helices is only visible for odd cumulenes [63]. However, we show in the continuation of the manuscript that helical MOs are present independently for the even or odd case.

## Helical molecular orbitals in even and odd cases

Most publications examine several **even-odd** cases and **parallel-perpendicular** chain ends.

Publications dealing with **even systems** and **perpendicular extremities** are numerous [9, 63, 69, 70, 62, 71]. There are several examples showing that the helical behaviour is observable only when the chain ends [9] are **twisted** at a certain angle (i.e. the ends are no longer flat [63]), and others depending on the nature of the substituents of the chain ends [9, 62, 55, 71]. The substituents listed may be -dimethyl [9, 71], -dichloro [55], or -nitrogen [62]. The cumulenes studied range from [*n*] = 2 [63, 69, 62, 55, 71] to [*n*] = 8 [70]. The case mentioned by Imamura et al. [70] shows a certain influence of the angle of twist of the $CH_2$ end groups in the formation of helixes. The presence of non-hydrocarbon substituents provides a break in symmetry with free pairs, allowing different interactions resulting in a helical character [62]. The **even parallel** cases seem to show helixes from a certain angle or even no helix for ketone substituents [72]. According to Garner et al. [56], the axial torsion angle is a crucial factor in the constitution of helical orbitals (between 30° and 150°).

The analysis of the **odd cases** between [*n*] = 3 to [*n*] = 9 with perpendicular chain ends, shows once again the impact of the torsion angle to see the existence of helixes [70]. **Parallel cases** show the same behaviour with the presence of helix at a certain angle [9, 56] and others not [63, 55].

It is obvious that the helical shape of an electronic system will depend on the molecular symmetry. But finally, it is also true that the definition of a cylindrical helical system is quite subjective, this is why it is imperative to define parameters to quantify them and that is what we try to do in this manuscript.
## 5.2 Helical states

### 5.2.1 Properties of helical states

#### General case

As discussed previously (section 5.1.2), the helical MOs are visible on the $[n]$-cumulenes. From a general point of view, we can formalize mathematically the **criteria of appearance** of these helical systems.

Let $E$ the energy of a molecule and $z$ the position of the atoms on the linear chain with $z = 0$ corresponding to the left end chain, and $z = N$ the right end chain.

We define the MO as $\psi(z)$ associated to the atoms as :

$$ \psi(z) = \begin{pmatrix} \psi_x(z) \\ \psi_y(z) \end{pmatrix} = MR(z\omega)v_0 \quad (5.1) $$

$v_0$ a fixed two-dimensional unit vector, $\omega \in \mathbb{R}$, $R(z\omega)$ the rotation matrix of angle $z\omega$ and $M$ a real symmetric matrix.

The helices mainly found are said to be **classical**, and are sometimes called **perfect helices** [56], but it is difficult to estimate the perfect cylindrical shape of a helix.

We can therefore define the **shape of a helix** as:

$$ \begin{aligned} x(t) &= r(t) \cos(t) \\ y(t) &= er(t) \sin(t) \\ z(t) &= P(t) \end{aligned} \quad (5.2) $$

with $P(t)$ a smooth increasing function and $r(t)$ an arbitrary function corresponding to a non-constant radius, so $r: \mathbb{R} \to \mathbb{R}^+$.

The best approximation of the helix is finally obtained in a **non-linear context**, i.e. by using polynomial functions $P(t)$ and $r(t)$.

There are different **morphologies** of helical MOs. Actually, helices are not necessarily "perfect" as we will see in the continuation of the manuscript.

#### Distributions of angles

Let a linear chain of atoms $C_i$ oriented according to $z$, we define the **distributions of angles** according to the **numerical** and **theoretical** point of view, which are respectively part of the continuous and discrete geometry.

**Numerically**, the distribution depends on the choice of **basis** such as $z \in [0, N]$, then, it exists the MOs $\psi(z)$ which give the **distribution $D_{Base}$** according to a chosen basis: $\{\psi(z_1)...\psi(z_N)\}$.

**Theoretically**, the distribution called **Hückel distribution $D_{Huckel}$**, valid on each atoms, we have $\psi(C_i)$ orbitals depending on $p_{xi}$ and $p_{yi}$. The Hückel distribution is such that $z = \{z_1...z_N\}$.

We can define the **distribution of a constant helix** such as:

$$ D_{helix} = \{\phi(z_i) = \frac{z_i}{b}\} \quad (5.3) $$
with b the constant pitch and i = 0,..., N.

We consider the **cumulated angle** $\phi_c$ as:

$$ \phi_c(i) = \sum_{1}^{i} | \phi(z_i) - \phi(z_{i-1}) | \qquad (5.4) $$

If we have $\phi_c(z_0) = 0$,

$$ \phi_c(z) = \frac{1}{b}z \qquad (5.5) $$

This corresponds to a straight line passing through the origin with a constant slope $\frac{1}{b}$.

We compare the set of points $\{(z_i, \phi_c(z_i)) \ (with \ i = 0, ..., N)\}$ for a given **helical distribution** with a set of helical points $(z_i, \frac{z_i}{b})$ by minimizing a parameter as:

$$ L(b) = \sum_{i=0}^{n} (\phi_c(i) - f_i)^2 \qquad (5.6) $$

with $f_i = \frac{z_i}{b}$ by using a **regression method** analysis to obtain a fitted straight line as:

$$ \phi_c = \alpha(z - z_0) + \phi_0 \qquad (5.7) $$

where $\alpha \in \mathbb{R}$ determined following the values of $\phi_c(z_i)$.

The shape of the line is imposed by the initial point $(z_0, \phi_c(0))$ which is the first value of the cumulative angle distribution.

Finally, the first atom is positioned at $z_0 = 0$ and $\phi_c(0) = 0$, thus, the optimization issue is reduced to:

$$ \phi = \alpha z \qquad (5.8) $$

Which fits the results of $\phi_c(z_i)$.

We based our work on the study carried out by the team of Solomon et al. [56], who define the **mean of the absolute deviation** as the **MAD** index. To be informative the MAD index is used to determine how much an orbital helix deviates from the so-called perfect helix. A high MAD means that the studied helix deviates strongly from a perfect helix and a low MAD value shows the MO is very close to the perfect helix.

The **mean of the absolute deviation** is given as:

$$ | \phi_c(z_i) - f_i | \qquad (5.9) $$

We give an **analytical solution** as:

$$ \alpha = \frac{\sigma(\phi_c(z), z) + \phi_c(z)z}{\sigma^2(z)} \qquad (5.10) $$

If we have two mathematical series: $x = (x_0, ..., x_N)$ and $y = (y_0, ..., y_N)$:

$$ \bar{x} = \frac{1}{N+1} \sum_{i=0}^{N} x_i \qquad (5.11) $$
$$ \sigma(x, y) = \sum_{i=0}^{N} \frac{(x_i - \bar{x})(y_i - \bar{y})}{N + 1} \qquad (5.12) $$

$$ \sigma^2(x) = \sum_{i=0}^{N} \frac{(x_i - \bar{x})^2}{N + 1} \qquad (5.13) $$

Each of these equations represent the mean of x ($\bar{x}$), the covariance of x and y ($\sigma(x, y)$) and the variance of x ($\sigma^2(x)$) respectively.

We also defined the **correlation factor** as:

$$ \rho(x, y) = \frac{\sigma(x, y)}{\sigma(x)\sigma(y)} \qquad (5.14) $$

By linear regression, we have the approximation approach such that $\rho(\phi_c(z), z)$, $\rho^2(\phi_c(z), z)$ must be close to 1.

For reminder, to determine the quality of a measurement we use linear regression by approximating a straight line, and the correlation coefficient has to be greater than 0.95 which represents a very good approximation.

We therefore propose an **indicator of helicity (HEL)** such as:

$$ HEL = \rho(\phi_c(z), z) \qquad (5.15) $$

We prove by this indicator that the correlation coefficient is sufficiently well-known to be easily interpreted even if it is subject to discussion, is well documented. Moreover, the values of HEL can easily be compared to the data of linear regressions since $HEL \in [0, 1]$. This is more convenient than the MAD index since the MAD values can be arbitrary which is not obvious to interpret and compare.

## Distribution of angles for twisted [n]-cumulenes

We studied the angle distribution of p-orbitals along [n]-cumulene twisted with a Hückel distribution. The twisted term used here corresponds to a rotation between 0 and $\pi/2$ of one of the chain ends. Indeed, we have shown before in the manuscript that helical MOs are visible in the case of rotation of one of the extremities of the chain (seen in section 5.1.2).

**Simple example:** First of all, let us take a simple example for the angle distributions obtained for equivalent [4]-cumulene version of Möbius systems when $n = 1$ is:

$$ D_{4,1} = \{0, 32, 45, 58, 90\} \qquad (5.16) $$

The associated cumulated distribution is:

$$ \phi_c = \{0, 32, 77, 135, 225\} \qquad (5.17) $$

The "best" perfect helix fitting this set of data and the correlation factor which shows very good agreement are:

$$ \phi(z) = 49.7z \qquad (5.18) $$

$$ \rho(z, \rho_c(z)) = 0.98872 \qquad (5.19) $$
<figure><img src="image_122.png" /><figcaption>Figure 5.4: Best fitted helix for distribution of equivalent [4]-cumulene version of Möbius system and n=1</figcaption></figure>

The same approach can be done for the distribution of the equivalent [4]-cumulene version of Möbius system when n=2. In this situation, the distribution is:

$$ \mathcal{D}_{4,2} = \{0, 58, 135, 212, 270\} \qquad (5.20) $$

The associated cumulated distribution is given by:

$$ \phi_c = \{0, 58, 193, 405, 675\} \qquad (5.21) $$

The "best" perfect helix fitting and the correlation factor of this set of data which again show a very good fit are:

$$ \phi(z) = 145.3z \qquad (5.22) $$

$$ \rho(z, \rho_c(z)) = 0.97942 \qquad (5.23) $$

<figure><img src="image_123.png" /><figcaption>Figure 5.5: Best fitted helix for distribution of equivalent [4]-cumulene version of Möbius system and n=2</figcaption></figure>

**Hückel distribution of** $0 < \theta < \frac{\pi}{2}$: The Hückel distribution for [n]-cumulene twisted is obtained by vectors as:

$$ \psi_n(z) = a_n R\left(\frac{\theta}{2}\right) D_{\frac{b_n}{a_n}} R(k_n z + \delta_n) \qquad (5.24) $$

with $D_{\frac{b_n}{a_n}}: \mathbb{R}^2 \to \mathbb{R}^2$.

$D_{\frac{b_n}{a_n}}$ corresponds to the **dilatation map of weight $\frac{b_n}{a_n}$** defined as $(x, y) \to (x, \frac{b_n}{a_n}y)$.
$\delta_n$ and $k_n$ are given as:

$$ \tan(\delta_n) = -\frac{a_n}{b_n} \tan\left(\frac{\theta}{2}\right) \qquad (5.25) $$

$$ \sin^2((N+1)k_n) = \cos^2(\theta) \sin^2(k_n) \qquad (5.26) $$

with $1 \le n \le 2N$.

Generally, it is not possible to obtain angle distribution values except in the cases $\theta = 0$ and $\theta = \frac{\pi}{2}$, one can only give indications on the structure with the following.

We can then define the **Hückel distribution** $D_{Huckel}$ according to N and $\theta$:

$$ \phi(N-z) - \phi\left(\frac{N}{2}\right) = \phi\left(\frac{N}{2}\right) - \phi(z) \qquad (5.27) $$

with $z = 0, ..., N$.

This angle distribution has an axis of symmetry along the $\phi(\frac{N}{2})$ vector.

For $\phi(0) = 0$ and $\phi(N) = \theta$, this axis will be at $\frac{\theta}{2}$ or $\frac{\theta}{2} + \frac{\pi}{2}$ in accordance with the number of turns and the direction of winding of the helix.

Finally, the distribution of angles obtained is from a circular distribution as:

$$ \phi_n(z) = k_n z + \delta_n \qquad (5.28) $$

Note that with respect to zero, the distribution is:

$$ \phi_n(N-z) = -\phi_n(z) \qquad (5.29) $$

with $z = 0, ..., N$.

We know that **rotations** are **angle-preserving** transformations, but **dilatations** do not preserve the **angles** but preserve the structure of a circular distribution.

In our case, the circular distribution provides the $D_{Huckel}(\theta)$ by dilatation ($D_a^b$) along the vectors: $v_n(z) = ae^{i\phi_n(z)}$ with $i^2 = -1$ followed by a rotation of $e^{i\frac{\theta}{2}}$.

The previous observation can be justified by the work of Gunasekaran et al. [68] with circular polarized MOs. They increase the coupling between the terminal p-orbitals from $t$ to $\sqrt{2}t$ rather than a constant coupling between atoms of the molecule and thus consider a modified Hückel matrix.

The **mathematical analysis** is simplified for distributions $\phi_n(0) = 0$ and $\phi_n(N) = \theta$:

$$ \phi_n(z) = k_n z \qquad (5.30) $$

$$ Nk_n = \begin{cases} \theta + m\pi, & n = 2m + 1 \\ m\pi - \theta, & n = 2m \end{cases} \qquad (5.31) $$

with $z = 1, ..., N-1$.

We have for $z \in \{1, ..., N-1\}$:

$$ \phi_n(z) = Nk_n \frac{z}{N} = \begin{cases} (\theta + m\pi) \frac{z}{N}, & n = 2m + 1 \\ (m\pi - \theta) \frac{z}{N}, & n = 2m \end{cases} \qquad (5.32) $$
We define for $z = \frac{N}{2}$:

$$ \phi_n\left(\frac{N}{2}\right) = \begin{cases} (\theta + m\pi)\frac{1}{2}, & n = 2m + 1 \\ (m\pi - \theta)\frac{1}{2}, & n = 2m \end{cases} \qquad (5.33) $$

The Hückel distribution $D_{Hckel}(\theta)$ satisfies:

$$ \phi_n\left(\frac{N}{2}\right) - \phi_n(z) = \begin{cases} (\theta + m\pi)\left(\frac{1}{2} - \frac{z}{N}\right), & n = 2m + 1 \\ (m\pi - \theta)\left(\frac{1}{2} - \frac{z}{N}\right), & n = 2m \end{cases} \qquad (5.34) $$

and,

$$ \phi_n(N - z) - \phi_n(\frac{N}{2}) = \begin{cases} (\theta + m\pi)\left(1 - \frac{z}{N} - \frac{1}{2}\right), & n = 2m + 1 \\ (m\pi - \theta)\left(1 - \frac{z}{N} - \frac{1}{2}\right), & n = 2m \end{cases} = \phi_n\left(\frac{N}{2}\right) - \phi_n(z) \qquad (5.35) $$

**Hückel distribution in the cases $\theta = 0$ or $\theta = \frac{\pi}{2}$:** As explained, the helical MOs are only observed when we impose a **deformation constraint or substituents** at the ends of the $[n]$-cumulenes. Thus, when the constraint is at $\theta = 0$ or $\frac{\pi}{2}$ there are no helices. In reality, we know that the helices are due to the coupling of the $\pi_x$ and $\pi_y$ systems in a $C_2$ symmetry. We note that the distribution in a **cyclic Möbius case is quasi-perfect**, but this is not valid for its $[n]$-cumulene analogue.

**Case $\theta = \frac{\pi}{2}$:** The $p_x$ part is defined from $z = 1$ to $z = N$ and, the $p_y$ part from $z = 0$ to $z = N-1$.

The **coefficients** related to the wave function for $z = 1$ to $N$ are given as:

$$ c_{N,n}(z) = \sqrt{\frac{2}{N+1}} a_{N,n}(z) \qquad (5.36) $$

with $a_{N,n}(z) = \sin(k_n z)$ and $z = 1, ..., N$.

and with:

$$ k_n = \frac{n\pi}{N+1} \qquad (5.37) $$

with $1 \le n \le N$.

In this case we have a **Hückel matrix** implying the coefficients $c_x(z)$ and $c_y(z)$:

$$ c_x(z) = \sqrt{\frac{2}{N+1}} a_{N,n}(z) \qquad (5.38) $$

with $z = 1, ..., N$.

$$ c_y(z) = \sqrt{\frac{2}{N+1}} a_{N,n}(z+1) \qquad (5.39) $$
with $z = 0, ..., N - 1$.

We notice that $c_x(0) = 0$ and $c_y(N) = 0$.

For proof we refer to [73], if we get a classical $N \times N$ matrix corresponding to a chain of atoms as:

$$A_N(w) = \begin{pmatrix} w & 1 & 0 & \dots & 0 \\ 1 & w & \ddots & \ddots & \vdots \\ 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & & & & 1 \\ 0 & \dots & 0 & & w \end{pmatrix} \qquad (5.40)$$

with $w = \frac{\alpha - \lambda}{\beta}$ and $S_N(\lambda)$ the secular determinant of the Hückel matrix $H_N$ as $S_N(\lambda) = \det(H_N - \lambda I_d)$ (with $I_d$ the identity matrix).

Then, the **secular determinant** is:

$$S_N(\lambda) = \beta^N (P_N(w))^2 \qquad (5.41)$$

with $P_N(w) = \det(A_N(w))$.

The solutions of $S_N(\lambda)$ are **symmetric** and **doubly-degenerated** such as:

$$\lambda_n = \alpha + 2\beta \cos(k_{N,n}) \qquad (5.42)$$

with $k_{N,n} = \frac{n\pi}{N+1}$.

In the case where $C_2$ **symmetry is reached**, we have wave functions $\psi_{+,n}$ and $\psi_{-,n}$ that can be described according to the direction of winding of the helix respectively to the right (+) and to the left (-):

$$\psi_{+,n} = \frac{1}{\sqrt{2}}(c_y(z)p_y + c_x(z)p_x) \qquad (5.43)$$

$$\psi_{-,n} = \frac{1}{\sqrt{2}}(c_y(z)p_y - c_x(z)p_x) \qquad (5.44)$$

If we focus on the $\psi_{+,n}$, $\psi_{-,n}$ is:

$$\psi_n(z) = \sqrt{\frac{2}{N+1}} \begin{pmatrix} a_{N,n}(z) \\ a_{N,n}(z+1) \end{pmatrix} \qquad (5.45)$$

Then, we deduce $\psi_n(N - z)$:

$$\psi_n(N - z) = \sqrt{\frac{2}{N+1}} \begin{pmatrix} a_{N,n}(N - z) \\ a_{N,n}(N - z + 1) \end{pmatrix} \qquad (5.46)$$

For more details, for $z = 0, ..., N$ and $N \ge 1$, we can prove that $a_n$:

$$a_{N,n}(N - z + 1) = (-1)^{n+1}a_{N,n}(z) \qquad (5.47)$$

We know that:

$$a_n(z) = \sin\left(\frac{zn\pi}{N+1}\right) \qquad (5.48)$$
Then,

$$
\begin{align*}
a_n(N - z + 1) &= \sin\left((N - z + 1)\frac{n\pi}{N+1}\right) \\
&= \sin\left(\frac{(N+1)n\pi}{N+1} - \frac{zn\pi}{N+1}\right) \tag{5.49} \\
&= \sin(n\pi)\cos\left(\frac{zn\pi}{N+1}\right) - \cos(n\pi)\sin\left(\frac{zn\pi}{N+1}\right) \\
&= -\cos(n\pi)\sin\left(\frac{zn\pi}{N+1}\right) = (-1)^{n+1}a_n(z)
\end{align*}
$$

Thus,

$$
\psi_n(z) = (-1)^{n+1} \sqrt{\frac{2}{N+1}} \begin{pmatrix} a_{N,n}(z+1) \\ a_{N,n}(z) \end{pmatrix} \qquad (5.50)
$$

By using an **axial symmetry** along $x = y$, we have the diagonal symmetry $S : (x, y) \rightarrow (y, x)$, the wave function $\psi_{+,n}(N - z)$ is given as:

$$
\psi_{+,n}(N-z) = (-1)^{n+1} S(\psi_{+,n}(z)) \tag{5.51}
$$

We get the Hückel distribution relation eq. 5.27.

We give an expression of the **angles** along the distribution according to the various parameters considered.

Thus, we note by $\mathcal{A}_{N,n,+0,z}$ the **angle** between $\psi_{+,n}(0)$ and $\psi_{+,n}(z)$.

The angle will have the formula:

$$
\mathcal{A}_{N,n,+0,z} = \cos^{-1} \left( \frac{\epsilon(a_{N,n}(1))\epsilon(a_{N,n}(z+1))}{\sqrt{1 + \left(\frac{a_{N,n}(z)}{a_{N,n}(z+1)}\right)^2}} \right) \quad (5.52)
$$

$$
\mathcal{A}_{N,n,+0,N} = (-1)^{n+1} \frac{\pi}{2} \qquad (5.53)
$$

with $z = 0, \dots, N-1$ and $\epsilon(x)$ is a function of the same sign of $x$, equal to $1$ when $x > 0$ and equal to
$-1$ when $x < 0$.

The proof of this relation is given in Appendix 2 section 24.

**Case θ = 0:** In this situation, the $p_x$ part will be between $z = 1$ and $z = N - 1$ and the $p_y$ part from $z = 0$ to $z = N$, thus, the $[n]$-cumulenes are composed by $N + 1$ atoms.

The following **coefficients** depend on the form of the Hückel matrix:

$$
c_y(z) = \sqrt{\frac{2}{N+2}} \sin \left( \frac{n\pi(1+z)}{N+2} \right) \qquad (5.54)
$$

$$
c_x(z) = \sqrt{\frac{2}{N}} \sin \left( \frac{(n-1)\pi z}{N} \right) \qquad (5.55)
$$

we note that $c_x(0) = 0$ and $c_x(N) = 0$.
The proofs of these relations are available in Appendix 2 section 25.

If we are interested in **MOs** with $C_2$ symmetry, $\psi_{+,n}$:

$$ \psi_{+,n} = \begin{pmatrix} \sqrt{\frac{2}{N}} \sin\left(\frac{(n-1)\pi z}{N}\right) \\ \sqrt{\frac{2}{N+2}} \sin\left(\frac{n\pi(z+1)}{N+2}\right) \end{pmatrix} \qquad (5.56) $$

with $z = 1, ..., N - 1$.

If we consider the values that $n$ can take we have the **angles**:

$$ \mathcal{A}_{N,n,+0,N} = \frac{(1 - (-1)^{n+1})}{2} \pi = \begin{cases} 0 & \text{if } n \text{ is odd} \\ \pi & \text{otherwise} \end{cases} \qquad (5.57) $$

In the continuation of the manuscript we show different examples of distributions as defined here.

## 5.3 Criteria for the existence of helices

As we have seen previously in this chapter, the literature shows specific cases in which helices appear. In this section, we indicate the necessary criteria to obtain helices following two approaches: physical and chemical.

### 5.3.1 By a physical approach : Löwdin partitioning technique

This section is based on the work of Gunasekaran and al. [68] which presents a discussion of the **existence** of helical states studying directly the eigenstates of the Schrödinger equation associated to the molecule. We prove using their approach that molecules admits **generically helical states**.

In terms of mathematics, we have to be interested in the so-called **Löwdin partitioning technique** [74][57] for matrices. This is applied to Hamiltonian matrices which we can write in the case of a cumulenic system. The technique is to compute a Hamiltonian matrix of the part of the molecule that we are interested in, which then allows to return to a Hamiltonian matrix taking into account the total space.

A **schematic** example to improve the meaning of the Hamiltonian matrix is given in the work of Jin and Song [57]:

<figure><img src="image_124.png" /><figcaption>Figure 5.6: Schematic illustration of a molecule to define a Hamiltonian square matrix in the Löwdin partitioning technique [57] (the graph consists of two branch graphs a and b in cyan and a center graph c in black - the dashed edges in red represents the connections between them, with A and B being the branch-root nodes).</figcaption></figure>
Then for a complete system we define the Hamiltonian square matrix as:

$$H = \begin{pmatrix} H_{aa} & H_{ac} & 0 \\ H_{ca} & H_{cc} & H_{cb} \\ 0 & H_{bc} & H_{bb} \end{pmatrix} \qquad (5.58)$$

Note that $H_{ca}^t = H_{ac}$ with t is the transpose.

As a reminder, our system is composed of a linear chain with 2N orbitals with a left (L) and right (R) part as we have shown in the schematic figure 5.2 and more generally in figure 5.6. The elements of the left part is noted a and those of the right part b. Thus, we have on the left $p_a = (p_1, ..., p_a)$ orbitals and on the right $p_b = (p'_1, ..., p'_b)$ orbitals. Then, we have $2(N - 2)$ orbitals as $p_{x,i}$ and $p_{y,i}$ (with $i = 1, ..., N-1$), constituting the vector $p_c$.

As we know the Schrödinger equation, we have:

$$\psi = \sum_{i=1}^{a} c_i p_i + c_L p_L + \sum_{i=1}^{n} c_{x,i} p_{x,i} + c_{y,i} p_{y,i} + c_R p_R + \sum_{j=1}^{b} c_j p'_j \qquad (5.59)$$

We get the linear system as follows:

$$\left\{ \begin{array}{l} H_{aa}p_a + H_{ac}p_c = E p_a \\ H_{ca}p_a + H_{cc}p_c + H_{cb}p_b = E p_c \\ H_{b}p_c + H_{bb}p_b = E p_b \end{array} \right. \qquad (5.60)$$

If we consider once again the identity matrix $I_d$, with $d > 0$ the $I_d$ matrix is of size $d$. For a given $E$ the matrices $EI_a - H_{aa}$ and $EI_b - H_{bb}$ are invertible, we get $p_a, p_b$ and $p_c$ as:

$$p_a = (EI_a - H_{aa})^{-1} H_{ac} p_c \qquad (5.61)$$

$$p_b = (EI_b - H_{bb})^{-1} H_{bc} p_c \qquad (5.62)$$

We then replace in the linear system eq. 5.60:

$$H_{ca}(EI_a - H_{aa})^{-1}H_{ac}p_c + H_{cc}p_c + H_{cb}(EI_b - H_{bb})^{-1}H_{bc}p_c = E p_c \qquad (5.63)$$

We can then, join the respective terms of the left and right parts of the molecule, it results in a Hamiltonian term called effective:

$$H_{eff} = \Sigma_L + H_c + \Sigma_R \qquad (5.64)$$

with $\Sigma_L(E)$ and $\Sigma_R(E)$ the $2N \times 2N$ matrix depending on the coupling of the terminal groups as:

$$\Sigma_L = H_{ca}(EI_a - H_{aa})^{-1}H_{ac} \qquad (5.65)$$

and

$$\Sigma_R = H_{cb}(EI_b - H_{bb})^{-1}H_{bc} \qquad (5.66)$$

Gunasekaran et al. [68] gives the following criterion of existence of helical MOs, it is a physical criterion: to obtain helical states it is necessary that the matrices $\Sigma_L$ and $\Sigma_R$ do not commute i.e.:

$$[\Sigma_L, \Sigma_R] \neq 0 \qquad (5.67)$$
where [ , ] denotes the brakets on matrices defines by [ A, B ] = AB − BA.

The condition eq. 5.67 is **open** because generically we have helices, it is only necessary to introduce the right substituents at the end of the chain.

It should be noted that the calculation of the 2 matrices Σ<sub>L</sub> and Σ<sub>R</sub> is relatively complex [57] and are
computable only in **very simple** cases.

**5.3.2 By chemistry: using molecular symmetries**

In this section, we bypass the computational problem linked with the matrices Σ<sub>L</sub> and Σ<sub>R</sub> by stating
a criterion based on the **symmetry group** of the molecule.

We consider a **linear chain molecule** $L_A$ containing $N$ atoms denotes by $A_i$ (with $i = 1, ..., N$).
We can denote by the set $\mathbf{A} = \{A_i\}_{i=1,...,N}$ whose coordinates depend on the $z$ axis.

<figure><img src="image_125.png" /><figcaption>Figure 5.7: Schematic representation of a linear chain depending on the z axis</figcaption></figure>

We get *n* atoms being two-dimensional, with a basis set *p*<sub>*x*</sub>, *p*<sub>*y*</sub>, *p*<sub>*z*</sub> orbitals participating in the *σ*-system.
For a linear molecule, there are two planes *P*<sub>*x*</sub> and *P*<sub>*y*</sub> which contain the family of *p* orbitals as *p*<sub>*x*</sub>(*z*)
and *p*<sub>*y*</sub>(*z*) orbitals according to their position on the *z* axis.

In this case, using symmetry, we have therefore **frontier orbitals** on the left (p<sub>y</sub> = p<sub>0</sub>) and on the
right (p<sub>N</sub>) of the linear chain at N = n + 2 atoms oriented in the *xy*-plane.

The **Hückel matrix** of a molecule as $L_A$ takes the form of a basis set $p_0 = p_x, p_x^i, p_y, p_y^i, p_N$ with $i = 1, ..., n$ such as:

<figure><img src="image_126.png" /></figure>

where the matrix $A_x$ and $A_y$ are given such as (the matrix of $A_y$ is given as example):
$$A_y = \begin{array}{c} \begin{array}{c} p_y^1 \\ \vdots \\ p_y^k \end{array} \\ \begin{pmatrix} \alpha & a & \cdots & a \\ a & \ddots & \ddots & 0 \\ \vdots & \vdots & \ddots & a \\ 0 & \cdots & a & \alpha \end{pmatrix} \end{array} \qquad (5.69)$$

We can decouple the **characteristic polynomial** of the Hamiltonian matrix in two situations:

* $\odot^{\parallel} = 0$ corresponds to the situation where $p_N$ is orthogonal to $p_x^N$ (i.e. $p_N = p_y$). The matrix of mixing is reduced to:

$$Mix^{\perp} = \begin{pmatrix} \alpha \\ 0 \\ 0 \end{pmatrix} \begin{matrix} \begin{array}{c} 0 \\ \alpha \\ \mathbf{\odot}^{\perp} \end{array} & 0 \\ \alpha & \mathbf{\odot}^{\perp} \end{matrix} \begin{pmatrix} \alpha \\ 0 \\ 0 \end{pmatrix} \qquad (5.70)$$

with the characteristic polynomial:

$$P(x) = P_A(x) P_{A^{\perp}, \odot^{\perp}}(x). \qquad (5.71)$$

* $\odot^{\perp} = 0$ corresponds to the situation where $p_N$ is orthogonal to $p_y^N$ (i.e. $p_N = p_x$). The mixing matrix is reduced to:

$$Mix^{\parallel} = \begin{pmatrix} \alpha \\ \mathbf{\odot}^{\parallel} \\ 0 \end{pmatrix} \begin{matrix} \begin{array}{c} \mathbf{\odot}^{\parallel} \\ \alpha \\ 0 \end{array} & 0 \\ \alpha & \mathbf{\odot}^{\parallel} \end{matrix} \begin{pmatrix} \alpha \\ 0 \\ 0 \end{pmatrix} \qquad (5.72)$$

with the characteristic polynomial is given by:

$$P(x) = P_{A^{\perp}}(x) P_{A, \odot^{\parallel}}(x) \qquad (5.73)$$

Thus, the first case is the orthogonal configuration of the ethylene and the other one to the planar configuration.

Finally in others configurations, the **necessary condition** to obtain helical MOs is to have a mixing of the components of the $p_x$ and $p_y$ orbitals.

Note that it is obvious that the presence of the $p_x$, $p_y$ and $p_z$ orbitals limit the method to certain elements of the periodic table, in particular to those of the p-block.

The **mixing** of these orbitals seems simple but the point to keep in mind is the **symmetry**. Indeed, the presence of a **helicogenic axis** [9], i.e. an axis of symmetry $C_2$ is necessary. This axis implies that: $C_2(p_x) = p_y$ and this axis is involved in the formation of helical orbitals.

For instance, we consider the case of **ethylene** which doesn't present helical MOs. In its **orthogonal** configuration, ethylene has **mirror plane symmetry**, and **3 C_2 axes of rotation**: one along the molecular chain and 2 axes (dihedral) helicogenics. In this situation, there is a mixing generated by the axes, but due to the mirror plane symmetry, the characteristic polynomial is factorized which induces a disconnection between the family of the $p_x$ and $p_y$ orbitals. By consequence, there are **no** helical MOs.

In its **planar** configuration, there are **3 C_2 axes** (one along the molecular chain and 2 passing through the centre directed the x and y axes), none of these axes are helicogenics. The symmetries don't induce mixing of orbitals then, we **cannot** wait helical MOs.
We deduce by this example, that the **absence of a mirror plane** in the molecule is necessary from
a symmetric point of view to obtain helical orbitals.

Other point of symmetry, in the linear case of [n]-cumulenes, which present helical MOs, [n]-cumulenes **do not** have a mirror plane, but a C₂ axis of symmetry. In addition, the absence of mirror plane, implies that the molecule is chiral.

*As a reminder, the chirality of a molecule indicates that the image of the molecule by a mirror plane is not invariant.*

We then define a **symmetry criterion for helical states** as:
for a linear chain $L_A$ chiral and satisfying the structural hypothesis, admitting a $C_2$ helicogenic axis
presents explicit helical MOs.

With these different criteria of symmetry we can refer to **Curie's principle** [75]. This principle stated
in a paper of 1894 by Pierre Curie proves the relationship between the **symmetry of an effect and
its cause**: "when certain causes produce certain effects, the elements of symmetry of the causes must
be found in the effects produced". The Curie's principle gives a physical support to the sentence of
Garner et al. [55]: "the formation of helical symmetry-adapted MOs requires chirality; not surprising
considering a helix is a chiral object" this is not surprising indeed and is a consequence of the Curie's
principle.

More details about the validity of the Curie’s principle are given in the work of Chalmers [76] and
Ismael [77].

5.3.3 In the non-planar linear molecules

As discussed, in **linear molecules** the criteria of appearance of helical MOs are **well-defined** but
what about in the case of other molecules?

In the case of molecules that do not naturally have helical MOs, the molecular main chain has **2 helical axes** $C_2$, **one axis** $C_2$ **along the chain** and **3 orthogonal planes**. The presence of **mirror planes** does not permits helices. It is therefore necessary to bypass these mirror planes. From the point of view of the symmetry of molecules, the orthogonal planes are noted mirror planes such as : $\sigma_{xy}$, $\sigma_{xz}$ and $\sigma_{yz}$.

Thus, to respect the **helicity criteria**, we need to break the mirror planes. For example, if we break
the $\sigma_{xz}$ plane, then the molecule is no more plane.

Each terminal fragment ($L_1, L_2$) and ($R_1, R_2$) (as terminal groups denoted in the figure 5.2) has to be
taken into account. Considering for example the fragments $L$ in the plane of symmetry $\sigma_{xz}$. Then, if
this mirror plane is broken the fragment $R$ will not be contained in the plane $\mathcal{P}_y$ defined in the previous
section.

These different actions will generate "mechanical" constraints that we see as torsions of the chain ends
(as we have seen in the literature). As we will see in the next chapter, we show that we can prove the
existence of helical MOs on this type of systems in their ground states.

We have to take into consideration that the symmetry of the molecule also depends on the energy
state.

Thus, if we examine a symmetry group whose symmetry elements are valid for the helicity criteria, it
is then possible to see helical MOs in energy states other than the fundamental.
## 5.4 Conclusion

This chapter is very dense in information. The first part of the chapter is very informative on Möbius systems following the work of R. Hoffmann. Based on the linear case of [*n*]-cumulenes and the curiosities of the **helical orbitals** that this type of carbon systems presents.

In the second part of the chapter, we provide a detailed description of the **properties** of the helical states such as the **angle distribution** or the **helicity index (HEL)**. We detail the **angle distribution** of [*n*]-cumulenes according to the rotation of one of the chain end substituents.

Finally, the last part presents the criteria of existence of helices: by physical and chemical approach. The **physical approach** is based on the **Löwdin partitioning technique** which shows the use of Hamiltonian matrices: in particular the **criterion of non commutation** of the matrices of each of the extremities of the molecular system. This computational step being complex, it can only be applied in very simple cases.

The **chemical approach** uses the **symmetries** of the molecules. The necessary condition to obtain helices is the **mixing** of the $p_x$ and $p_y$ orbitals but also the presence of a helical axis: that is to say an axis of symmetry of type $C_2$. The last point related to symmetry is the **absence of a mirror plane**. The link with **Curie's principle** is then established with respect to the criterion of symmetry of helical states. Finally, in the case of non-planar linear molecules, the criterion of helicity is applicable in **any energy state**.
Chapter 6

Applications and visualizations

In this chapter, we use the points discussed in the previous chapter 5 and present our results in an
illustrated way.

We expose results on:

* [n]-cumulenes (with n = 3, 4, 11): in the fundamental and excited states,

• [n]-hetero-cumulenes (with n = 2, 3): in the fundamental state,

* [B=N] or [N=B]-cumulenes: in the fundamental and excited states,

* DPBD, tolanophane and molecules with metals.

The results illustrate among other the work of R. Hoffmann [9], Escudié et al. [78], Toyota et al. [79], Garner et al. [55], Caviglasso et al. [80], Cretu et al. [81] and our preprint [44].

Our studies present in a simple way the possibility of reaching **helical orbitals** within **simple native
systems** but summarize most of the cases. The systems considered are [n]-**cumulenes** with [n] =
3 and [n] = 4 without substituents at the ends of the chains. The interest focuses on the terminal
hydrogen and on the singlet or triplet energy state of the [n]-cumulene, which is clearly not mentioned
in the bibliography.

Then, we display the presence of such helices in the case of [n]-cumulenes composed of **hetero-elements** and more complex carbon molecules as **diphenylbutadiyne** and **tolanophane**.

Our results prove the presence of helix orbitals in all the cases studied, all the properties are represented by our systems. The bibliography does not rely on simple cases to answer precisely to the questions.

Most publications refer to DFT calculations to determine the presence of helical orbitals. Our work
is based for the ground and excited-states geometries, both on DFT calculations (B3LYP exchange-
correlation functional with 6-311G(d.p) basis set using Gaussian 09 program package) [82] but also on
a more solid calculation program with results obtained by Molpro CASPT2 (10.8/6-311G(d.p) using
Molpro program packages) [15]. These computations allow us to confirm the validity of the DFT results.

*The procedure followed for the calculations is given in detail in Appendix 1.*
## 6.1 Results on [n]-cumulenes

We describe the results obtained for **[n]-cumulenes** with [n] = 3, [n] = 4 and [n] = 11 (i.e. containing 3, 4 or 11 successive double bonds).

We study **several cases**: either the ends of the chains are **parallel**, or **perpendicular** and into energetic states **singlet** or **triplet**.

### 6.1.1 Fundamental states

We distinguish the **positions of the chains ends**, by imposing **rotation from 0° to 90°**. We observe the shape of the orbitals and the evolution of changes throughout the rotations. We note that the calculations are carried out in a $C_2$ symmetry to satisfy the helicity criteria.

We notice that the investigation of the helical states required a long research work, since the helical MOs are generally visible between HOMO-1 and LUMO+1.

#### [3]-cumulene

**DFT computations:** To describe the molecule: the core linear chain composed by 3 double bounds and 4 carbon atoms, has 2 $C_2$ **helicogenic axes** and a $C_2$ **axis** along the chain.

In addition, the linear chain possesses **3 orthogonal mirror planes** ($\sigma_{xy}$, $\sigma_{xz}$ and $\sigma_{yz}$).

<figure><img src="image_127.png" /><figcaption>Figure 6.1: Representation of helical MOs of a [3]-cumulene at different states of rotation (0, 50°, 90° carried out by DFT computations) in the ground state</figcaption></figure>
<figure><img src="image_128.png" /><figcaption>Figure 6.2: Molecular orbitals and energies of a [3]-cumulene associated for rotations from 0° to 90° (carried out by DFT computations) in the ground state</figcaption></figure>

As we expose in the chapter 5, the figure 6.1 illustrates perfectly the fact that the **torsional constraint** of one of the chain ends induces the appearance of a helix. Indeed in the extreme cases, at 0° plane and 90° perpendicular case we see the electronic clouds on both sides of the plane of the molecule. But at 50° when there is mixing of the $p_x$ and $p_y$ orbitals the helices are perfectly defined. It should obviously be noted that as we see in the previous figure (6.2), all cases of intermediate rotation (the calculations of the rotations with steps of 1° were carried out) also show helices.

The **helical orbital criterion** is satisfied, in fact we have broken the mirror planes symmetries by torsional constraints.

### [4]-cumulene

**DFT computations:** In the case of [4]-cumulene we are in the same situation and the same observations can be reported as for the [3]-cumulene case, but the molecule contains 4 double bounds and 5 carbon atoms.
<figure><img src="image_129.png" /><figcaption>Figure 6.3: Representation of helical MOs of a [4]-cumulene at different states of rotation (0, 50°, 90° carried out by DFT computations) in the ground state</figcaption></figure>

<figure><img src="image_130.png" /><figcaption>Figure 6.4: Molecular orbitals and energies associated for rotations from 0° to 90° (carried out by DFT computations) in the ground state</figcaption></figure>
Once again as in the case of [3]-cumulene the **helical orbital criterion** is satisfied.

For more details in these 2 cases presented, the **criterion is satisfied** thanks to the **breaking of the symmetries** of the mirror planes. Actually, it depends on the fragments of the chain ends and it is done in several conditions:

* the mirror plane $\sigma_{xz}$ is broken which means that the molecule is no longer a planar linear chain,

* considering the left fragment L of the molecule which is contained in the $\sigma_{xz}$ plane which is broken implies that the right fragment R is not contained in the $P_y$ plane. This, also implies the breaking of the $\sigma_{xy}$ mirror plane.

There is therefore an axial torsion of the terminal group of the linear chain which generically induces helical MOs.

### [11]-cumulene

**DFT calculations:** We considered the case of a **longer [n]-cumulene** and at only 10° of rotation of one of the chain ends. We also see in this case the presence of helices and even if the constraint is not very high (10° of rotation), the **helical orbital criterion is satisfied**.

*Note for a better visualization the transparency of the helices.*

<figure><img src="image_131.png" /><figcaption>Figure 6.5: Helical molecular orbitals for a rotation of 10° (carried out by DFT computations) in the ground state</figcaption></figure>
### 6.1.2 Excited states

As in the previous section, we performed the same DFT calculations, but in a **triplet excited state**, which has not been proven in the literature. This excited state results in the MOs being in 2 different spins $\alpha$ and $\beta$.

#### [3]-cumulene

**DFT computations:** We are in the case where the [n]-cumulene is in an **excited state**, i.e. in a triplet state (multiplicity $\mu = 3$). We notice here also that as in the ground state situations, when the rotations are of 0° and 90°, we **do not observe helices** but at 50° of rotation the **helices** are clearly present under the 2 multiplicities.

*Note that the shape, the number of turns and the direction of winding of the helices are not the same in the α and β-spin state.*

<figure><img src="image_132.png" /><figcaption>Figure 6.6: Representation of helical MOs of a [3]-cumulene at different states of rotation (0, 50°, 90° carried out by DFT computations) in a triplet state.</figcaption></figure>
### [4]-cumulene

**DFT computations:** The same behaviours are observed.

<figure><img src="image_133.png" /></figure>

Figure 6.7: Representation of helical MOs of a [4]-cumulene at different states of rotation (0, 50°, 90° carried out by DFT computations) in a triplet state.

We note in the 2 situations of the figures 6.6 and 6.7, that the helical MOs are more defined under α-spin than β-spin.

We know that the symmetry group of the MOs of a molecule depends on its **energy state**. Therefore if the symmetry group of the molecule does not have helical MOs but contains symmetry elements that satisfy the criterion for the existence of helices, then there will be excited states of the molecule with helical MOs that will fit the symmetry. In the same way, this statement also applies for the **electronic multiplicities**.
## 6.2 Molecules with hetero-elements and more examples

### 6.2.1 [n]-hetero-cumulenes

[n]-cumulenes being molecules with a linear carbon chain, we chose to work with more varied [n]-cumulenes. By referring to the literature [78] where 15 families of [n]-cumulenes are reviewed, we studied [n]-cumulenes containing in the principal linear chain (from atoms C₀ to Cₙ): **silicon, germanium, phosphorus, arsenic or more simply oxygen or nitrogen**.

We resume all the studied [n]-hetero-cumulenes in the following table:

<table><caption>Table 6.1: Set of studied [n]-hetero-cumulenes</caption><thead><tr><th></th><th>C<sub>0</sub> / C<sub>N</sub></th><th>L<sub>1</sub></th><th>L<sub>2</sub></th><th>R<sub>1</sub></th><th>R<sub>2</sub></th></tr></thead><tbody><tr><td>[n]-cumulene</td><td>C / C</td><td>H</td><td>H</td><td>H</td><td>H</td></tr><tr><td>[2]-heterocumulene:</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Phosphaallene</td><td>P / C</td><td>H</td><td>-</td><td>H</td><td>H</td></tr><tr><td>Diphosphaallene</td><td>P / P</td><td>H</td><td>-</td><td>H</td><td>-</td></tr><tr><td>Phosphaazallene</td><td>P / N</td><td>H</td><td>-</td><td>H</td><td>-</td></tr><tr><td>Arsaallene</td><td>As / C</td><td>H</td><td>-</td><td>H</td><td>H</td></tr><tr><td>Arsaphosphaallene</td><td>As / P</td><td>H</td><td>-</td><td>H</td><td>-</td></tr><tr><td>Diarsaallene</td><td>As / As</td><td>H</td><td>-</td><td>H</td><td>-</td></tr><tr><td>Siaallene</td><td>Si / C</td><td>H</td><td>H</td><td>H</td><td>H</td></tr><tr><td>Phosphasilaallene</td><td>Si / P</td><td>H</td><td>H</td><td>H</td><td>-</td></tr><tr><td>Silaketene</td><td>Si / O</td><td>H</td><td>H</td><td>-</td><td>-</td></tr><tr><td>Germaallene</td><td>Ge / C</td><td>H</td><td>H</td><td>H</td><td>H</td></tr><tr><td>Germaphosphaallene</td><td>Ge / P</td><td>H</td><td>H</td><td>H</td><td>-</td></tr><tr><td>[3]-heterocumulene:</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Phosphabutatriene</td><td>P / C</td><td>H</td><td>-</td><td>H</td><td>H</td></tr><tr><td>Diphosphabutariene</td><td>P / P</td><td>H</td><td>-</td><td>H</td><td>-</td></tr><tr><td>Arsabutariene</td><td>As / C</td><td>H</td><td>-</td><td>H</td><td>H</td></tr><tr><td>Silabutariene</td><td>Si / C</td><td>H</td><td>H</td><td>H</td><td>H</td></tr></tbody></table>

For more details concerning the table: we note [2]-hetero-cumulene and [3]-hetero-cumulene constituted by hetero-elements (P: phosphorus, N: nitrogen, As: arsenic, Si: silicon, Ge: germanium, O: oxygen) and the presence or absence of hydrogen in terminal groups.

#### [2]-hetero-cumulenes

**DFT calculations:** The representation of the MOs of the [2]-hetero-cumulenes from the table 6.1, shows the **existence of helices** between the **HOMO-1** and the **LUMO+1**, for a **rotation of 25°** of only one of the terminal group.
<table><caption>Figure 6.8: Representation of helical MOs of [2]-hetero-cumulenes at 25° of torsion by DFT computations in the ground state</caption><thead><tr><th></th><th>HOMO-1</th><th>HOMO</th><th>LUMO</th><th>LUMO+1</th></tr></thead><tbody><tr><td>Phosphaallene</td><td></td><td></td><td></td><td></td></tr><tr><td>Diphosphoaallene</td><td></td><td></td><td></td><td></td></tr><tr><td>Phosphaazaallene</td><td></td><td></td><td></td><td></td></tr><tr><td>Arsaallene</td><td></td><td></td><td></td><td></td></tr><tr><td>Arsaphosphaallene</td><td></td><td></td><td></td><td></td></tr><tr><td>Diarsaallene</td><td></td><td></td><td></td><td></td></tr><tr><td>Silaallene</td><td></td><td></td><td></td><td></td></tr><tr><td>Phosphasilaallene</td><td></td><td></td><td></td><td></td></tr><tr><td>Silaketene</td><td></td><td></td><td></td><td></td></tr><tr><td>Germaallene</td><td></td><td></td><td></td><td></td></tr><tr><td>Germaphosphaallene</td><td></td><td></td><td></td><td></td></tr></tbody></table>

We observe for most of the molecules that the helices exist for both HOMO and LUMO but the shapes and windings of LUMO and LUMO+1 seem more complex.

### [3]-hetero-cumulenes

**DFT calculations:** Here, also for a 25° of torsion, we represented the MOs between the **HOMO-1** and the **LUMO+1**.

In this case, the helices are existing and better defined for HOMO-1 and HOMO.
<table><thead><tr><th></th><th>HOMO-1</th><th>HOMO</th><th>LUMO</th><th>LUMO+1</th></tr></thead><tbody><tr><td>Phosphabutatriene</td><td></td><td></td><td></td><td></td></tr><tr><td>Diphosphabutatriene</td><td></td><td></td><td></td><td></td></tr><tr><td>Arsabutatriene</td><td></td><td></td><td></td><td></td></tr><tr><td>Silabutatriene</td><td></td><td></td><td></td><td></td></tr></tbody></table>

Figure 6.9: Representation of helical MOs of [3]-hetero-cumulenes at 25° of torsion by DFT computations in the ground state

In the 2 figures presented 6.8 and 6.9 we observe that the helical MOs are **less well-defined** than in the cases of [2] and [3]-hetero-cumulenes.

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

After using hetero-elements, we chose to design linear chains as [n]-cumulenes from **boron** and **nitrogen** only, based on the work of Cretu et al. [81]. These linear boron nitride chains highlight the existence of **non-standard helices**.

*Note that boron nitride nanostructures are known to present a wide diversity of chemical and physical properties.*

Here, we see a **large variety of MOs in helices** according to the number of atoms and the alternation between boron and nitrogen in the ground and excited states.

<table><tbody><tr><td rowspan="4">δ = 0<br>Doublet</td><td>α-HOMO-1</td><td>α-HOMO</td><td>α-LUMO</td><td>α-LUMO+1</td></tr><tr><td>β-HOMO-1</td><td>β-HOMO</td><td>β-LUMO</td><td>β-LUMO+1</td></tr><tr><td>α-HOMO-1</td><td>α-HOMO</td><td>α-LUMO</td><td>α-LUMO+1</td></tr><tr><td>β-HOMO-1</td><td>β-HOMO</td><td>β-LUMO</td><td>β-LUMO+1</td></tr><tr><td rowspan="4">δ = 0<br>Quadruplet</td><td>α-HOMO-1</td><td>α-HOMO</td><td>α-LUMO</td><td>α-LUMO+1</td></tr><tr><td>β-HOMO-1</td><td>β-HOMO</td><td>β-LUMO</td><td>β-LUMO+1</td></tr><tr><td>α-HOMO-1</td><td>α-HOMO</td><td>α-LUMO</td><td>α-LUMO+1</td></tr><tr><td>β-HOMO-1</td><td>β-HOMO</td><td>β-LUMO</td><td>β-LUMO+1</td></tr></tbody></table>

Figure 6.10: Representation of the molecular orbitals of the **B=N=B=N=B-cumulene** (by DFT computations) for a rotation of 30° (charge 0 - doublet and quadruplet states)
<figure><img src="image_134.png" /><figcaption>Figure 6.11: Representation of the molecular orbitals of the N=B=N=B=N-cumulene (by DFT computations) for a rotation of 30° (charge 1 - singlet and triplet states)</figcaption></figure>

<figure><img src="image_135.png" /><figcaption>Figure 6.12: Representation of the molecular orbitals of the B=N=B=N=B=N-cumulene (by DFT computations) for a rotation of 30° (charge 0, 1 and 2 - singlet, doublet and triplet states)</figcaption></figure>

### 6.2.3 Other examples

For other examples, there is also interest in diphenylbutadiyne (DPBD) [83] which has a linear chain combining single and triple bonds and phenyl end groups.

The tolanophane molecule is also interesting because it is composed by 2 units of diphenylethyne with 2 linear chains [79].
## DPBD molecule

The following figure 6.13 presents the representation of the MOs from HOMO-1 to LUMO+1 according to the $C_2$ symmetry for rotations from 0° to 90° which highlights the **helices**. The helices are only present on the linear chain where the helical orbital criterion can be applied.

We note that the helices are well defined for the **LUMO+1** between 20° and 80° of rotation and once again that the **rotation** of one of the chain ends induces the appearance of **helices**.

<figure><img src="image_136.png" /><figcaption>Figure 6.13: Representation of the molecular orbitals of the DPBD molecule (by DFT computations)</figcaption></figure>

## Tolanophane molecule

In the following figure, the molecule of symmetry $C_2$ is **rotated** by one end of the chain. As indicated at 0° there are no helices but for intermediate angles helices are present.

Beyond 65° the helices are less well defined. We note that the helices are **clearly visible** only on the **LUMO+3** orbitals and once again along the linear chains where the criterion of helical orbitals is applicable.
<figure><img src="image_137.png" /><figcaption>Figure 6.14: Representation of the molecular orbitals LUMO+3 of the tolanophane molecule (by DFT computations)</figcaption></figure>

### Molecules with metals

As discussed in the previous chapter, to study the helical MOs, we focused on the *p*-orbitals of the systems. But it is also possible to be interested in these same states in the case of *d*-orbitals [56]. For this type of orbitals, it is the **metals** that carry them thus, we have to be attentive to the **metal-cumulenes**.

The study of MOs in helices in molecules with metal-like atoms is **more confidential**. In a recent paper by Garner et al. [56] the research team highlights the possibility of obtaining helical states with *d*-type orbitals. They propose **complex systems** based on ruthenium such as:

$$ \text{trans} - [EtC = (C =)_4C = Ru = (C =)_4CMe]^2+ $$

To try to prove this as well, we have carried out different calculation tests with metals, such as $[M_2X_8]^{2-}$ [80]. We have worked on metal complexes from rhenium $[Re_2H_8]^{2-}$. We have chosen to carry out our analysis in a **lower energy configuration** with eclipsed hydrogen atom ligands, thus
in $D_{4h}$ symmetry. This configuration provides a better overlap of the sigma systems between metals. The formation of the helical MOs is observed at the region of the rheniums and leads to the breaking of the applied symmetry.

<figure><img src="image_138.png" /><figcaption>Figure 6.15: Representation of the molecular orbitals HOMO and HOMO-5 of the rhenium complex $[Re_2H_8]^{2-}$ (by DFT computations)</figcaption></figure>

## 6.3 Conclusion

In this 6th chapter, we **applied** the elements presented and proven in chapter 5 with concrete examples: $[n]$-cumulenes, $[n]$-hetero-cumulenes and more complex molecules especially with metals.

We show the results of DFT calculations for $[n]$-cumulenes with 3, 4, and 11 double bonds in **ground** and **excited** states following chain end rotations.

In parallel, we have carried out CASPT2 type calculations which also confirm the presence of helices, and proves that the formation of helices **does not depend** on the computation method.

We give illustrations of the formation of helices for each of the examples except for the 0° and 90° rotation cases. The so-called criterion for obtaining helices is thus satisfied.

In the case of $[n]$-hetero-cumulenes the helices are studied for a rotation of 25° and helices are obtained in most situations. We note that for silaketene the helices are not clearly visible between HOMO-1 and LUMO+1.

The cases of $[B=N]$ and $[N=B]$-cumulenes show non-standard helices of the different energetic states.

The examples of **DPBD** and **tolanophane** are complementary. They show that the formation of helices depends on the mixing of $p_x$ and $p_y$ induced by the multiple bonds of the central linear chain during the rotation of one of the chain ends, whether it is the extremities are complex or not.

Finally to complete the study, we studied *d*-type orbitals with **metal-cumulenes**. These metallic complexes present **pseudo-helices** but further investigations for these situations could be considered.

To conclude on these results, we note that the influence of the rotation of the chain ends is essential in the existence of helices and that the complexity of the substituents and the number of atoms or double bonds (odd or even) are not determining factors.

Finally, the formation of such helices provides a different electronic distribution which certainly has an influence on the molecular reactivity. This gives access to new perspectives that require investigations beyond this manuscript.
Conclusion of the part III

This part III is composed of 3 chapters very dense in information.

Chapter 4 is **theoretical** and identifies all the elements and tools necessary to understand aromaticity for linear and cyclic systems in different situations.

Chapter 5 focuses on **helical states** which interested us a lot and required some resources. We expose the properties of these helical states and the ways to characterize them (HEL, $A_{N,n,\pm,0,z}$ for example). Moreover, we determine and prove the **criteria** of existence of helices based on mathematical (Löwdin) and chemical (Hückel and symmetries) works.

Chapter 6 gives a series of **illustrated examples** to summarize what we have shown by theory and to validate hypotheses concerning the nature of the substituents and the atoms constituting the system and on the constraints to be applied to provide helices.
<figure><img src="image_139.png" /></figure>
# Part IV

## General conclusion
This manuscript is the result of three years of work, which required a great investment and a close collaboration between the work of chemists and mathematicians to provide general tools easily usable by the scientific community.

The **3 parts** of this manuscript are complementary and were detailed for a good understanding.

The **reminder** gives a general basic review of what is needed to understand in this work. The reminders related to the Schrödinger equation, the Hückel matrices, the atomic and molecular orbitals are essential for the subjects studied in this thesis. In addition, the computational methods used for this work are also largely explained.

A **second part** of this thesis work is based on the understanding of the **reactivity** and its representation by the π-Orbital Axis Vector (POAV) theory initially proposed by R. C. Haddon.
We have worked to implement an adapted, clarified and generalized response of the tools related to POAV1. This answer gives the possibility to avoid heavy computational methods in the case of complex molecules, and to predict the reactivity, essential information for chemists.

All of this work also, gave us the opportunity to publish our first article on the subject of the pyramidalization angle, the POAV1 and the associated tools. This work has been illustrated with fullerenic, non-fullerenic and nitrile imines molecules by the means of the *Pychemcurv* software developed in the context of the thesis and which is detailed in chapter 2. This illustrative dense chapter shows clearly the extent of the work that has been done and the investments of our research team.

**POAV2 theory's** work has not been fully finalized and will certainly require additional time for analysis and research, particularly the quantified difference of POAV2 versus POAV1. The theoretical difference between the two theories has clearly been established by our work, however the results of comparisons are not sufficient enough to make clear conclusions at this stage. This leads to the following question, is the computational cost of the algorithm compensated by the precision's degree obtained? It doesn't, meaning that POAV1 can be used without any problem. The struggle being majorly on the reactivity cartographies of molecules, the result is important because they can be carried out without difficulties on large molecules. At this time, a preprint of our work on POAV2 is in progress and we should have solutions to the unsolved questions.

The **third part** of this manuscript is focused on **aromaticity** and **helical states**.

The study of aromaticity is closely related to the reactivity issues encountered in part II and aromatic molecules will be privileged for many useful reactions.

The work on the **aromaticity criteria** has required an important research work by using old works of Hückel and Zimmerman. We have established with the help of mathematics all the definitions and relations in order to make the presentation and the link between the visions of Hückel and Möbius, to establish relations and energetic comparisons. Moreover, we make the distinction according to the electronic configuration between the cyclic and linear cases in a complete way by many parameters.

The work related to aromaticity has emphasized the curiosity of the **helical orbitals** studied by R. Hoffmann. We have spent a lot of time for understanding these particular electronic states. All the properties of these states such as the distribution of angles, could be studied in different situations in particular according to the rotational constraint applied to the molecular ends which proved to be essential. In a physical and chemical way, we have established the criteria of existence of these helices related to the molecular symmetry. The last chapter illustrates this important work and gives the opportunity to validate properties and observations.

The manuscript provides an important contribution of information on the theoretical level and gives simple tools to establish the link between different major notions as curvature, reactivity and aromaticity. It therefore, opens perspectives for the determination of the reactivity and the characterization of new materials.
# Part V

## Appendices
# Appendix 1 : Chemistry part

*This first appendix presents the detailed computational techniques used to perform this work and, the different molecule representations used for the quantum calculations of the third part of this thesis manuscript chapter 6.*

## Computational techniques

The different simulations of molecules carried out in this manuscript were possible thanks to the cartesian coordinates (x, y, z) of each molecule under a precise symmetry of sections 2.2, and 2.3, and in a precise torsion angle in the particular case of the third part chapter 6. We make available in a supplementary information file all the cartesian data that we have generated to perform the simulations.

For all the simulations carried out in the application part of the database of Tománek and Frederick (section 2.2), the cartesian coordinates were available in free access [5].

However, for the coordinates used in the applications to the compounds resulting from the work of R. C. Haddon (section 2.3), and in the case of [n]-cumulenes and the other examples associated to chapter 6 it was necessary to create molecular structures (in 3D) files.

By consequence, the data files were generated by modeling the molecules, by creating each of the molecular structures in 3D by the **Avogadro** or VMD [84][40] software. For each molecule, their geometry was optimized on the software. Then, the cartesian data of the molecules are extracted in a **.xyz** file. These data are the cartesian coordinates (x, y, z) of each molecule under precise constraints.

*Note that a lot of work has been necessary to select and design all these molecules.*

When all the cartesian data were obtained, we used them for the Gaussian [12] quantum calculations. For each of the molecules, we created **.com** type calculation launching files in order to obtain all the necessary information in terms of energy, frequency, and orbitals in **.out** and **.fchk** type output files. The exploitation (in particular the visualization of the orbitals) of the **.out** files was carried out by **Molden** [85] and the **.fchk** files by the **Gaussview** software [86] which give access to 3D structural data of the molecules, energy and orbital data but also give information on the viability of the molecules designed. All the calculations were performed by DFT at a B3LYP-6-311G* level but also CASPT2 calculations have completed and validated certain of the previous results.

All the simulations of computations were carried out on the scientific computing cluster Pyrene of the university. All the information related to the cluster is easily accessible [87].

*Notice that the theoretical details about the quantum computation methods are given in the reminder part of the manuscript 0.3.*

## Representation of the molecules used for DFT computations in chapter 6

The representations of this section, concern the molecules used for the DFT calculations presented for all of the [3]-cumulene, [4]-cumulene, [11]-cumulene, [2]-heterocumulenes, [3]-heterocumulenes, [B=N]-cumulenes, [N=B]-cumulenes, DPBD molecule, and tolanophane molecule following the detailed method presented in the first section of the appendix.

All the representations of the molecules are provided for molecules with 0°, 25°, 30° of rotation.
**[n]-cumulenes:**

**[3]-cumulene**

<figure><img src="image_140.png" /><figcaption>Figure a1.16: Representation of the [3]-cumulene in the fundamental state at 0°.</figcaption></figure>

**[4]-cumulene**

<figure><img src="image_141.png" /><figcaption>Figure a1.17: Representation of the [4]-cumulene in the fundamental state at 0°.</figcaption></figure>

**[11]-cumulene**

<figure><img src="image_142.png" /><figcaption>Figure a1.18: Representation of the [11]-cumulene in the fundamental state at 0°.</figcaption></figure>

**[2]-heterocumulenes:**

**Phosphaallene**

<figure><img src="image_143.png" /><figcaption>Figure a1.19: Representation of the phosphaallene in the fundamental state at 25°.</figcaption></figure>

**Diphosphoaallene**

<figure><img src="image_144.png" /><figcaption>Figure a1.20: Representation of the diphosphoaallene in the fundamental state at 25°.</figcaption></figure>

**Phosphaazaallene**

<figure><img src="image_145.png" /><figcaption>Figure a1.21: Representation of the phosphaazaallene in the fundamental state at 25°.</figcaption></figure>
### Arsaallene

<figure><img src="image_146.png" /><figcaption>Figure a1.22: Representation of the arsaallene in the fundamental state at 25°.</figcaption></figure>

### Arsaphosphaallene

<figure><img src="image_147.png" /><figcaption>Figure a1.23: Representation of the arsaphosphaallene in the fundamental state at 25°.</figcaption></figure>

### Diarsaallene

<figure><img src="image_148.png" /><figcaption>Figure a1.24: Representation of the diarsaallene in the fundamental state at 25°.</figcaption></figure>

### Silaallene

<figure><img src="image_149.png" /><figcaption>Figure a1.25: Representation of the silaallene in the fundamental state at 25°.</figcaption></figure>

### Phosphasiaallene

<figure><img src="image_150.png" /><figcaption>Figure a1.26: Representation of the phosphasiaallene in the fundamental state at 25°.</figcaption></figure>

### Silaketene

<figure><img src="image_151.png" /><figcaption>Figure a1.27: Representation of the silaketene in the fundamental state at 25°.</figcaption></figure>

### Germaallene

<figure><img src="image_152.png" /><figcaption>Figure a1.28: Representation of the germaallene in the fundamental state at 25°.</figcaption></figure>
## Germaphosphaallene

<figure><img src="image_153.png" /><figcaption>Figure a1.29: Representation of the germaphosphaallene in the fundamental state at 25°.</figcaption></figure>

### [3]-heterocumulenes:

#### Phosphabutatriene

<figure><img src="image_154.png" /><figcaption>Figure a1.30: Representation of the phosphabutatriene in the fundamental state at 25°.</figcaption></figure>

#### Diphosphabutatriene

<figure><img src="image_155.png" /><figcaption>Figure a1.31: Representation of the diphosphabutatriene in the fundamental state at 25°.</figcaption></figure>

#### Arsabutatriene

<figure><img src="image_156.png" /><figcaption>Figure a1.32: Representation of the arsabutatriene in the fundamental state at 25°.</figcaption></figure>

#### Silabutatriene

<figure><img src="image_157.png" /><figcaption>Figure a1.33: Representation of the silabutatriene in the fundamental state at 25°.</figcaption></figure>

### $[B = N]$ or $[N = B]$-cumulenes:

#### B=N=B=N=B cumulene (charge = 0 - doublet and quadruplet states)

<figure><img src="image_158.png" /><figcaption>Figure a1.34: Representation at 30° of rotation.</figcaption></figure>
N=B=N=B=N cumulene (charge = 1 - singlet and triplet states)

<figure><img src="image_159.png" /><figcaption>Figure a1.35: Representation at 30° of rotation.</figcaption></figure>

B=N=B=N=B=N cumulene (charge = 0 - singlet state)

<figure><img src="image_160.png" /><figcaption>Figure a1.36: Representation at 30° of rotation.</figcaption></figure>

B=N=B=N=B=N cumulene (charge = 1 - doublet state)

<figure><img src="image_161.png" /><figcaption>Figure a1.37: Representation at 30° of rotation.</figcaption></figure>

B=N=B=N=B=N cumulene (charge = 2 - triplet state)

<figure><img src="image_162.png" /><figcaption>Figure a1.38: Representation at 30° of rotation.</figcaption></figure>

## DPBD molecule

<figure><img src="image_163.png" /><figcaption>Figure a1.39: Representation at 30° of rotation.</figcaption></figure>

## Tolanophane molecule

<figure><img src="image_164.png" /><figcaption>Figure a1.40: Representation at 30° of rotation.</figcaption></figure>
Rhenium complex $[Re_2H_8]^{2-}$

<figure><img src="image_165.png" /><figcaption>Figure a1.41: Representation of the rhenium complex.</figcaption></figure>
# Appendix 2: Technical and proofs part of the results used

For the convenience of the reader, proofs of the results are detailed in this appendix and some of the results in this thesis are provided from our article and preprints [4, 44, 36].

## 1-Proofs associated to the section 0.2.4: Characteristic polynomial of H

Note that by using the graph approach for Hückel's calculations, we are in the case of uniform distribution:

$$ \mathbf{H} = \alpha \mathbf{Id}_N + \beta \cos(\phi) \mathbf{A}(\mathbf{G}) \quad (a2.1) $$

Taking $\sigma_1, \dots, \sigma_n$ the real ordered eigenvalues of $\mathbf{A}(\mathbf{G})$, since $\mathbf{A}(\mathbf{G})$ is symmetrical, we obtain the relation between the eigenvalues of $\mathbf{H}$ and $\mathbf{A}(\mathbf{G})$:

$$ \lambda_i = \alpha + \beta \cos(\phi) \sigma_i \quad (a2.2) $$

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

Assuming that $\mathbf{H}$ is diagonalizable, $P$ is the passing matrix between the initial basis and the eigenvector basis such that:

$$ \mathbf{D} = P^{-1}\mathbf{H}P \quad (a2.3) $$

where $\mathbf{D}$ is the diagonalizable matrix.

$$ \mathbf{D} = \begin{pmatrix} \lambda_1 & 0 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & \dots & 0 & \lambda_N \end{pmatrix} \quad (a2.4) $$

If we multiply on each side the relation a2.1 and using $P^{-1}P = \mathbf{Id}_N$, we have:

$$ \mathbf{D} = \alpha \mathbf{Id}_N + \beta \cos(\phi) P^{-1} \mathbf{A}(\mathbf{G}) P \quad (a2.5) $$

The matrix $P^{-1}\mathbf{A}(\mathbf{G})P$ must be diagonal and of eigenvalues $\sigma_i$, $i = 1, \dots, N$ which results in the relation on the eigenvalues of $\mathbf{H}$ and $\mathbf{A}(\mathbf{G})$.

## 2-Proofs associated to the section 0.2.4: Total energy of a molecule

This relation is simplified when we assume a uniform distribution $\phi_i = \phi$ because $\nu_i(\phi) = \cos(\phi)\sigma_i$ where the $\sigma_i$ are the eigenvalues of the adjacent matrix of the graph $G$ of the molecule:

$$ E = \pi(S)\alpha + \beta \cos(\phi) \sum_{i=1}^{N} g_i \sigma_i \quad (a2.6) $$

Note that in most interesting cases, we can relate the value of $g_i$ to the sign of $\sigma_i$ via the rule $g_i = 2$ if $\sigma_i > 0$ and $g_i = 0$ if $\sigma_i < 0$.

In this case, the energy is: $E = \pi(S)\alpha + 2\beta \cos(\phi) \sum_{i=1}^{N} \sigma_i \mathbf{1}_{\sigma_i>0}$ where $\mathbf{1}_{\sigma_i>0}$ is the indicator of $\sigma_i > 0$. By separating the eigenvalues $\le 0$, noted $\sigma_1, \dots, \sigma_k$, for a certain $k$, from the positive eigenvalues $\sigma_{k+1}, \dots, \sigma_N$ and by using the relation $(\sigma_1 + \dots + \sigma_k) + (\sigma_{k+1} + \dots + \sigma_N) = 0$, we rewrite the energy as:
$$
\begin{align*}
E &= \pi(S)\alpha + 2\beta \cos(\phi) \sum_{i=1}^{N} \sigma_i \mathbf{1}_{\sigma_i>0} \\
&= \pi(S)\alpha + 2\beta \cos(\phi) \sum_{i=k+1}^{N} \sigma_i \\
&= \pi(S)\alpha + \beta \cos(\phi) \left[ \sum_{i=k+1}^{N} \sigma_i + \sum_{i=k+1}^{N} \sigma_i \right] \tag{a2.7} \\
&= \pi(S)\alpha + \beta \cos(\phi) \left[ \sum_{i=k+1}^{N} \sigma_i - \sum_{i=1}^{k} \sigma_i \right] \\
&= \pi(S)\alpha + \beta \cos(\phi) \sum_{i=1}^{N} |\sigma_i|
\end{align*}
$$

The quantity $E(G) = \sum_{i=1}^{N} |\sigma_i|$ is called energy of the graph $G$ of the molecule. We have then the relation:

$$E = \pi(S)\alpha + \beta \cos(\phi)E(G) \quad (a2.8)$$

which explains that in some simple configurations, the $G$ graph contains all the information of the orbitals.

### 3-Proofs associated to the section 1.1.1: POAV$_{\epsilon}$(A)

The POAV$_{\epsilon}$(A) vector is only defined by the **local geometry** of the molecule in a given atom A and precisely only on the relative angles of the bonds starting in A. By construction, all regularized vector $\overline{AReg_{\epsilon}(B)}$ for $B \in \star(A)$ satisfy the relation (1.4) meaning that if POAV$_{\epsilon}$(A) makes a constant angle with all the vectors $\overline{AReg_{\epsilon}(B)}$ for $B \in \star(A)$, this vector makes also a constant angle with all vectors in $\overline{AReg'_{\epsilon}(B)}$ with $B \in \star(A)$ for any $\epsilon' > 0$. As a consequence, the vector POAV(A)$_{\epsilon}$ vector is independent of $\epsilon$.

### 4-Proofs associated to the section 1.2.1: Condition of normalization

The normalization on the $s$ component of $(h_1, h_2, h_3, h_{\pi})$ gives:

$$c_{\pi}^{2} + 3\mu^{2}\lambda_{\pi}^{2} = 1, \qquad (a2.9)$$

and the one on the $p_z$ component gives:

$$\lambda_{\pi}^{2} + 3\mu^{2}c_{\pi}^{2} = 1. \qquad (a2.10)$$

We then obtain by addition of the two previous equations:

$$(c_{\pi}^{2} + \lambda_{\pi}^{2})(1 + 3\mu^{2}) = 2 \qquad (a2.11)$$

As $c_{\pi}^2 + \lambda_{\pi}^2 = 1$, this gives:

$$\mu = \frac{1}{\sqrt{3}} \qquad (a2.12)$$

This concludes the proof.
## 5-Proofs associated to the section 1.2.1: Hybridization coefficients and POAV1

The POAV1 vector is such that the angle between the $h_π$ orbital and each orbital $h_1, h_2, h_3$ is:

$$ \frac{\pi}{2} + \text{Pyr}(A) $$

The vector associated to $h_π$ is $v_π = (0, 0, \lambda_π)$ and the one with $h_1$ is $v_1 = (c_{1,2}, 0, \frac{-c_π}{\sqrt{3}})$.

The scalar product between the two vectors gives $v_π \cdot v_1 = -\frac{1}{\sqrt{3}}c_π\lambda_π$.

This scalar product is also equal to $\|v_π\| \|v_1\| \cos(v_π, v_1)$.

As $\|v_π\|^2 = \lambda_π^2$ and $\|v_1\|^2 = \frac{3c_{1,2}^2+c_π^2}{3}$, we obtain taking the square on each side of the equality:

$$ \frac{\lambda_{\pi}^{2} c_{\pi}^{2}}{3} = \frac{3c_{1,2}^{2} + c_{\pi}^{2}}{3} \lambda_{\pi}^{2} \cos^{2}\left(\frac{\pi}{2} + \text{Pyr}(A)\right) \quad (a2.13) $$

We then have:

$$ \lambda_{\pi}^{2} \left[ c_{\pi}^{2} - (3c_{1,2}^{2} + c_{\pi}^{2}) \cos^{2}\left(\frac{\pi}{2} + \text{Pyr}(A)\right) \right] = 0 \quad (a2.14) $$

As $\lambda_π \neq 0$, we obtain:

$$ c_{\pi}^{2} \left( 1 - \cos^{2}\left(\frac{\pi}{2} + \text{Pyr}(A)\right) \right) = 3c_{1,2}^{2} \cos^{2}\left(\frac{\pi}{2} + \text{Pyr}(A)\right) \quad (a2.15) $$

Using the fact that:

$$ 1 - \cos^2\left(\frac{\pi}{2} + \text{Pyr}(A)\right) = \sin^2\left(\frac{\pi}{2} + \text{Pyr}(A)\right) $$

and the equality:

$$ \cos\left(\frac{\pi}{2} + \text{Pyr}(A)\right) = -\sin(\text{Pyr}(A)) $$

and

$$ \sin\left(\frac{\pi}{2} + \text{Pyr}(A)\right) = \cos(\text{Pyr}(A)) $$

We deduce:

$$ c_{\pi}^{2} \cos^{2}(\text{Pyr}(A)) = 3c_{1,2}^{2} \sin^{2}(\text{Pyr}(A)) \quad (a2.16) $$

Assuming that $\text{Pyr}(A) \neq \frac{\pi}{2}$, we obtain:

$$ c_{\pi}^{2} = 3c_{1,2}^{2} \frac{\sin^{2}(\text{Pyr}(A))}{\cos^{2}(\text{Pyr}(A))} = 3c_{1,2}^{2} \tan^{2}(\text{Pyr}(A)), \quad (a2.17) $$

which concludes the proof.

## 6-Proofs associated to the section 1.2.1: sp³ hybridization conditions

This follows from $c_{2,3}^2 + c_{3,3}^2 = 2c_{2,3}^2 = 1$, which gives for example $c_{2,3} = \pm \frac{1}{\sqrt{2}}$ and $c_{3,3} = \pm \frac{1}{\sqrt{2}}$.

The second normalization gives $c_{1,2}^2 + 2c_{2,2}^2 = 1$.

The symmetry condition (eq. 1.23) with $c_{2,3}^2 = \frac{1}{2}$ leads to the relation $c_{2,2}^2 + \frac{1}{2} = c_{1,2}^2$. Putting this expression in the normalization equation gives $c_{2,2}^2 + \frac{1}{2} + 2c_{2,2}^2 = 1$.

We deduce that $3c_{2,2}^2 = \frac{1}{2}$ or $c_{2,2}^2 = \frac{1}{6}$ and so $c_{1,2}^2 = \frac{2}{3}$.
## 7-Proofs associated to the section 1.3: Admissible molecules

The proof of this angle is well-defined, and follows the same line as in the trivalent case. Let $\epsilon > 0$ be given and denoted by $z_{A,\epsilon}$, the distance $O_{\epsilon}A$. Then, we have for a given bond AB, $B \in \star(A)$, as:

$$ \sin(Pyr(A)) = \frac{z_{A,\epsilon}}{\epsilon} \qquad (a2.18) $$

This quantity does not depend on B and is, in fact, independent of $\epsilon$. Indeed, let us consider $\epsilon' = \lambda\epsilon$, then $z_{A,\epsilon'} = \lambda z_{A,\epsilon}$ and $\frac{z_{nA,\epsilon'}}{\epsilon'} = \frac{z_{A,\epsilon}}{\epsilon}$.

The previous definition is, of course, far from being satisfied as it works for a very restrictive class of molecules. We discuss more precisely the restriction associated with the use of the pyramidalization angle in the section 1.3.

## 8-Proofs associated to the section 1.4.1: Non-linear relationship between the spherical curvature and the pyramidalization angle

B $\in \star(A)$ and I the middle between A and B and, O the center of the osculating sphere of radius $R_A$.

We have $AB = a$, $AI = \frac{a}{2}$ and $AO = R_A$.

In the triangle AIO, we have the angle $\widehat{OAI} = \frac{\pi}{2} - Pyr(A)$.

By consequence, $\cos(\frac{\pi}{2} - Pyr(A)) = \frac{AI}{AO} = \frac{a}{2R_A}$,

thus, $\cos(\frac{\pi}{2} - Pyr(A)) = \sin(Pyr(A))$.

## 9-Proofs associated to the section 1.4.2: Spherical curvature in non-regular case

By definition, in the figure 1.8 the line $\mathcal{L}$ corresponds to the set of points at equal distance of B, C and D. The center of the osculating sphere must belong to $\mathcal{L}$. The center $O_z$ is then of the form $O_z = (0, 0, z)$ with $z \in \mathbb{R}$ in the reference frame $\mathcal{R}$.

We look for a point $O_z$ such that $O_zA = O_zB = O_zC = O_zD$.

As $O_zB = O_zC = O_zD$ by construction, we have only to ensure $O_zA = O_zB$.

By the Pythagorean theorem, we have $O_zB^2 = z^2 + l^2$ and,

$$ O_zA^2 = O_zO_z^2 + O_zA^2 = L^2 + (z_A - z)^2 = L^2 + z_A^2 + z^2 - 2zz_A \text{ thus,} $$

$$ O_zB^2 = O_zA^2 \text{ implies } 2zz_A = L^2 + z_A^2 - l^2. $$

As $A \notin \mathcal{P}$ then $z_A \neq 0$ and $z = \frac{L^2+z_A^2-l^2}{2z_A}$.

This concludes the proof.

## 10-Proofs associated to the section 1.5: Relationship angular defect and pyramidalization angle

By definition, we have:

$$ \theta_1(A) + \theta_2(A) + \theta_3(A) = 2\pi \qquad (a2.19) $$
By construction, we have $OB_1 = OB_2 = OB_3 = R$, with R the radius of the **circumcircle** for the triangle $B_1B_2B_3$.

The triangle $AOB_1$ is rectangle in O by construction. By definition of the *Pyr*, we have:

$$OB_1 = l \cos(Pyr(A)) \qquad (a2.20)$$

Then,

$$R = l \cos(Pyr(A)) \qquad (a2.21)$$

Using this quantity, one can compute the quantity $B_1B_2 + B_2B_3 + B_3B_1$ which is the perimeter of the triangle $B_1B_2B_3$ in $P(A)$. We have two ways to compute this quantity:

- First, using the fact that in the triangle $B_1OB_2$, the length $B_1B_2$ is given by $2R \sin(\frac{\theta_1(A)}{2})$ and similar expressions for $B_2B_3$, $B_3B_1$. We then obtain:

$$B_1B_2 + B_2B_3 + B_3B_1 =$$

$$2R \left[ \sin \left( \frac{\theta_1(A)}{2} \right) + \sin \left( \frac{\theta_2(A)}{2} \right) + \sin \left( \frac{\theta_3(A)}{2} \right) \right] \qquad (a2.22)$$

- Second, using the fact that in the triangle $B_1AB_2$, the length $B_1B_2$ is given by $2l \sin(\frac{\alpha_1(A)}{2})$ and similar expressions for $B_2B_3$, $B_3B_1$. We then obtain:

$$B_1B_2 + B_2B_3 + B_3B_1 =$$

$$2l \left[ \sin \left( \frac{\alpha_1(A)}{2} \right) + \sin \left( \frac{\alpha_2(A)}{2} \right) + \sin \left( \frac{\alpha_3(A)}{2} \right) \right] \qquad (a2.23)$$

Replacing R by its expression, and writing the equality of these two expressions, we deduce:

$$\cos(Pyr(A)) \left[ \sin \left( \frac{\theta_1(A)}{2} \right) + \sin \left( \frac{\theta_2(A)}{2} \right) + \sin \left( \frac{\theta_3(A)}{2} \right) \right] = \\ \left[ \sin \left( \frac{\alpha_1(A)}{2} \right) + \sin \left( \frac{\alpha_2(A)}{2} \right) + \sin \left( \frac{\alpha_3(A)}{2} \right) \right] \qquad (a2.24)$$

The *Pyr* is then understood as a measure of the difference between the angles $\theta_i(A)$ and $\alpha_i(A)$, $i = 1, 2, 3$.

Denoting by $I_1$ the middle of the segment $B_1B_2$ and by using the triangle $AOI_1$ which is rectangle in O, we obtain using the Pythagorean theorem:

$$\cos \alpha_F(A) = \cos^2 Pyr(A) \cos \theta_F(A) + \sin^2 Pyr(A) \qquad (a2.25)$$

This concludes the proofs.
## 11-Proofs associated to the section 3.1.2: Conditions of orthogonality

We have by definition: s orbitals and the set {$p_x, p_y, p_z$} of orbitals as: $u_i = u_{i,2}p_x + u_{i,3}p_y + u_{i,4}p_z$ and $u_\pi = u_{\pi,2}p_x + u_{\pi,3}p_y + u_{\pi,4}p_z$ where $u_{i,k}$ and $u_{\pi,k}$, $k = 1, 2, 3$ are scalars.

We identify the orbital $u_i$ with the vector ($u_{i,2}, u_{i,3}, u_{i,4}$) of $\mathbb{R}^3$ as well for $u_\pi$.

By definition, we have $\langle s, s \rangle = 1$, $\langle p_x, p_x \rangle = 1$, $\langle p_y, p_y \rangle = 1$, $\langle p_z, p_z \rangle = 1$ and $\langle p_x, p_y \rangle = \langle p_x, p_z \rangle = \langle p_y, p_z \rangle = 0$.

We then deduce that: $\langle u_i, u_i \rangle = 1$ and $\langle u_\pi, u_\pi \rangle = 1$ as the vectors $u_i$ and $u_\pi$ are unitary.

Moreover, we have by definition: the angle $\theta_{i,j}$ as $\langle u_i, u_j \rangle = \cos(\theta_{i,j})$. In the same way, we obtain $\langle u_i, u_\pi \rangle = \cos(\theta_{i,\pi})$.

As a consequence, we obtain:

$$ \begin{aligned} \langle h_i, h_j \rangle &= c_i c_j \langle s, s \rangle + \lambda_i \lambda_j \langle u_i, u_j \rangle \\ &= c_i c_j + \lambda_i \lambda_j \cos(\theta_{i,j}) \end{aligned} \quad (a2.26) $$

As $\langle h_i, h_j \rangle = 0$ by orthogonality conditions, we obtain the first set of conditions ($H_{i,j}$).

The set of conditions ($H_{i,\pi}$) follows the same lines.

## 12-Proofs associated to the section 3.1.2: Relation between the angles $\theta_{i,j}$

This is a simple computation, we have orthogonality between $h_\pi$ and the $h_i$:

$$ \lambda_\pi = - \frac{c_i c_\pi}{\lambda_i \cos(\theta_{i,\pi})}, $$

thus,

$$ \frac{c_i}{\lambda_i \cos(\theta_{i,\pi})} = \frac{c_j}{\lambda_j \cos(\theta_{j,\pi})} $$

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

We deduce:

$$ c_j \lambda_i \cos \theta_{i,\pi} = c_i \lambda_j \cos \theta_{j,\pi} \quad (a2.27) $$

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

Moreover, the orthogonality condition between the $h_i$ orbitals leads to:

$$ c_i = - \frac{\lambda_i \lambda_k \cos(\theta_{i,k})}{c_k} \quad \text{for all } i \neq k \in \{1, 2, 3\}. \quad (a2.28) $$

Replacing $c_i$ and $c_j$ in eq. a2.27 by these expressions for $k \neq i$ or $j$, we obtain:

$$ \frac{\lambda_j \lambda_i \lambda_k}{c_k} \cos(\theta_{j,k}) \cos(\theta_{i,\pi}) = \frac{\lambda_i \lambda_j \lambda_k}{c_k} \cos(\theta_{i,k}) \cos(\theta_{j,\pi}) \quad (a2.29) $$

for $i, j \in \{1, 2, 3\}$,

which leads to:

$$ \cos(\theta_{j,k}) \cos \theta_{i,\pi} = \cos(\theta_{i,k}) \cos(\theta_{j,\pi}) \quad (a2.30) $$

This concludes the proof.
## 13-Proofs associated to the section 3.1.2: $C_{3v}$ symmetry

If the angles between the vector $u_\pi$ and the bonds are equal, then we obtain directly using eq. 3.5 that $\theta_{i,k} = \theta_{j,k}$ and the molecule possesses a $C_{3v}$ symmetry.

If the molecule possesses a $C_{3v}$ symmetry, we have $\theta_{i,k} = \theta_{j,k}$ and as a consequence using relation 3.5, $\theta_{i,\pi} = \theta_{j,\pi}$ for all i, j.
This concludes the proof.

## 14-Proofs associated to the section 3.1.2: Components of the $u_\pi$ vector solutions of linear system

The proof is a simple computation.

We have $u_i \cdot u_\pi = x_i x_\pi + y_i y_\pi + z_i z_\pi = \cos \theta_{i,\pi}$, (with $i=1, 2, 3$) by definition of the angles $\theta_{i,\pi}$ and the fact that the $u_i$ and $u_\pi$ are unitary.

Multiplying the previous equation for $i$ by $\cos \theta_{j,k}$ and subtracting for $j$ multiplied by $\cos \theta_{i,k}$, we obtain the relation:

$$ (x_i \cos \theta_{j,k} - x_j \cos \theta_{i,k})x_\pi + (y_i \cos \theta_{j,k} - y_j \cos \theta_{i,k})y_\pi + (z_i \cos \theta_{j,k} - z_j \cos \theta_{i,k})z_\pi \quad (a2.31) \\ = \cos(\theta_{j,k}) \cos(\theta_{i,\pi}) - \cos(\theta_{j,\pi}) \cos(\theta_{i,k}) $$

which is equal to zero by eq. 3.5.

## 15-Proofs associated to the section 3.1.2: Linear system $M \cdot u_\pi = 0$

The determinant of $M$ is zero. Indeed, if we denote by $M = (m_{i,j})_{i,j \in \{1,2,3\}}$, we observe that $m_{2,i} - m_{1,i} = m_{3,i}$ for $i=1, 2, 3$. As a consequence, the matrix $M$ has a kernel of dimension 1 or 2.

A condition for the kernel to be of dimension 2 is that the rank of the matrix $M$ is 1. This can be done if all the 2 by 2 minor of the matrix have a determinant equal to zero. A direct computation of these determinants lead to three conditions corresponding to the determinant of three minors.

Precisely, we have:

$$ \begin{vmatrix} x_3 \cos \theta_{1,2} - x_2 \cos \theta_{3,1} & y_3 \cos \theta_{1,2} - y_2 \cos \theta_{3,1} \\ x_1 \cos \theta_{2,3} - x_2 \cos \theta_{3,1} & y_1 \cos \theta_{2,3} - y_2 \cos \theta_{3,1} \end{vmatrix} $$

$$ = \cos \theta_{1,2} \cos \theta_{2,3} (x_3 y_1 - y_3 x_1) + \cos \theta_{1,2} \cos \theta_{3,1} (y_3 x_2 - y_2 x_3) + \cos \theta_{2,3} \cos \theta_{3,1} (x_1 y_2 - y_1 x_2) \quad (a2.32) $$

In the same way, we obtain:

$$ \cos \theta_{1,2} \cos \theta_{2,3} (y_3 z_1 - z_3 y_1) + \cos \theta_{1,2} \cos \theta_{3,1} (z_3 y_2 - z_2 y_3) + \cos \theta_{2,3} \cos \theta_{3,1} (y_1 z_2 - z_1 y_2) \quad (a2.33) $$

and

$$ \cos \theta_{1,2} \cos \theta_{2,3} (x_3 z_1 - z_3 x_1) + \cos \theta_{1,2} \cos \theta_{3,1} (z_3 x_2 - z_2 x_3) + \cos \theta_{2,3} \cos \theta_{3,1} (x_1 z_2 - z_1 x_2) \quad (a2.34) $$

As $u_1 \wedge u_3 = (y_1 z_3 - y_3 z_1, x_3 z_1 - x_1 z_3, x_1 y_3 - x_3 y_1)$ we can resume the previous equality by the following vector relation:

$$ \cos \theta_{1,2} \cos \theta_{2,3} u_1 \wedge u_3 + \cos \theta_{1,2} \cos \theta_{3,1} u_2 \wedge u_3 + \cos \theta_{2,3} \cos \theta_{3,1} u_1 \wedge u_2 = 0 \quad (a2.35) $$
This relation can be rewritten as:

$$u_1 \wedge (\cos \theta_{1,2} \cos \theta_{2,3} u_3 + \cos \theta_{2,3} \cos \theta_{3,1} u_2) = - \cos \theta_{1,2} \cos \theta_{3,1} u_2 \wedge u_3. \quad (a2.36)$$

The vector $u_2 \wedge u_3$ is orthogonal to the plane $P_{2,3}$ generated by $u_2$ and $u_3$.

As $w = \cos \theta_{1,2} \cos \theta_{2,3} u_3 + \cos \theta_{2,3} \cos \theta_{3,1} u_2$ belongs to $P_{2,3}$, the vector $u_1 \wedge w$ is perpendicular to the plane $P_{2,3}$ if and only if $u_1$ belongs to this plane, meaning that $u_1, u_2$ and $u_3$ belongs to $P_{2,3}$ and the molecule is planar.

## 16-Proofs associated to the section 3.1.3: Hybridization numbers $n_1, n_2, n_3$

We give the proof only for $n_1$, the other ones following the same strategy.

We remind that due to eq. 3.3 and 3.4, we have the following system of relations:

$$ \begin{aligned} c_1 c_2 + \lambda_1 \lambda_2 \cos \theta_{1,2} &= 0 \\ c_1 c_3 + \lambda_1 \lambda_3 \cos \theta_{1,3} &= 0 \\ c_2 c_3 + \lambda_2 \lambda_3 \cos \theta_{2,3} &= 0 \end{aligned} \quad (a2.37) $$

The first equation gives:

$$ \frac{\lambda_1}{c_1} = - \frac{c_2}{\lambda_2} \frac{1}{\cos \theta_{1,2}} \quad (a2.38) $$

Using the third equation, we obtain the relation:

$$ \frac{c_2}{\lambda_2} = - \frac{\lambda_3}{c_3} \cos \theta_{2,3} \quad (a2.39) $$

it can be used to replace $\frac{c_2}{\lambda_2}$ in eq. a2.38, then we have:

$$ \frac{\lambda_1}{c_1} = \frac{\lambda_3 \cos \theta_{2,3}}{c_3 \cos \theta_{1,2}} \quad (a2.40) $$

Using the second equation, we explicit $\frac{\lambda_3}{c_3}$ as:

$$ \frac{c_3}{\lambda_3} = - \frac{\lambda_1}{c_1} \cos \theta_{1,3} \quad (a2.41) $$

Replacing in eq. a2.40, we finally obtain:

$$ \frac{\lambda_1}{c_1} = - \frac{c_1}{\lambda_1} \frac{\cos \theta_{2,3}}{\cos \theta_{1,2} \cos \theta_{1,3}} \quad (a2.42) $$

which leads to:

$$ \frac{\lambda_1^2}{c_1^2} = - \frac{\cos \theta_{2,3}}{\cos \theta_{1,2} \cos \theta_{1,3}} \quad (a2.43) $$

This concludes the proof.

## 17-Proofs associated to the section 3.1.3: sp³ normalization

The sp³ normalization condition (eq. 3.10) can be rewritten as:

$$ \sum_{i=1}^{3} \frac{1}{1+n_i} + \frac{m}{m+1} = 1 \quad (a2.44) $$

By factorizing $c_i^2$ in the third first terms and $\lambda_\pi^2$ in the last one.
We then obtain:

$$ \frac{m}{m+1} = 1 - W_{\sigma} \qquad (a2.45) $$

using the definition of $W_{\sigma}$, which leads to:

$$ m = \frac{1 - W_{\sigma}}{W_{\sigma}} \qquad (a2.46) $$

## 18-Proofs associated to the section 4.3.5: Study of the degeneracies of the energy levels in the even case $\pi(S_N^1) \equiv 0$ [mod.4] for the Möbius case

For proof, the solutions correspond to i and j such as:

$$ \cos \left( (2k + 1)\pi \frac{i}{N} \right) = \cos \left( (2k + 1)\pi \frac{j}{N} \right) $$

This leads to,

$$ (2i + 1)\frac{\pi}{N} = -(2j + 1)\pi\frac{j}{N} + 2l\pi \qquad (a2.47) $$

Thus,

$$ 2i + 1 = -2j - 2 + lN $$

As $i$ and $j \in 0, ..., N-1$, there is $l = 1$.

We then, have:

$$ j = \frac{N}{2} - 1 - i \qquad (a2.48) $$

if $N$ is an even number.

## 19-Proofs associated to the section 4.3.5: Study of the degeneracies of the energy levels in the even case $\pi(S_N^1) \equiv 0$ [mod.4] for the ribbon case

The solutions of $\lambda_i = \lambda_j$ correspond to i and j as:

$$ \cos \left( \frac{2\pi i}{N} \right) = \cos \left( \frac{2\pi j}{N} \right) $$

This last equation gives:

$$ \frac{2\pi i}{N} = -\frac{2\pi j}{N} + 2l\pi $$

Thus, $i = -j + lN$.

As, $i, j \in 0, ..., N-1, l = 1$, therefore, $j = N - i$ and $1 \le i \le N - 1$.

## 20-Proofs associated to the section 4.3.5: Structure of molecules $S_N^1 C_2$ invariant, property of torsion distribution

$$ \phi_{N-i+1} = \phi_i \qquad (a2.49) $$

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

To prove this, it is a simple computation. Each vector $u_i$ (with $i = 1, ..., N$), is obtained in a reference frame centered in the atom $C_i$ by giving the tangent vector $n_i$ in the direction of the curve and the normal plane in which we note by convention the coordinates by $(x, y)$, and the coordinate $x$ being normal to the plane of the molecule. Each vector $u_i$ is then, in its adapted reference frame of the form
$(a_i, b_i, 0)$ which is identified with a vector of $\mathbb{R}^2$ noted $U_i = (a_i, b_i)$. By convention, we have $U_1 = (1, 0)$.

By definition of the $\phi$ distribution, the angle $\phi_i$ is such as $U_{i+1} = e^{-j\phi_i}U_i$. The $C_2$ symmetry of axis $z$ refers the atom $C_i$ to the atom $C_{N-i+1}$. By invariance we also have $U_i = U_{N-i+1}$. Moreover, the $C_2$ symmetry being an isometry, it maintains the angles and we have:

$$ \phi_i = \widehat{U_i, U_{i+1}} = \widehat{U_{N-i+1}, \widehat{U_{N-i}}} = \phi_{N-i+1} \quad (a2.50) $$

with $i = 1, \dots, N-1$.

In other words, we obtain the relation:

$$ \phi_i = \phi_{N-i+1} \quad (a2.51) $$

with $i = 1, \dots, N-1$.

## 21-Proofs associated to the section 4.3.9: Torsion angle and properties - $\phi_{1,n}$ relation

This angle is computed in the plane $(x, y)$ by the formula:

$$ \cos(\phi_{1,n}) = \frac{u_1 \cdot u_n}{\| u_1 \| \| u_n \|} = \frac{a_1 a_n}{| a_1 | \sqrt{b_n^2 + a_n^2}} \quad (a2.52) $$

then,

$$ \phi_{1,n} = \arccos \left( \frac{a_1 a_n}{| a_1 | \sqrt{b_n^2 + a_n^2}} \right) \quad (a2.53) $$

An other way to write this formula is:

$$ \phi_{1,n} = \arccos \left( \epsilon(a_1) \epsilon(a_n) \frac{|\alpha_n|}{\sqrt{1 + \alpha_n^2}} \right) \quad (a2.54) $$

where $\epsilon(a_1)$ et $\epsilon(a_n)$ give the sign of $a_1$ et $a_n$ respectively and $\alpha_n$ is given by:

$$ \alpha_n = \frac{a_n}{b_n} \quad (a2.55) $$

This concludes the proof.

## 22-Proofs associated to the section 4.3.9: Torsion angle and properties - $\phi_{l,l+1}$

Since the norms of the vectors involved in the calculation of the angle are preserved, we just have to check if the scalar product is the same.

We have:

$$ \begin{align} & u_{n+1-l} \cdot u_{n+1-(l-1)} \notag \\ &= b_{n+1-l}b_{n+1-(l-1)} + a_{n+1-l}a_{n+1-(l-1)} \tag{a2.56} \\ &= (a_{l+1}a_l + b_{l+1}b_l)(-1)^{2k+2} \notag \\ &= u_l \cdot u_{l+1} \notag \end{align} $$

This gives the result.
## 23-Proofs associated to the section 4.3.9: Orbital cases of the form $(b_n, a_n): \phi_{\infty,n}$

We can study the asymptotics of this angle by investigating the evolution of $\alpha_n$.

We have:

$$ \alpha_n = \frac{a_n}{b_n} = \frac{\sin\left(\frac{nk\pi}{n+1}\right)}{\sin\left(\frac{(n-1)k\pi}{n+1}\right)} \quad (a2.57) $$

As we have:

$$ \begin{aligned} & \sin\left(\frac{nk\pi}{n+1}\right) \\ &= \sin\left(\frac{(n-1)k\pi}{n+1} + \frac{\pi k}{(n+1)}\right) \tag{a2.58} \\ &= \sin\left(\frac{(n-1)k\pi}{n+1}\right) \cos\left(\frac{\pi k}{n+1}\right) + \cos\left(\frac{(n-1)k\pi}{n+1}\right) \sin\left(\frac{\pi k}{n+1}\right) \end{aligned} $$

We get:

$$ \alpha_n = \cos\left(\frac{\pi k}{n+1}\right) + \cos\left(\frac{(n-1)k\pi}{n+1}\right) \frac{\sin\left(\frac{\pi k}{n+1}\right)}{\sin\left(\frac{(n-1)k\pi}{n+1}\right)} \quad (a2.59) $$

For all $n$ and $k$ fixed, we have $N$ going to infinity: $\frac{\pi k}{n+1}$ and $\frac{(n-1)k\pi}{n+1}$ which tend to zero.

We can then use the equivalents:

$$ \sin\left(\frac{\pi k}{n+1}\right) \equiv \frac{\pi k}{n+1}, $$

and,

$$ \sin\left(\frac{(n-1)k\pi}{n+1}\right) \equiv \frac{(n-1)\pi k}{n+1} \quad (a2.60) $$

which gives by using the fact that: $\cos(\frac{\pi k}{n+1})$ tends to 1 when $N$ tends to infinity:

$$ \lim_{N \to \infty} \alpha_n = 1 + \frac{1}{n-1} \quad (a2.61) $$

By adding back into the formula of $\phi_{1,n}$, we finally get:

$$ \lim_{N \to \infty} \phi_{1,n} = \arccos\left(\frac{n}{\sqrt{n^2 + (n-1)^2}}\right) \quad (a2.62) $$

by using the fact that the sign of $a_1$ and $a_n$ are positives for large values of $N$.

## 24-Proofs associated to the section 5.2.1: $A_{N,n,+,z}$ Angle between $\psi_{+,n}(0)$ and $\psi_{+,n}(z)$

For proof, we have:

$$ A_{N,n,+,0,z} = \cos^{-1} \left( \frac{\psi_n(0) \cdot \psi_n(z)}{\|\psi_n(0)\| \|\psi_n(z)\|} \right) \quad (a2.63) $$

$\psi_n(0)$ and $\psi_n(z)$ are two vectors such as:

$$ \psi_0(z) = \sqrt{\frac{2}{N+1}} \begin{pmatrix} 0 \\ a_n(1) \end{pmatrix} \quad (a2.64) $$

$$ \psi_n(z) = \sqrt{\frac{2}{N+1}} \begin{pmatrix} a_n(z) \\ a_n(z+1) \end{pmatrix} \quad (a2.65) $$
For more details, the scalar product and the norm of the vectors give:

$$ \psi_n(0) \cdot \psi_n(z) = \frac{2}{N+1} a_n(1) a_n(z+1) \qquad (a2.66) $$

$$ \| \psi_n(0) \| = \sqrt{\frac{2}{N+1}} | a_n(1) | \qquad (a2.67) $$

$$ \| \psi_n(z) \| = \sqrt{\frac{2}{N+1}} \sqrt{(a_n(z))^2 + (a_n(z+1))^2} \qquad (a2.68) $$

Then,

$$ \mathcal{A}_{N,n,+0,z} = \cos^{-1} \left( \frac{a_n(1)a_n(z+1)}{|a_n(1)| \sqrt{(a_n(z))^2 + (a_n(z+1))^2}} \right) \qquad (a2.69) $$

We know that $a_n(z+1) \neq 0$ and $z = 1, \dots, N-1$, then:

$$ \mathcal{A}_{N,n,+0,z} = \cos^{-1} \left( \frac{\epsilon(a_n(1))\epsilon(a_n(z+1))}{\sqrt{1 + \left(\frac{a_n(z)}{a_n(z+1)}\right)^2}} \right) \qquad (a2.70) $$

For $z = N$ we have:

$$ a_n(N+1) = \sin(n\pi) = 0 \qquad (a2.71) $$

thus,

$$ \mathcal{A}_{N,n,+0,N} = \cos^{-1}(0) = \pm \frac{\pi}{2} \qquad (a2.72) $$

For $z = N$, we get:

$$ \mathcal{A}_{N,n,+0,N} = (-1)^{n+1} \frac{\pi}{2} \qquad (a2.73) $$

The sign will depend on the symmetry (see eq. 5.51).

## 25-Proofs associated to the section 5.2.1: Hückel distribution coefficients in the case $\theta = 0$

The coefficients are detailed and depend on the form of the Hückel matrix:

$$ c_y(z) = \sqrt{\frac{2}{N+2}} a_{N+1,n}(z+1) = \sqrt{\frac{2}{N+2}} \sin \left( \frac{n\pi(1+z)}{N+2} \right) \qquad (a2.74) $$

$$ c_x(z) = \sqrt{\frac{2}{N}} a_{N-1,n-1}(z) = \sqrt{\frac{2}{N}} \sin \left( \frac{(n-1)\pi z}{N} \right) \qquad (a2.75) $$

with $c_x(0) = 0$ and $c_x(N) = 0$.

For more details, we consider the secular determinant of the Hückel matrix $S_N(\lambda)$:

$$ S_N(\lambda) = \beta^N P_{N+1}(w) P_{N-1}(w) \qquad (a2.76) $$

Note that we have for solution $2N$ non-degenerate roots. $P_{N-1}(w)$ and $P_{N+1}(w)$ are symmetric roots corresponding respectively to the systems $p_x$ and $p_y$ and $\beta$ being as we have previously seen the total overlap.
<figure><img src="image_166.png" /></figure>
# Part VI

## Abstract
English version:

# Title

Structural and Electronic Chemistry: Role of Symmetries and Curvature

# Abstract

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

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

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

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

The use of the tools resulting from this work will facilitate the simulation of molecules without size limitation and for systems whose modelisation by ab-initio or DFT calculations remains inaccessible.
We also tried to show the relation which could exist between the geometry of the systems and the orbital information. We were interested in many systems as examples most of them are carbon-based and potentially aromatic.

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

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

# Keywords

Theoretical Chemistry - Simulation - Electronic Structure - Symmetry - Curvature - Materials - Pyramidalization Angle - POAV - Helical Orbitals
# Titre

Chimie Structurale et Electronique: Rôle des Symétries et de la Courbure

# Résumé

Cette thèse est le fruit de 3 années de travail reposant sur le sujet intitulé : "Chimie Structurale et Electronique: Rôle des Symétries et de la Courbure".

L'originalité de la thématique abordée dans ce travail se trouve dans la pluridisciplinarité du sujet qui fait appel à la fois aux chimistes théoriciens et aux mathématiciens. Une importante partie des travaux se base sur la compréhension de la chimie théorique et de son utilisation par modélisation.

Un des objectifs alloué à ce travail, est la compréhension des systèmes moléculaires par des outils géométriques principalement initiés par Robert C. Haddon dès 1997. Nous avons tout d'abord proposé une clarification et une généralisation de ces outils basées sur des matériaux carbonés comme les fullerènes dont l'étude des caractéristiques topologiques est largement exploitée dans ce manuscrit.

L'ensemble des points théoriques et des outils développés (l'angle de pyramidalisation, la courbure sphérique, le défaut angulaire, et l'hybridation) ont été largement illustrés avec des cartographies par l'utilisation d'outils de programmation et de logiciels qui ont permis l'étude de la déformation, de la courbure et l'étude des propriétés chimiques et physiques des systèmes.

L'utilisation des outils issus de ce travail devrait permettre de pouvoir faciliter la modélisation de molécules sans limitation de tailles et pour des systèmes dont la modélisation par des calculs ab-initio ou DFT reste inaccessible.

Nous avons également cherché à montrer le lien pouvant exister entre la géométrie des systèmes et l'information orbitalaire. Nous nous sommes intéressés à titre d'exemples à de nombreux systèmes pour la plupart carbonés et potentiellement aromatiques.

Enfin, nous avons proposés l'étude des critères d'aromaticité suivant les théories énoncées entre autre par Hückel. Par cette étude, nous nous sommes penchés sur des systèmes linéaires moins connus i.e. les [n]-cumulènes qui présentent en particulier la curiosité de posséder des orbitales hélicoïdales. Nous avons établis les critères d'apparition de ce type d'hélices en fonction de la symétrie moléculaire.

Le but *in fine* de ce travail, était de réussir à faire le lien entre des caractéristiques topologiques et la réactivité chimique de ces molécules et matériaux.

# Mots clefs

Chimie Théorique - Modélisation - Structure Electronique - Symétrie - Courbure - Matériaux - Angle de Pyramidalisation - POAV - Orbitales Hélicoïdales
<figure><img src="image_167.png" /></figure>
List of Figures

<table>
  <tbody>
    <tr>
      <td>1.1</td>
      <td>Representation of the pyramidalization angle in a regular case</td>
      <td>27</td>
    </tr>
    <tr>
      <td>1.2</td>
      <td>Construction in a trivalent regular case [4]</td>
      <td>28</td>
    </tr>
    <tr>
      <td>1.3</td>
      <td>Angle between a bond and the POAV1(A) vector [4]</td>
      <td>28</td>
    </tr>
    <tr>
      <td>1.4</td>
      <td>Representation of the regularized ⋆(A) [4]</td>
      <td>29</td>
    </tr>
    <tr>
      <td>1.5</td>
      <td>Representation of the pyramidalization angle [4].</td>
      <td>36</td>
    </tr>
    <tr>
      <td>1.6</td>
      <td>Geometry of an admissible molecule [4].</td>
      <td>37</td>
    </tr>
    <tr>
      <td>1.7</td>
      <td>Representation of the osculating sphere and the spherical curvature [4]</td>
      <td>38</td>
    </tr>
    <tr>
      <td>1.8</td>
      <td>Spherical curvature in non-regular case [4]</td>
      <td>39</td>
    </tr>
    <tr>
      <td>2.1</td>
      <td>Representation of the number of atoms in function of the number of isomers in fullerenes [4]</td>
      <td>44</td>
    </tr>
    <tr>
      <td>2.2</td>
      <td>Boxplot representation of the pyramidalization angle (in degrees) in function of the number of atoms [4]</td>
      <td>46</td>
    </tr>
    <tr>
      <td>2.3</td>
      <td>Cartographies of the pyramidalization angle for the fullerenes C<sub>20</sub>, C<sub>60</sub>, C<sub>70</sub>, C<sub>180</sub></td>
      <td>47</td>
    </tr>
    <tr>
      <td>2.4</td>
      <td>Cartographies of the angular defect for the fullerenes C<sub>20</sub>, C<sub>60</sub>, C<sub>70</sub>, C<sub>180</sub></td>
      <td>47</td>
    </tr>
    <tr>
      <td>2.5</td>
      <td>Cartographies of the spherical curvature for the fullerenes C<sub>20</sub>, C<sub>60</sub>, C<sub>70</sub>, C<sub>180</sub></td>
      <td>48</td>
    </tr>
    <tr>
      <td>2.6</td>
      <td>Cartographies of the hybridization for the fullerenes C<sub>20</sub>, C<sub>60</sub>, C<sub>70</sub>, C<sub>180</sub></td>
      <td>48</td>
    </tr>
    <tr>
      <td>2.7</td>
      <td>Representation of the molecule 1,6-methano-[10]-annulene (C<sub>11</sub>H<sub>10</sub>)</td>
      <td>49</td>
    </tr>
    <tr>
      <td>2.8</td>
      <td>Cartographies of the parameters of 1,6-methano-[10]-annulene (C<sub>11</sub>H<sub>10</sub>)</td>
      <td>49</td>
    </tr>
    <tr>
      <td>2.9</td>
      <td>Representation of the molecule 1,5-methano-[10]-annulene (C<sub>11</sub>H<sub>10</sub>)</td>
      <td>49</td>
    </tr>
    <tr>
      <td>2.10</td>
      <td>Cartographies of the parameters of 1,5-methano-[10]-annulene (C<sub>11</sub>H<sub>10</sub>)</td>
      <td>50</td>
    </tr>
    <tr>
      <td>2.11</td>
      <td>Representation of the molecule 7bH-cyclopenta-[cd]-indene (C<sub>11</sub>H<sub>8</sub>)</td>
      <td>50</td>
    </tr>
    <tr>
      <td>2.12</td>
      <td>Cartographies of the parameters of 7bH-cyclopenta-[cd]-indene (C<sub>11</sub>H<sub>8</sub>)</td>
      <td>51</td>
    </tr>
    <tr>
      <td>2.13</td>
      <td>Representation of the molecule syn-1,6,8,13-bismethano-[14]-annulene (C<sub>16</sub>H<sub>14</sub>)</td>
      <td>51</td>
    </tr>
    <tr>
      <td>2.14</td>
      <td>Cartographies of the parameters of syn-1,6,8,13-bismethano-[14]-annulene (C<sub>16</sub>H<sub>14</sub>)</td>
      <td>52</td>
    </tr>
    <tr>
      <td>2.15</td>
      <td>Representation of the molecule trans-15,16-diethyldihydropyrene</td>
      <td>52</td>
    </tr>
    <tr>
      <td>2.16</td>
      <td>Cartographies of the parameters of trans-15,16-diethyldihydropyrene</td>
      <td>52</td>
    </tr>
    <tr>
      <td>2.17</td>
      <td>Representation of the molecule bicyclo-[2.2.1]-hept-2-ene</td>
      <td>53</td>
    </tr>
    <tr>
      <td>2.18</td>
      <td>Cartographies of the parameters of bicyclo-[2.2.1]-hept-2-ene</td>
      <td>53</td>
    </tr>
    <tr>
      <td>2.19</td>
      <td>Representation of the molecule C<sub>12</sub>H<sub>16</sub></td>
      <td>53</td>
    </tr>
    <tr>
      <td>2.20</td>
      <td>Cartographies of the parameters of Tricyclo[4.2.2.2<sup>2.5</sup>]dodecane</td>
      <td>54</td>
    </tr>
    <tr>
      <td>2.21</td>
      <td>Representation of the molecule C<sub>28</sub>H<sub>16</sub></td>
      <td>54</td>
    </tr>
    <tr>
      <td>2.22</td>
      <td>Cartographies of the parameters of 9,9',10,10'-tetrahydrodianthracene</td>
      <td>55</td>
    </tr>
    <tr>
      <td>2.23</td>
      <td>Representation of the molecule sesquinorbornatriene (bridged down)</td>
      <td>55</td>
    </tr>
    <tr>
      <td>2.24</td>
      <td>Cartographies of the parameters of sesquinorbornatriene (bridged down)</td>
      <td>55</td>
    </tr>
    <tr>
      <td>2.25</td>
      <td>Representation of the molecule sesquinorbornatriene (bridged to the top)</td>
      <td>56</td>
    </tr>
    <tr>
      <td>2.26</td>
      <td>Cartographies of the parameters of sesquinorbornatriene (bridged to the top)</td>
      <td>56</td>
    </tr>
    <tr>
      <td>2.27</td>
      <td>Representation of the molecule Trans-cyclooctene</td>
      <td>56</td>
    </tr>
    <tr>
      <td>2.28</td>
      <td>Cartographies of the parameters of C<sub>8</sub>H<sub>14</sub>: Trans-cyclooctene</td>
      <td>57</td>
    </tr>
    <tr>
      <td>2.29</td>
      <td>Representation of the molecule C<sub>24</sub>H<sub>24</sub></td>
      <td>57</td>
    </tr>
    <tr>
      <td>2.30</td>
      <td>Cartographies of the parameters of C<sub>24</sub>H<sub>24</sub></td>
      <td>58</td>
    </tr>
  </tbody>
</table>