Abad Galán

L ; Riobé

F ; Placide

V ; Guy

L ; Zinna

Di Bari

L ; Bensalah

Context and objectives

The team of Theoretical Inorganic Chemistry (CTI) at the Institute of Chemical Sciences in Rennes (ISCR) is a research group gathering 14 researchers of the CNRS, the University of Rennes 1 and the Engineering School of Chemistry of Rennes. 1 The research group makes use of computational chemistry tools to unravel structural and physical properties of molecules and solid-state systems. Its research topics are more specifically dealing with the rationalization using quantum chemical tools of chemical structures and physical properties of: i) complex architectures (i.e., large clusters, glasses) (axis 1), ii) solid-state materials for energy (axis 2), and iii) multifunctional molecular systems for opto-electronics or spintronics (axis 3).

The present project is part of the research axis 3 and aims at contributing to the development of technologies in molecular electronics, opto-electronics or spintronics using a computational approach. The main interest of these devices is to provide high efficiency at the nanometric scale (with a small number of atoms and low energy use) that is fully in line with nowadays environmental challenges. Molecular materials have demonstrated their potential applications in a wide group of sectors like data storage, information technologies, medicine, food, sensing, energy (solar energy harvesting, lighting…). The recent achievements of the CTI group performed in collaboration with experimentalists of the ISCR and with several foreign research teams clearly demonstrate that quantum chemical studies provide a clear added value to the understanding of physico-chemical properties in such systems and strongly help in the design of functional inorganic and organometallic molecular compounds for molecular (opto)electronics and spintronics. 2,3,4,5,6,7 The CTI group aims at contributing to these research efforts using computational chemistry both as a comprehensive tool to understand physical properties, and as a molecular screening tool to access specific functionalities in the aim of conceiving efficient molecular-based devices. The thesis research project presented in this manuscript is devoted to two applications that take advantage of transition metal-containing molecules: low-cost lighting and (thermal) energy harvesting at the nanoscale (more specifically thermoelectricity at the nanoscale).

For the first objective, our effort has been driven toward copper containing molecules which are appealing systems because of the availability, low-toxicity and low cost of copper. The general objective of this research is to provide an alternative to heavy metal and lanthanide systems which are the most commonly developed molecular systems for phosphorescent light emitting diodes (PhOLED) or lighting applications which are usually more toxic, rare and therefore relatively expansive. The use of heavy metal and lanthanide containing compounds is rationalized by their capacity to facilitate phosphorescence or late fluorescence and thus to lead to emission properties usually much more interesting than those of most of the pure organic materials. The group of Lescop at the ISCR has recently shown that, by a careful choice of its ligand sphere and an innovative synthesis procedure, highly interesting luminescent coppercontaining compounds can be obtained. The computational study which is presented in this manuscript consists in the rationalization of the optical properties of different copper (I) complexes systems by employing a carefully selected theoretical approach.

A similar research approach will be applied to the in-silico design of molecular systems for nano-thermoelectric applications with the aim of harvesting and converting wasted energy (heat) in nano-devices into electricity (and to prevent over-heating damages as well). This project is developed together with the laboratory of "Chemistry of Novel Materials" at the University of Mons in Belgium for the computational aspects and different experimental partners:

• the research team of Rigaut at the ISCR, specialist in the synthesis of coordination systems containing transition metals or rare earth atoms mainly for molecular design for targeted applications and switchable molecules.

• the research team of Scheer of the Department of Physics of the University of Konstanz, specialist in mesoscopic physics with an accent on the field of nanoelectronics in which charge transport phenomena are explored, notably with the physical characterization of single molecule systems using the Mechanically Controlled Break Junctions (MCBJ) technique.

• the research team of Vuillaume at the Institute for Electronics, Microelectronics and Nanotechnology, specialist in the design and characterization of self-assembled monolayers and molecular-based nano-electronic devices using different scanning probe microscopy techniques (e.g., CP-AFM).

Based on our objectives, the manuscript is split in two parts as follows:

Part I: Computational study of multi-nuclear copper (I) luminescent complexes

The first chapter aims at developing the different terms and basic concepts that define the luminescence of molecules, notably the thermally activated delayed fluorescence (TADF) mechanism and the interest for molecules containing transition metals in that domain, more particularly copper-containing systems. This description is followed by a presentation of the coordination driven supramolecular chemistry developed by Lescop and coworkers to synthetize innovative multi-nuclear copper (I) complexes presenting interesting luminescent properties for some of them which are studied in the next chapters. To date, Density Functional Theory (DFT) is the only first-principles computational tool that is able to treat molecules containing several hundreds of atoms; consequently, this technique has been employed in these studies. The choice of the computational procedure, particularly the functional and the atomic basis set, is explained on the basis of literature. The second chapter is dedicated to the study of two di-nuclear complexes, [Cu2(X)3] 2+ with X corresponding to bridging diphosphine ligands.

The relatively small size of these systems was interesting to evaluate the precision of the computational methodology when increasing the atomic basis set and when adding the treatment of relativistic effects in comparison to the experimental data. The third chapter aims at rationalizing the photo-physical properties of two much larger copper (I) complexes, a hexanuclear and an octa-nuclear complex using the same computational methodology as in the previous chapter.

Part II: Study of metal-containing molecular junctions

The second part of the manuscript aims at developing a computational protocol to evaluate the thermoelectric properties of coordination compounds in molecular junction configurations and ultimately at performing a molecular design to enhance these properties. The thermoelectric ability of a system describes its potential to convert a thermal gradient into an electrical current (Seebeck effect), or reversely to cool down a device by applying an electrical potential (Peltier effect). The first chapter reports an introduction to the field of molecular electronics and the main experimental equipment used to characterize molecular junctions: Scanning Tunneling Microscope (STM), CP-AFM and MCBJ. This section is followed by a basic development of the theoretical background of molecular junctions for electronic transport as well as of the computational tools and details necessary to compute and rationalize the properties of the molecular systems studied in the next chapters. This chapter is concluded with the introduction of theoretical concepts of thermoelectric transport in molecular junctions. The second chapter introduces elements and parameters to allow for the comparison between experimental data obtained from current-voltage curves (I-V curves) of molecular junction using different models (Single Level Model, Transition Voltage Spectroscopy) and additional measurements (Ultraviolet Photoelectron Spectroscopy). An overview of both theoretical and experimental values of the thermal phonon conductance ph in a molecular junction involved in the thermoelectric effects is also given. Finally, the calculation of electronic transmissions is achieved to study the influence of the electrode geometries. The main aim of the third chapter is to computationally evaluate the thermoelectric ability of organometallic systems inserted in a junction and to determine the factors which could enhance it. This domain of research is an emerging field in the group of CTI that is triggered by the early results obtained by the consortium about a molecular junction showing photo-switchable electronic transport. 8 The chapter four gathers experimental data obtained by our collaborators on organometallic junctions in different configurations and experimental conditions. They are analyzed by applying the different models described in chapter two. A computational analysis of the conducting properties of the corresponding model junctions and a comparison with the experimental data are reported. This analysis reveals the need of faster computational tools able to better take into consideration the diversity of geometries revealed by the dispersion observed in the experimental measurements. The increase in the speed of the calculations would also be beneficial for molecular design purpose. The chapter five also gathers experimental data obtained by the group of Scheer on organic systems, namely the gold / oligophenyl-diethynyl / gold junctions which anchoring groups are molecules alkynyl units (Au-C anchoring) -OPAn (n = 2, 3, 4), where n is the number of phenyl units. A computational approached based on DFT/NEGF and DFTB was performed to understand and rationalize the electrical properties of such systems and analyze the potential of direct bonding between carbon atoms and electrodes.

Finally, as mentioned in the previous chapters, we also reach the conclusion that there is a need to address a large number of contact geometries in order to better compare the computational results to the experimental measurements which show usually important dispersion. The cost of the DFT/NEGF method is for the moment prohibitive for such studies. This issue is addressed preliminarily in the last chapter by using a tight-binding Density Functional Theory (DFTB) method which has been used successfully in organic systems and is evaluated there in the domain of organometallics and coordination chemistry.

I -Introduction

The relationship between the black-body radiation, fluorescent lamps, light emitting diodes (LEDs) and more and more applications (sensors, biomedical imaging...) is that all take advantage of the luminescent properties of materials. The research of new luminescent constituents has been growing exponentially since the early 60s to reach more than 23 000 articles containing the words "luminescent materials" in the article in 2020 (Figure 1). This growing interest is related to the global warming issues that drive innovation toward sustainable energy in which luminescent materials are particularly attractive. Different means of excitation are generating luminescence of systems of large diversity (bulk, molecules, atoms), allowing for a wide range of applications (Table 1). Here, we will focus on lighting applications for which copper (I) complexes are particularly attractive as discussed below. Our interest will be essentially focused on the photoluminescence properties of those systems in the context of a close collaboration with the experimental team of Lescop, INSA, Rennes.

II -Concepts of luminescence II.1 -General notions of luminescence

The luminescence is a general term used to describe an electromagnetic radiationphoton emission from a material which is not induced by a thermal process. The phenomenon is triggered by the de-excitation of a system from an excited state to a more stable state. 2 The word "luminescence" is attributed to the physicist Wiedemann in 1888. It is derived from the Latin word "lumen" which means "the light". 3 As said previously, our research will be focused on photoluminescence, which is the emission of light induced by photo-excitation. Indeed, upon light absorption (energy of the photons hν; h = Planck constant; ν = frequency), a molecule can undergo a change in its electronic configuration leading to the promotion of an electron to a higher-lying unoccupied electronic level. The energy gap between these two states matches the photon energy. From this excited state, several de-excitation processes exist to return to the ground state, such as typically internal conversion (IC) (loss of energy by phonon emission/vibration), or photon emission i.e. luminescent de-excitation. 2 Photoluminescence is usually divided in two categories: fluorescence and phosphorescence.

The differentiation is based on the nature of the electronic properties of the excited state from which the emission occurs. 4 In the case of a system whose ground state is a closed-shell singlet state, the emission which arises from a singlet excited state is called fluorescence. In certain cases, a non-radiative intersystem crossing (ISC) between two different spin states, typically between a singlet excited state (Sn) and a triplet excited state (Tn), can be non-negligible as shown in Figure 2. For numerous organic molecules in which the spin-orbit coupling (SOC) energies are negligible, ISC does not occur. The selection rules 1,5 which state that a change of

spin multiplicity is prohibited during optical transitions are indeed based on this hypothesis.

The SOC corresponds to relativistic interactions that are usually found in non-planar aromatic molecules and/or heavy atom-containing systems. A second consequence of the SOC is that the lifetime of a triplet state is longer than that of a singlet state. In that context, nuclear relaxation can occur, leading to a modification of the excited state geometry. Spontaneous emission of photons from the triplet excited state to the local ground state (constant geometry) eventually occurs after a certain lifetime determined by SOC. This corresponds to the second type of luminescent process which is called phosphorescence.

Experimentally, considering a continuous excitation, a main observable difference between fluorescence and phosphorescence is the characteristic time scale for the light emission to decay. It should be noted that for some metal-containing complexes, the time decay does not provide a clear distinction between fluorescence and phosphorescence. The various electronic processes induced by the absorption of light by a luminescent material (also called luminophore or luminochrome) can be, in a first approximation, represented by a Jablonski-type diagram (Figure 2). The fundamental singlet state, S 0 , the first excited states S 1 and T 1 , and higher excited states S A and T B are represented. Each of these states includes several vibrational energy sublevels. At room temperature (RT) the thermal energy (k B T ≈ 25 meV) is in most of the cases not sufficient to populate the first excited states. The absorption of light can promote the system from the ground state to an excited state S A if the incoming photon energy ℎ𝜈 matches the energy gap between the two states. The absorption usually takes place from the lowest vibrational sub-level of the ground state S 0 because it is the most occupied vibrational state.

After absorption, the excited molecule can:

• follow Kasha's rule 6 , and rapidly relax by IC from the S A excited state to the S 1 state before: i) emitting a photon (fluorescence); ii) non-radiative vibrational de-excitation (exothermic processes) or iii) undergoing an ISC to a triplet state if the SOC is substantial and emit a photon from the T1 state (after nuclear geometry relaxation).

• emit a photon from the S A excited state or from a lower-energy S A-x state above S 1 that is populated upon IC. This does not follow Kasha's rule and usually occurs when the S 1 state is largely separated from the rest of the excited states.

• follow an alternative route: the system can undergo ISC from the populated S A-x to a triplet T B excited state whose lifetime usually allows geometrical relaxation and/or IC.

Eventually, emission of light (phosphorescence) followed by non-radiative relaxation to the ground state geometry will follow.

Those different relaxation routes can be competing and thermal population of a state can also occur (Figure 2). Other external competing phenomena such as energy transfer between fluorophores or other molecules (fluorescence inhibitors) may also take place in the excited state. This competition between processes can reduce the number of emitted photons compared to the number of absorbed photons (quenching of the luminescence). Since the excited triplet state is stabilized by additional exchange energy compared to the excited singlet state, the energy released by phosphorescence is lower than in the fluorescence process when the emission occurs from the lowest excited state. The phosphorescence emission is in that case shifted towards longer wavelengths relative to the fluorescence emission (bathochromic shift).

Furthermore, the phosphorescent decay ranges between 10 -6 s and 10 s, while a decay of 10 -10 s to 10 -7 s is measured for the fluorescence. 22 Because of this decay time, phosphorescence has the characteristic to be eye-perceived upon extinction of the light source if the wavelength is in the visible region.

The second and third row transition metal containing systems have been intensively studied.

The heavy atom effect is indeed rendering those compounds really attractive for emission properties. 7,8 They have the advantage, compared to organic phosphors, that the efficiency of the phosphorescence is increased. The lower-lying triplet excited states of a coordination complex have in most of the cases half-life times several orders of magnitude shorter than in organic molecules due to the much higher rate of spontaneous emission. For example, the emissive triplet state of naphthalene has a half-life duration of 2 s, while that of the organometallic [Ru(bpy)3] 2+ complex is approximately 1 μs. 9 They do offer a variety of excited states in which the ligands play an important role. The electronic transitions that can occur can be classified into three major families:

• Metal-centered transitions (MC). These transitions are occurring between metalcentered levels (d orbitals). They are called d-d transitions and often result in low intensity bands because of the selection rules. They are located in the visible or near-IR (12000 -28000 cm -1 / 1.5 -3.5 eV).

• Transitions with a large charge transfer character (CT) which include:

o Ligand to metal charge transfer (LMCT) or metal to ligand charge transfer (MLCT). The corresponding absorption bands can be strong in intensity and are also localized in the UV or in the visible region of the spectra. MLCT are often observed in transition metals whose electronic configurations are d 6 , d 8 or d 10 , with metals having a large reducing character and when ligands have low energy orbitals with σ* or π*-acceptor character such as poly-pyridine ligands.

o Ligand to ligand charge transfer (LLCT) or ligand to another type of ligand charge transfer (LL'CT).

In general, the electronic transitions in a metallic complex containing conjugated or aromatic ligands are enlarged to:

• Transitions between levels mainly localized on ligands. These are the n → π* or π → π* transitions. The absorption bands corresponding to the π → π* transitions are relatively intense. They are generally located in the UV range.

Metal complexes with 4d 6 , 5d 6 or 5d 8 electronic configurations are extensively used for optoelectronic applications (with Ru, Re, Os, Ir, Pt or Au). 10 The efficiency of the singlet-triplet ISC is the key mechanism in photophysical properties. It is due to the heavy atom effects fostered by the important atomic SOC constant of those metals (between 400 to 4000 cm -1 depending on the metal ion and its electronic state). 11 The resulting triplet emissive states, most

of the time 3 MLCT states, are usually highly emissive for the same reason. It makes these systems attractive for several types of applications such as organic light-emitting diodes (OLEDs), light-emitting electrochemical cells, photovoltaic cells, chemical sensors, or bioimaging. 12,13 The disadvantage of these compounds is that they are composed of metals with low to very low abundance in the Earth's crust and are thus really expensive and generate geopolitical tensions (Table 2). For this reason, creating photoactive molecular complexes composed of abundant elements instead (like Cu, Fe or Cr) is an appealing solution. Cu(I) containing systems are for some of them MLCT emitters with an absence of d-d excitations due to the 3d 10 electronic configuration. The quite low atomic SOC (between 200 to 900 cm -1 depending on the electronic structure) 14,15 that decreases the ISC and the spontaneous emission probability has some advantages since it allows for a competition between different processes like ISC, reverse inter-system-crossing (RISC) or thermal activation thanks to the longer life lifetime of the excited states.

II.2 -Thermally Activated Delayed Fluorescence -TADF

Among all possible de-excitation schemes, a competition between several paths can occur. This is the case in thermally activated delayed fluorescence (TADF). Intense research has been conducted on TADF, both in organic and in coordination chemistry. It offers a large range of emission properties. 17 The concept of TADF is not new, as it has been reported for the first time by Perrin et al. in 1929, 18 and was studied by a few others during the 20 th century. 19 In 2012,

Adachi and coworkers 20 used the TADF mechanism to create an efficient OLED without phosphorescence signal, which generated an extensive attention in this competitive domain.

This topic is currently studied by several research groups around the world.

A simplified description of the TADF mechanism is given in Figure 3. In a TADF emitter, the 𝑆 𝐴 and 𝑇 𝐵 energy levels which are usually 𝑆 1 and 𝑇 1 are sufficiently coupled to allow an ISC between the two states. If the SOC is sufficient, the ISC time scale to populate 𝑇 𝐵 will become competitive with the reverse thermal ISC process, i.e., repopulation of 𝑆 𝐴 from 𝑇 𝐵 and eventually fluorescence. The TADF occurs when the energy separation (Δ𝐸 𝑆𝑇 ) and the associated thermal energy barrier are sufficiently small. Nevertheless, to obtain interesting TADF properties at room temperature, Δ𝐸 𝑆𝑇 should be lower than ~1000 cm -1 (125 meV). 22 This small energy gap allows for a RISC from the 𝑇 𝐴 state to the 𝑆 𝐵 state in a thermally activated process. Once in the 𝑆 𝐴 state, the de-excitation to the ground state eventually occurs via fluorescence and associated non-radiative relaxation paths. Since RISC is a slow process, the fluorescence from harvested triplet excitons occurs with a latency compared to direct fluorescence. Therefore, this fluorescence is called thermally-activated delayed fluorescence.

At room temperature, the compounds exhibiting such a property are leading to a high quantum yield of emission and emission decay time of the order of few microseconds (Figure 3). For organic systems exhibiting TADF, no direct fluorescence is generally observed. 21 Since the SOC is small in organic molecules, phosphorescence from the triplet states are also rarely observed. In metal-containing complexes, phosphorescence can also take place when the temperature is too small to induce RISC. Indeed, at low temperature, the system is trapped in the triplet excited state and undergoes phosphorescence only if the SOC is sufficient. It becomes a competing deactivation process to TADF at higher temperature. The emission lifetime is thus strongly influenced by the temperature, ranging from a few microseconds when the TADF process dominates to hundreds of microseconds up to milliseconds at low temperature. The reduction of Δ𝐸 𝑆𝑇 has a leading effect on the TADF lifetime decay. However, this is no more the case when Δ𝐸 𝑆𝑇 is lower than 25-40 meV ( 200 -300 cm -1 ); in this situation, the El Sayed's rule 22,23 shows that the singlet excited state and the triplet excited state have very similar charge transfer/local exciton character leading to the reduction of the RISC coupling. Depending on the targeted applications, the radiative decay should be fast as for OLED applications or slow as for lighting and sensing. Within this TADF framework, the lifetime variation as a function of the temperature follows the relation: 22 𝜏(𝑜𝑏𝑠) = 3 + exp (-𝛥𝐸 𝑆𝑇 𝑘 𝐵 𝑇 ) 3𝑘(𝑇 𝐴 ) + 𝑘(𝑆 𝐵 ) exp (-𝛥𝐸 𝑆𝑇 𝑘 𝐵 𝑇 )

Eq. 1

where 𝜏(𝑜𝑏𝑠) is the observed lifetime at a given temperature, 𝑘(𝑇 𝐴 ) and 𝑘(𝑆 𝐵 ) the decay rates of the excited state TA and SB, respectively. 𝑘 𝐵 , the Boltzmann constant, T the temperature of the system and ΔE 𝑆𝑇 the energy difference between the TA and SB state.

The lifetime of the SA and TB states are directly related to their decay rate by 𝑘(𝑇 𝐴 ) = 1/𝜏(𝑇 𝐴 ) and 𝑘(𝑆 𝐵 ) = 1/𝜏(𝑆 𝐵 ). TADF is observed for organic molecular systems 24 and for coordination complexes. An example of TADF-related photophysical properties of a Cu(I) complex is depicted in Figure 4. 25 Part I

III -Cu(I) complexes

Copper is a transition metal naturally present and abundant in earth's crust (see Table 2). 27 In solution, it is generally found in two oxidation degrees: +I and +II. Due to a special combination of electronic and structural characteristics (e.g. flexible geometric structure, chromophoric character, luminescence, electrochemical behavior), Cu(I) complexes have stimulated a great interest in several fields such as supramolecular chemistry, molecular machinery, catalysis, ultrafast spectroscopy, solar energy conversion, opto-electronics and luminescence. 25-a) As stated before, the optical properties of Cu(I) complexes are directly related to their d 10 electronic configuration. This configuration favors an electronic transition from the metal to ligands (MLCT), the formally populated ligand orbitals being often π anti-bonding orbitals (𝜋 * ). The Cu(I) ionic metal center is thus used to build coordination complexes. Several series of complexes are presenting small singlet-triplet energy gap (Δ𝐸 𝑆𝑇 ) often leading to TADF emission upon harvesting of the triplet excitons, which is of great interest in OLED applications. These emission properties that are function of temperature have also other applications like lighting, optical thermometry, sensing and imaging for instance.

Commonly, copper complexes adopt a (distorted-)tetrahedral geometry around the Cu(I) atom (Figure 5). Unfortunately, pseudo-tetrahedral assemblies often exhibit low luminescence due to the important non-radiative de-excitation mechanisms produced by the geometrical distortion

that can occur in the excited states. To overcome this problem, an interesting way to design highly luminescent Cu(I) complexes using a TADF mechanism is to introduce bulky chromophoric ligands around the Cu(I), thus, limiting the possibilities of geometric reorganization and protecting the metallic center from its environment. Other possible ways include the preparation of coordination complexes where the Cu(I) metal center exhibits a trigonal-planar or a linear geometry with rigid ligands to hamper the geometric distortion in the excited states. It is also possible to build complex polynuclear structures of Cu(I) where the metal centers are introduced in complex polycyclic and geometrically constrained supramolecular structures.

IV -Coordination-driven supramolecular chemistry (CDS)

Among all possible synthetic routes, supramolecular chemistry consisting in the association of precursor molecules to create new assemblies has emerged recently. It involves the creation of weak interactions like electrostatic forces, hydrogen bonds, van der Waals forces or hydrophobic interactions between molecules. These interactions play an important role in various fields. They are for example the reason of the double helix shape of DNA (deoxyribonucleic acid). These interactions were theorized by the chemist Fischer in the 19 th century. 29 The first statement of supramolecular chemistry was found in the Webster dictionary in 1903, 30 and it was then redefined in 1978 by the chemist Lehn (Figure 6). 31 Nowadays, supramolecular chemistry is widely explored in various fields like molecular machines, 32 Part I Chapter I 23 molecular sensors, 33 gas adsorption, 34 nanoreactors, 35 chemical catalysis 36 and many others. It can be seen as a cross-disciplinary science, which relies on organic chemistry, coordination chemistry, polymer chemistry, material science, and biological sciences. 37 One of the important concepts of supramolecular chemistry is the auto-assembly faculty, 39 which is the ability of individual components to establish, via intermolecular interactions the generation of functional assemblies. In that context, Cu(I) copper complexes used in supramolecular chemistry are generally performing poorly, notably because of the insufficient rigidity of the molecular clips and precursors.

Supramolecular chemistry based on coordination (coordination-driven supramolecular chemistry, CDS) is based on a similar synthetic scheme but the association of molecular elements is done in that case via coordination bonds between pre-defined building-blocks. It is the molecular recognition which is the driving force in the synthetic mechanism. 40 Yields are usually quite high for this one-pot synthesis in which the precursors have a pre-determined connectivity. In this field, the work of Stang, 41 Fujita, 42 Scheer, 43 Nitschke, 44 Lehn, 45 Severin, 46 Ward, 47 Raymond, 48 Clever, 49 and others 50 showed, in an independent manner, that a huge variety of metallic complexes and connecting ligands can be assembled to create supramolecular architectures. A number of supramolecular derivatives have been synthesized

from "molecular clips" based on metal ions such as Pt(II) 51 , Pd(II) 52 , Au(I) 53 and Ag(I) 54 with Part I Chapter I 24 rigid ligands. Interestingly, a limited number of supramolecular derivatives based on the Cu(I) ion has been developed. 55 Indeed, the Cu(I) ion is, a priori, not really suitable to be included in a supramolecular association process guided by coordination chemistry. The main reasons are that Cu(I)-ligand bonds are usually quite labile in solution and that the coordination sphere of Cu(I) shows usually a conformational flexibility that prevents directional interactions.

However, the work carried out in the group of Lescop, at INSA Rennes, has shown that it is possible to circumvent this limitation by using dimeric Cu(I) precursors based on phosphol assembler ligands in which the inter-metallic distances are relatively small. This type of preorganized and stable copper(I) precursor complexes have been successfully used to synthesize supramolecular complexes. 39-f,56 The group has enlarged the concept to polyphosphinyl compounds to conceive large assemblies by supramolecular chemistry guided by coordination.

Interestingly, contrarily to the organophosphate series, most of the phosphinyl-based supramolecular systems synthetized by Lescop and coworkers are luminescent. Among them, some are presenting thermochromic and mechanochromic luminescent properties. These interesting optical properties which are attributed to TADF in some cases are due to a combination of factors for which computational studies are highly valuable to understand the electronic features which are driving such behaviors.

V -Computational studies of copper molecules

The Density Functional Theory (DFT) and Time-Dependent DFT (TD-DFT) methods have currently become universal tools to apprehend and predict the behavior of an extensive variety of chemical, physical, and biological phenomena, including in the coordination chemistry field. 57 The DFT formalism describes the different electronic states of atoms and molecules from the three-dimensional electron density. Such a paradigm allows DFT to be several orders of magnitude faster than the wave function theory (WFT) for a reasonable accuracy level.

Indeed, the WFT methods which are providing more quantitative results are not affordable to treat large (poly-) metallic systems (geometry optimization, description of excited states, excited state relaxation…).

Nowadays, a large number of papers with DFT calculations dealing with copper systems have been devoted to the study of a wide range of properties. They offer reliable information on the geometric and electronic structures, the electronic absorption and emission spectra, and

vibrational spectra of copper coordination complexes. Several benchmark articles deal with DFT computational protocols applied to Cu(I) and Cu(II) containing systems. The density functional (DF) and basis set combination is the key factor to acquire an overall satisfactory comprehension of the properties of Cu(I) and Cu(II) centers for a reasonable computational cost. 58 The size of the studied complexes is the element which will drive the choice of the theoretical methodology.

In the paper of Sousa in reference 58, 18 Density Functionals (DFs) and 14 different basis sets were tested (selection of DFs and basis sets in Table 3) for a total of 119 combinations of functional/basis set to determine the geometric properties of a set of 50 copper complexes. 24

Cu(I) and 26 Cu(II) systems were investigated. Globally, as the complexity of the DFT method increases, the better is the description of the systems compared to experimental data. Certain DFs stand out:

• double Hybrid-Generalized Gradient Approximation (DH-GGA), notably the mPW2PLYPD density functional associated with the 6-31+G(d) basis set.

• Regarding TD-DFT calculations and optical simulations, the paper of Herres-Pawlis and coworkers, reference 57, reports a computational benchmarking for the structural and the optical properties for a bis(chelate)copper(I)guanidine-quinoline complex. They found that the exact Hartree-Fock exchange quantity used in the functionals and the nature of the basis set have a huge influence on their results. A shift of the metal-to-ligand bands up to 1.0 eV is calculated in the absorption spectrum depending on the quantity of exact exchange used. The DF/basis set combination that better fits the experimental data is the BP86/6-311G(d) and B3LYP/def2-TZVP. Moreover, their TD-DFT excitation energies diverge from the Many Body Perturbation Theory (MBPT) results by about 0.5 eV. Selected results are collected in Table 4.

VI -Methodology and software

On the basis of these computational benchmarking and tests performed on a first series of large systems, the best DF/basis set combination that could be afforded to treat the ground and excited states is PBE0/SVP. For sake of comparison, all studied copper compounds will be treated at this level of theory as a first step. The Gaussian 2016 software 62 A.03 revision was used for all geometry optimizations and the TD-DFT calculations. The Amsterdam Density Functional (ADF 2019) software from Software for Chemistry & Materials B.V. (SCM) has been used for the SOC calculations. 63 The PBE0 has been settled on the basis of the pure-GGA functional proposed by Perdew, Burke and Ernzerhof (PBE) in 1994 and adapted into a hybrid functional by Adamo (H-GGA). 64 This functional integrates 25% of Hartree-Fock exact exchange and 75% of DFT exchange. The basis set used for all systems is the Split Valence Polarization (SVP) from Ahlrichs et al. 65 Unfortunately, it was not conceivable to use a higher level of DFT, using LC-H-GGA or DH-GGA functionals as proposed in reference 58 or larger basis sets (reference 61) because of the large size of the studied copper complexes.

The protocol employed for the upcoming computational studies is the following:

• Geometry optimization in the ground state.

• Calculations of the absorption spectra (vertical TD-DFT), and vertical triplet states.

• Geometry optimization in the excited states (TD-DFT).

• Rationalization of the electronic and geometric properties.

The simulated absorption spectra were convoluted by using the GaussView software. The calculation of the molar attenuation coefficients is achieved as follows:

𝜖(𝜈 ̃) = ∑ 𝜖 𝑖 (𝜈 ̃) = ∑ (1.3063.10 8 𝑓 𝑖 𝜎 exp [-( 𝜈 ̃-𝜈 ̃𝑖 𝜎 ) 2 ] ) 𝑛 𝑖=1 𝑛 𝑖=1
Eq. 2

where i runs from the first to the n th electronic excitation. f is the dimensionless oscillator strength, σ the Full-Width at Half Maximum (FWHM), ṽi the excitation energy corresponding to the electronic excitation of interest, and εi the molar attenuation coefficient.

The evaluation of the atomic percentage contribution on each molecular orbital was performed with the method proposed by Ros and Schuit. 66 Part I

Chapter II -Study of diphosphino-Cu 2 coordination complexes

I -Introduction

As previously detailed, Cu(I)-based systems are attractive candidates to conceive cheap and efficient lighting devices. Our research is specifically devoted to the understanding of the unusual photophysical properties of Cu(I) polymetallic arrangements that are synthesized via the "one-pot" CDS protocols developed by Lescop and coworkers. For these systems, a subtle balance is found between the different processes and a computational study allows to better understand them. Our collaborators intentionally develop their research toward accessible and cheap molecular building blocks. The bidentate diphosphine bridging ligand has been one of the chosen ligands. This choice was driven because of their short-bite angle which confers a relatively short Cu-Cu distance in the complex and often promotes enhanced spin-orbit coupling. Additionally, most of those ligands like the diphenyl-diphosphino-methane (dppm) present a good compromise in rigidity and fluxionality (bridging or chelating coordination modes) that is necessary for CDS assembling. 1 The di-phosphine Cu(I) complexes have been extensively studied and a variety of coordination sphere around the metal has been identified

from tetrahedral to linear conformations with a variety of luminescent properties. 2,3,4,5,6,7,8 Interestingly, no full characterization of the dimer systems with diphosphinyl-only bridging ligands have been published to date. These systems are interesting compounds since they are the smallest systems of the target series in which several Cu-Cu dimers will be present. A combined experimental and computational study is thus of high interest for later comparisons. (Figure 1). The main experimental angles and distances obtained by single-crystal X-ray diffraction analysis are reported Table 1. The numbering of the atoms is indicated in Scheme 1. It was chosen in such way that the dppm arrangements between the phenyl groups are similar for all the considered geometries.

Compound

The Cu-Cu bond is compatible with a cuprophilic interaction, i.e., a distance shorter than about 3.00 Å. 9,10 The close Cu-Cu distance could also result from the chelating effect of the dppm bridging ligands. The sum of the angles P-Cu-P for each copper is 360° reflecting a trigonal planar arrangement. This trigonal arrangement is importantly deviating from the D3h highest symmetry with angles ranging from 107.3° to 136.7°. The excitation spectrum in Figure 2b was recorded considering an emission wavelength (em) of 478 nm. Absorption above 400 nm does not lead to emission. The lowest energy efficient excitation is situated near 340 nm at a wavelength slightly lower than the maximum absorption peak.

II.1.3 -Experimental solid-state emission spectra

The solid-state emission spectra of A TEF were also recorded by the group of Lescop at several temperatures between 80 to 300 K (ex = 375 nm). At RT, the maximum of emission is centered at 478 nm. The emission peak is shifted to lower wavelengths upon lowering of temperature.

This global hypsochromic shift of emission upon temperature increase from 489 nm at 80 K to 479 nm at 300 K is not linear. Indeed, the maximum wavelength is measured between 160-170 K with a maximum of emission at 497 nm. Considering the literature of copper (I) emissive systems, 12 this behavior corresponds to a TADF mechanism associated with phosphorescence at low-temperature and a thermal activation of the RISC that populates the lowest singlet excited state at higher temperature. The study of the emission lifetimes is necessary to support this hypothesis. to technical issues, the measurements could not be performed above 370 K and the expected plateau was not reached. A fitting using the TADF equation (Eq. 1, chapter I, section II.2) was applied and provided a EST of 0.129 eV (1040 cm -1 ) between the S1 and the T1 and respective lifetimes of 350 ns and 428 s; however, the absence of a plateau surely leads to fairly imprecise values. Computational studies will be performed to give some more credits to this hypothesis. 5). A single crystal X-ray diffraction analysis could be performed for both.

The main experimental angles and distances are reported in Table 2. (2.606 Å vs. 2.635 Å). This difference could be attributed to crystal packing for the [Cu2(dppa)3] 2+ ions or a reinforcement of the cuprophilic interactions. Steric hindrances are enforced to the phenyl groups due to hydrogen bonding interactions with the counterions and the presence of solvent. Hydrogen bonds are found for one of the three amino groups. Small modifications in the orientation of the indented phenyl groups can be detected but their extent is too small for further interpretation.

II.2.2 -Experimental solid-state absorption and excitation spectra

In order to evaluate the influence of the intermolecular interactions on the optical properties of the [Cu2(dppa)3] 2+ systems, experimental measurements were performed on two different solidstate arrangements, B BF4 and B PF6 . The absorption spectra of both systems were conducted in the solid state and are shown in Figure 6. The first low-energy intense absorption peak is located around 400 nm for both systems but a noticeable absorption is observed at lower energy. This shoulder located around 500 nm is more marked for B PF6 . The system containing the BF4 - counterion (B BF4 ) has 2 peaks at 417 nm and 317 nm; in the other structure (B PF6 ), two peaks are also present, but at 406 nm and 319 nm. It represents a difference of 80 meV (417 vs 406 nm) and 25 meV (319 vs 317 nm) depending on the counterion.

The absorption properties are slightly different depending on the counterion and solvent forming the crystals. The maximum of absorption is blue shifted compared to the parent A TEF revealing either electronic modifications due to the substitution of the CH2 of the dppm by NH and/or by geometrical modifications due to differences in the crystal packing.

II.2.3 -Experimental solid-state emission spectra

The maximum emission wavelengths of both compounds B BF4 and B PF6 change as the temperature rises. The maximum of emission decreases from 516 nm to 493 nm for B BF4 , and from 551 nm to 520 nm for B PF6 going from 80 K to 300 K. A probable explanation considering the previous results reported in the literature 12 and the experimental results obtained for the A TEF system is a combination of phosphorescence at low temperature followed by a TADF process at higher temperature. Here again, the study of the evolution of the emission lifetime is necessary to support this hypothesis. The experimental temperature dependent decay time of the emission was measured in Figure 9 from 80 K to 370 K for the two compounds B BF4 and B PF6 . Both lifetimes follow a Boltzmanntype curve which can reveal a TADF process. As previously, using Eq. 1 (chapter I, section II.2) one can extract the lifetime of the excited triplet and singlet state involved, and EST.

Again, this fit could underestimate the ΔEST and overestimate the τ(S1) value if, at high temperature, the lifetime plateau is not reached. This is the case for both system since the lifetime still decreases upon increase of the temperature up to 370 K.

III -Theoretical study

A computational study at the DFT level was performed on both isolated ions [Cu2(dppm)3] 2+ (A theo ) and [Cu2(dppa)3] 2+ (B theo ) in order to get insights on the intrinsic emissive properties of the ions and possibly deduce information on the role of the solid-state environment. Indeed, the previous section reveals that the emission properties can be altered depending on the counterion (B BF4 and B PF6 ).

III.1 -Geometry optimization

We first performed the geometry optimizations in the ground state of the A theo and B theo compounds in vacuum at 0 K using the DFT protocol detailed in the chapter I (Figure 10). The starting geometries were extracted from the crystallographic X-ray structures to keep the same arrangement between the phenyl groups for sake of comparison between systems and with experimental results (A TEF and B BF4 ). It appears interesting to evaluate the influence of the deprotonation of the amine groups of the B cation on the electronic and photophysical properties. X-Ray resolution does not provide sufficiently precise information to quantitatively calculate the strength of the hydrogen bonding that induces a weakening of the N-H bond by interaction with the fluorine atoms of the counterions. Thus, we choose to simulate the effect of the extreme case, i.e. deprotonation, on the electronic and photophysical properties. The main distances and angles are given in Chapter II 48 small differences is due to a different bite angle imposed by the dppm and dppa ligand, 63° and 64°, respectively, in a monometallic coordination. 13 No study exists on the evaluation of the angle imposed by di-phosphine ligand in bimetallic bridging configuration but we can presume similar constraints for dppa and dppm. Nevertheless, the Cu-Cu distance is 2.643 Å in B theo and 2.799 Å in A theo revealing that electronic effects are to be considered. The Cu-P mean distance is slightly shorter in B theo (2.351 Å vs. 2.360 Å), a feature also found in the X-ray experimental structures (2.294 Å vs. 2.304 Å). The Cu-P bond distance deviation between experimental and optimized structures are ranging from 1.5 % to 3.5 % which is in good agreement with the experimental structure considering that metal-P distances are always slightly underestimated in DFT (GGA and hybrid functionals). 14 The experimental Cu-Cu bond is well reproduced by DFT calculations for both dppa-containing systems (B PF6 and B BF4 ) with a difference of less than 2 %. It is less the case for the dppm system with an experimental value of 2.913 Å in A TEF and 2.799 Å in A theo (-3.9 %). In order to evaluate the impact of deprotonation of the amine groups on the electronic and optical properties, the mono and bis deprotonated systems [Cu2(dppa)2dppa-H+] 2+ (B theo/-H+ ) and [Cu2(dppa)(dppa-H+)2] 2+ (B theo/-2H+ )

were optimized. The impact on the Cu-P distances and angles is moderate. The mean Cu-P distances are only slightly shorter (-0.009 Å and -0.018 Å for dppa-H+ and dppa-2H+ respectively). The N-P distances are also affected by the deprotonation (-0.07 Å in both cases).

III.2 -Electronic structures

The electronic structures of A theo and B theo are given in Figure 11. Both systems are thermodynamically stable with a large energy gaps between the occupied and the virtual MOs (4.54 and 4.25 eV for A theo and B theo , respectively). In both cases, the LUMO electronic level is well separated from other unoccupied energy levels and shows a Cu-Cu bonding as main character. The HOMO is mainly localized on copper atoms and phosphorous atoms (∼20 % Cu, ∼35 % P), the rest being delocalized over the phenyl rings. Globally, the LUMO is mostly metalcentered but the LUMO of B theo presents a substantial participation of the amine group (23 % Cu, 18 % N, 15 % P). This can explain its stabilization compared to the LUMO of A theo . The HOMO-1 and the LUMO+1 of both systems are represented in Figure 12. The nature of the HOMO-1 is similar to the HOMO. In contrast, the LUMO+1 differs from the LUMO since it is * phenyl-centered. It is also holding true for the closeby LUMOs. The MOs energetic diagram of the mono-deprotonated system B theo/-H+ is represented in Figure 13. The deprotonation changes importantly the electronic structures. Indeed, the HOMO is significantly localized on the nitrogen non-bonding doublet of the amine group of the deprotonated dppa ligand (∼20 %) and not on the other amine groups. As for B theo , the HOMO has a substantial metallic character (∼19 %), the rest being localized on the phosphorus atoms (25 %) and the phenyl groups of the dppa. The LUMO is highly similar to that of B theo with slightly more delocalization on the phenyl rings.

The HOMO-1 and LUMO+1 are localized on the Cu2-dppa-H+ part for the former and on the two non-deprotonated dppa for the latter. The HOMO-LUMO gap is mainly decreased (3.57 for B theo/-H+ vs 4.25 eV for B theo ) because of the nitrogen lone-pair contribution in the HOMO.

These electronic structure modifications will have an impact on the electronic excitation properties.

A theo B theo

III.3 -Electronic excitation energies, simulated UV-Vis absorption spectra

Computational details

The lowest 15 excitation energies were calculated using the Time-Dependent Tamm-Dancoff Approximation DFT (TDA-DFT) 15,16 approach at the same level of theory as the geometry optimization (SVP atomic basis set, PBE0 functional), as implemented in Gaussian 16. This

level of calculation is called level Gaus1 next. The effect of the reduced SVP basis set was evaluated by expanding the basis to the 6-311G (3df, 3pd) atomic basis set as implemented in Gaussian (level Gaus2 ). Since the bimetallic systems are much smaller than the CDS polymetallic compounds of the next chapter, more sophisticated calculations were manageable. We also shed light on additional processes such as the ISC notably, and the SOC influence for some systems.

To date, SOC is not implemented for DFT in Gaussian 16; we thus used the ADF program of Amsterdam Modeling Suite. 17 In this case, the calculations were performed using the hybrid PBE0 functional with a large atomic basis set (triple- and two polarization functions (TZ2P, all electrons)) 18 at the scalar Zero Order Regular Approximation (ZORA). 19 This computational method will be called level ADF . We have tested for one system the influence of the scalar-ZORA correction by first removing it (level ADF-w/o-ZORA ). A better level of theory, effectively including SOC, was also tested in the case of B theo (level ADF-SOC ). We have also calculated the SOC between excited states by the perturbative TDA-SOPERT approach (level ADF-SOPERT ), 20 The absorption spectra can be simulated on the basis of these excitation energies. They are given in Figure 14. The lowest absorption peak observed is at 359 nm and at 387 nm for A theo and B theo and is followed by several more intense peaks. The experimental solid-state absorption spectra are given in Figure 2 and Figure 6 for A TEF , and B BF4 and B PF6 respectively.

A direct comparison of the intensities is not possible but the comparison of the energies can be performed. For A TEF , the onset of the absorption peak is in the same region as predicted by the

calculations (considering a calculation error of ~0.15 eV), while it is overestimated for B BF4

and B PF6 whose absorption peaks are roughly rising from 500 nm to lower wavelengths. This could be attributed to the difference in geometry between the optimized geometry B theo and the experimental one (for which the hydrogen atom positions are not available), as often stated in the literature. We also have simulated the absorption spectrum of the [Cu2(dppa)3] 2+ geometry extracted from the X-Ray structure of B PF6 with the hydrogen atom positions optimized (blue curve in Figure 14). The most intense absorption peak is slightly shifted to lower wavelengths and a shoulder appears without improving the comparison with experimental observations. The origin of the discrepancies is thus different.

and the participation of the lone pair of the deprotonated nitrogen atom in the HOMO. The simulated absorption spectra of B theo/-H+ and B theo/-2H+ are given in Figure 14 (green and red curves, respectively). As expected from the study of the electronic structures, the first absorption bands are found at 462 and 475 nm, respectively, followed by several absorption bands from 400 nm to the UV region. These simulations allow to speculate that the hydrogen bonds present in the crystal are leading to partial deprotonations that lower the energies of the first excited states.

The nature of the first excited states is particularly important to understand the properties of the excited states, and hence the emission in the present case. The main MO transition in all cases lead to the population of the LUMO which has mainly Cu-Cu bonding character and the depopulation of one of the highest MOs with Cu-P bonding character. The natural transition orbitals (NTOs) describing these excitations are represented for S1-vert (Figure 15 and16). As expected, they highly resemble the HOMO and LUMO. The nature of the electronic transition which describes the T1-vert excitation is similar to S1-vert as shown in Figure 15 and 16. Another way of describing the nature of the excitations is to plot the difference between the total density of the considered excited state with respect to the ground state (Figure 17 for B theo ). It allows to visualize the increase of density between the two copper atoms and the high similarities between S1-vert and T1-vert vertical excited states. The description of the emission properties necessitates to consider the relaxed excited states (see next section). A strengthening of the Cu-Cu interaction is expected in the relaxed geometries of S1-vert and T1-vert in all cases compared to the ground state. The fast-non-radiative de-excitation process which follows the excitation encompasses by IC between singlet excited states followed by an ISC to a triplet excited state. The Kasha's rule establishes that the IC processes should lead to the lowest excited state (S1), and then the ISC occurs between S1-vert and T1-vert. The ISC is favored by spin-orbit coupling. The SOC matrix elements between excited states were calculated at the level ADF-SOPERT for A theo and B theo in the S0 geometry (considering that the geometry relaxation occurs at a slower time scale) and collected in Table 8. The S1-vert to T1-vert ISC is largely disfavored due to negligible SOC between the two states. For both, S1-vert to T2-vert ISC is not possible since T2-vert is higher in energy. For A theo , this difference in energy is small (+88 cm -1 ), which is within the calculation precision at this level of theory. When increasing the level of calculation to level ADF-SO , the T2vert is even higher in energy (+1081 cm -1 ) ruling out this possibility of ISC from S1-vert to T2-vert. The most important SOC, in both compounds, is between S2-vert and T1-vert (112 and 143 cm -1 , respectively). However, their energy difference is more than 3000 cm -1 which is a really high value and one would need to know about the superposition between vibrational states to be Part I Chapter II 60 conclusive. Considering this analysis, non-radiative de-excitation processes are competing but more accurate quantum methods are necessary to draw more precise descriptions. The T 1-vert is eventually populated and geometrical relaxations will occur modifying the energy ordering between excited states. 18. The ground state S0 that is represented as background image allows to visualize the small changes that occur in the geometry. This is similar in the A theo series. This is beneficial to the RISC processes in which the difference in energy and the SOC also have to be considered. The energy diagram of the relaxed states compared with the potential energy surface (PES) of the ground state for the two series A theo and B theo are represented in Figure 19. The numbering of the relaxed excited states is given as a function of their relative energies but the methodology used to relax the geometry of the excited states cannot confirm that they are kept in the same ordering as the vertical excited states (possibility of PES exchange).

The experimental study of the luminescent properties of A TEF shows that the maximum of emission which is 489 nm at 80 K, increases to 497 nm between 160 and 170 K, and then decreases to 479 nm at 300 K (Figure 3). The evolution of the lifetime also shows a decrease followed by an increase up to 160 K and then a usual TADF-like decay (Figure 4). This behavior cannot be explained by a simple TADF process. The energies of the excited states and the emission wavelengths of A theo states are given in Figure 19 (top). The T1 relaxed state is low in energy and separated by 2073 cm -1 from the S1. The emission from the T1 state is calculated at 593 nm, really far from the experimental values. The emission wavelengths of the relaxed T2 and T3 states of 522 nm and 452 nm better agree with the experimental emission wavelength below 170 K. Moreover, T2 is rather close in energy to S1 (1121 cm -1 ) and T3 to S2 (2016 cm - 1 ). Several temperature-dependent non-radiative pathways involving both S1 and S2 can be invoked, associated with a dual combined phosphorescence. Additional experimental data with another counter-anion, study of the emission as the function of the excitation wavelength and time-resolved measurements for instance would be needed. The TADF behavior that appears to occur after 170 K should be studied differently by de-correlating the first process. Because the plateau at high temperature is not reached, a wavelength lower than 479 nm can be conjectured for the fluorescent species. Based on the computational results, the TADF process cannot involved the S1 state whose emission wavelength is 512 nm and would more likely take place from the relaxed S2 state. From the experimental results of Figure 8 (different excitation wavelengths used), the dppacontaining systems are showing an evolution of the maximum going from 516 nm to 493 nm for B BF4 , and from 551 nm to 520 nm for B PF6 upon the increase of temperature from

𝜆 = 𝐸 𝑆 1 /𝑇 1 -𝐸 𝑆 1 /𝑆 1 Eq. 2
For B theo , at 300 K, the calculated 𝜆 = 0.06 eV and ES1-T1 = 0.24 eV which leads to 𝜌 𝐹𝐶𝑊𝐷 = 2.4.10 -6 eV -1 . Finally, using the Fermi's golden rule can be used to estimate kRISC: 21

𝑘 𝑅𝐼𝑆𝐶 = 2𝜋 ℏ |𝑉 𝑆𝑂𝐶 | 2 . 𝜌 𝐹𝐶𝑊𝐷 Eq. 3
The theoretical T1 to S1 kRISC for B theo is 4.7.10 2 s -1 (1/kRISC = 2.1.10 -3 s). This value has to be compared to the electric dipole spontaneous radiative lifetime of T1 calculated using level ADF- SO .  = 1.2.10 -2 seconds. At RT, using simple expressions to estimate kRISC, it appears that the RISC is favored compared to the spontaneous emission of the triplet state (one order of magnitude). Thus, our computational studies reveal that if the [Cu2(dppa)3] 2+ could be crystallized without hydrogen bonding interactions with the counterions or included solvent, the TADF process should be operative.

IV -Conclusion

The study of fairly simple bimetallic copper(I) systems coordinated by diphosphinyl-only bridging ligands was performed initially to better evaluate the magnitude of uncertainty that is obtained by employing the level Gauss . Indeed, the largest systems that have been studied in the group which are synthesized by our collaborator Lescop 22,23,24,25 do not allow increasing the level of theory. Remarkably, their ground state and excited state properties are much complex than in the larger systems studied previously in the group. A theo emission properties are revealing at least two separate processes depending on the temperature. An unusual increase of the maximum of emission wavelength is observed upon increase of temperature from 80 K to 160 K, whose lifetime fits with phosphorescence, and a TADF like feature above 160 K. The computational studies show that the S1 and T1 states are low in energy and quite separated from the other excited states. Their emission wavelengths are higher compared to the experimental measurements; in this respect, we stress that in previous studies on much larger systems, the Part I

Chapter II 66 differences between experimental and computational emission wavelengths were below 0.15 eV. 22 The energy spectrum shown in Figure 19 is in favor of temperature dependency and most probably excitation wavelength dependency of the population for the T2 and T3 triplet states at low temperature, which would compete with a TADF process at higher temperature. A global explanation based on the computational results cannot be drawn and would necessitate to study conical intersections and the dynamic of the excited states. Additional experimental measurements are definitely required such as the study of the emission properties as the function of the excitation wavelength and time-resolved measurements.

The modification of the CH2 group by NH in the diphosphine ligand in A theo complicates the problem by introducing hydrogen bond interactions with the fluorine atoms of the counter-ions.

Indeed, B PF6 and B BF4 have different absorption and emission properties which are attributed to the differences in the intermolecular interactions in the crystal. The weakening effect of the N-H bond due to hydrogen interactions has an effect on the absorption properties, as assessed theoretically by the electronic excitation properties of the deprotonated systems B theo/-H+ and B theo/-2H+ . This induces a red shifting of the absorption in agreement with experimental results.

Unfortunately, a computational study of the excited states would need to be done in the solidstate to be comparable with the experimental results. Considering that [Cu2(dppa)3] 2+ could be possibly crystallized with a non-interacting ligand, a study of the excited states of B theo was done. Our simulations are indeed revealing that the TADF process would be operative in that configuration.

Overall, this theoretical study opens up interesting perspectives to study non-conventional emission properties of high interest at a fundamental point of view. The combination between experimental and computational results is highly desirable to tackle the problem.

Chapter III -Study of Cu 6 CN 4 and Cu 8 CN 4 coordination complexes

I -Introduction

The use of polyatomic Cu(I) TADF luminescent precursors acting as pre-organized precursors to conduct general CDS toward emissive coordination-driven supramolecular assemblies has been recently reported by Lescop and coworkers. 1,2 Cu(I) ion is usually described as a labile and non-directional coordination sphere that is a priori not sufficiently directional to allow to conduct selective CDS. Our collaborators from the INSA Rennes have recently shown that this limitation can be overcome using pre-assembled Cu(I) bimetallic molecular clips exhibiting short intermetallic distances, allowing general and versatile CDS preparation of a large family of stacked supramolecular assemblies. 1 They recently extended this CDS approach to the use of a pre-assembled TADF Cu(I) precursor, the tetrametallic metallacycle. 3 The synthetic mechanism based on the adaptive CDS preparation of polyatomic Cu(I) supramolecular assembly 1 from the tetrametallic compound A is given in Figure 1. This methodology can afford much complex architectures which can include other metal ions and which present original solid-state luminescent properties. 4 In this chapter, combined experimental (group of Lescop Precursor A was reacted with CuCN in a 1:2 ratio (Figure 1). X-ray diffraction studies performed at low temperature by Lescop and coworkers on single crystals revealed two different crystals: one being a blue luminophore, corresponding to the non-reacted precursor A while the second is a green luminophore, the new hexa-metallic Cu(I) metallacycle 1 (Figure 1). In this Cu6CN4 metallacycle 1b).

II.2 -Computational study

Computational details

The DFT calculations were carried out using the Gaussian 16 program and using ADF for the SOC calculations. The different systems in their ground state configuration were fully optimized using the X-Ray data as guess geometries with the hybrid PBE0 functional and SVP atomic basis set. The high computational cost needed for full excited state investigations prevents us to increase the atomic basis set. A tight value of 10 -10 u.a. was applied for energy convergence (vibrational frequency calculations could not be performed).

Several isomers are possible for 1, as shown in Scheme 1, and the X-ray characterization does not allow to firmly choose between them. A computational study was thus performed to calculate the relative energies of each isomer. The arrangement 1a presented in Scheme 1 is the most probable isomer according to DFT results. Nevertheless, the relative energies of those four isomers are small and do not exclude their existence on a thermodynamic point of view.

Scheme 1 -Isomers of Cu6CN4 configurations with their relative energies

Considering the massive computational cost that a complete excited state investigation of each isomer would generate, we have focused our efforts, at a first stage, on the study of the excited state properties of 1a, the most stable isomer, and 1c, which presents a nitrogen-only coordination sphere for the bridging bi-coordinated Cu atoms (most probable scenario from a synthetic mechanism point of view). In the most stable S0 arrangement, 1a is diverging more importantly from the experimental X-Ray geometry than 1c (bent Cu-NC-Cu-CN-Cu chains (167-174°) not experimentally observed). The first excited-state Sx and Tx (x = 1, 2, 3)

geometries of 1a and 1c were optimized using TD-DFT. At that stage, the isomer 1a was excluded from the rest of the computational study since its first triplet excited state is higher in energy than the one of 1c (+3.39 kcal.mol -1 / +0.147 eV) and its associative emission energy at 478 nm diverges from the experimental measurements (∼520 nm). Thus, only the 1c isomer was considered for the rest of the study and the use of labelling 1 in the main text stands for this isomer.

The main results obtained for the ground state 1-S0 and its lowest singlet and triplet relaxed excited states (1-S1 and 1-T1) are summarized in Figure 3 and Table 1. The DFT optimized geometry of S0 compares well with the X-Ray structure with intra-dimer Cu-Cu distances of 2.854 Å (exp. 2.881 Å, -0.9 % error) and Cu-Cu contact between the NC-Cu-CN connecting moieties of 3.840 Å (exp. 3.780 Å, 1.6 % error). The calculation of the first singlet-singlet excitation using TD-DFT reveals that this vertical excitation consists of an electronic density transfer mainly from the Cu-ligands to the interstitial Cu-Cu region. It induces a strengthening of the Cu-Cu interaction in each dimer, and, to a lesser extent, to the CN bridges (Figure 3). This constrained optimization gave rise to the system 1-T1-const (Figure 3 and Table 1). The emission energies calculated from these relaxed excited state structures nicely explain the experimental optical measurements. Indeed, the calculated wavelength of fluorescence (from 1-S1, 496 nm) agrees with the maximum of emission measured at 300 K (490 nm). At 80 K, the

experimental wavelength is 522 nm and its emission lifetime indicate a phosphorescence process. The calculated emission wavelength of 1-T1-const is 514 nm. The discrepancies between experimental and calculated emission energies is averaging 0.04 eV which is in line with the level of theory used. The calculated energy differences between (1-S1, 1-T1) and (1-S1, 1-T1const) are 533 cm -1 and 1703 cm -1 respectively. An intermediate value of 1023 cm -1 was extracted from the temperature dependent lifetime fit. This suggests that indeed the 1-T1 excited adopts a conformation in the solid state in which the metallacycle swollen moderately. This supports a TADF process for 1. The moderate value of calculated SOC energies given in Table 2 agrees with a competition between ISC/RISC and thermal population of S1 from T1 which present both close geometries. DFT SOC calculations were performed with the ADF package on the optimized Sx and Tx structures. For these calculations, the same hybrid functional PBE0 was used, with an all-electron triple-zeta Slater basis set with two additional polarization functions (TZ2P). Calculations relying on the Tamm-Dancoff approximation (TDA) of the full TD-DFT equations were used to evaluate the lifetime of the excited states and the oscillator strengths.

Table 2 -Spin-orbit coupling energy (cm -1 ) between states of 1 at the T1, T1const and S1

geometries.

Geometry S 0 /T 1 S 0 /S 1 S 1 /T 1 1-T1

III -Experimental and computational study of a [Cu 8 (CN) 4 ]-based metallacycle III.1 -Summary of the experimental study -Group of C. Lescop

The reaction of the precursor A at RT with NaN3 or KN(CN)2 in, respectively, 1:2 and 1:1 ratio, was performed by Lescop and coworkers. It led to colorless solutions from which were obtained the D complex (Figure 4) as a polycrystalline powder. The same synthetic protocol can be used to synthesize a large variety of extended systems as shown in Figure 4, including heteropolyatomic structures in which another functionality coming from the included metal ion can be added (magnetism for instance). Our computational effort has been devoted to the study of D because of time limitation but the other systems will surely be studied in the group in the future. X-ray diffraction studies on single crystals obtained at RT revealed that D is a {Cu8(DCM)2(CN)4dppm8}(PF6)2 system. This octa-nuclear derivative can be described as resulting of the connection of four [Cu2(dppm)2] fragments with four ditopic cyano ligands forming a Cu8(CN)4 metallacycle system with a slightly twisted rectangular shape (Figure 5). Crystalline samples of D also display at 300 K also a yellowish luminescence upon excitation at 365 nm (Figure 6). This corresponds to a RT emission band with an emission maximum em at 500 nm associated with a bi-exponential emissive lifetime 1 = 52 s and 2 = 18 s. Upon cooling to 80 K, the emission spectrum of D presents a gradual enhancement of its emission band associated with a moderate red-shift of the emission maximum (em = 512 nm at 80 K).

The detailed study of the temperature dependence of the emission lifetime (Figure 6

III.2 -Computational study

Quantum chemical calculations at the DFT level were performed for the compound D using the same methodology as for 1. The geometry optimization of the ground state S0 reveals that the metallacycle frame is preserved in vacuum which confirms the stability of such nanosized scaffold. The first vertical excitations of the lowest singlet and triplet excited states were calculated by TD-DFT (Table 3 andTable 4). Their electronic structures are highly similar (Figure 7) and correspond to electronic transitions from the metal-dppm centered molecular orbitals (MOs) to exclusively ligand-centered MOs involving both cyano and DCM ligands together with the phenyl rings of the dppm ligands. This demonstrates that introduction of different pseudo-halide ligands impacts not only the geometries and the self-assembling processes but also the photophysical properties of CDS Cu(I) assemblies. Figure 8 depicts the computationally optimized Cu8CN4 and the atom labeling used for the next tables. In Table 4 are presented the optimized Cu-Cu distances and angles. Compared to the experimental value dCu-Cu(exp) of 3.430 Å, we found a mean intermetallic distance dCu-Cu(theo) of 3.395 Å (-1.0 % error). The mean experimental dCu-NDCM = 2.090 Å is in excellent agreement with the computed value dCu-NDCM(theo) = 2.090 Å. Table 3 -Calculated vertical singlet electronic excited state energies of D and associated oscillator strength f. The energy is given in eV and the wavelength λ in nm. The difference in the total density with S0 is given in Figure 7 for the description of S1. backbone representation of the optimized system and the used labeling for Tables 5 and6.

State Energy

Table 5 -Cu-Cu distances in Å and main angles in degrees in the optimized geometry of the Cu8(CN)4 complex.

Cu

IV -Conclusion

Adaptive CDS conducted from the Cu(I) bi-metallic precursors [Cu2(dppm)2] discovered by

Lescop and coworkers gives a straightforward access to unprecedented polyatomic derivatives (1 and D) bearing attractive solid-state luminescence properties. In the view of the originality and the complexity of the supramolecular architectures obtained in these one-step syntheses from simple and commercially available building blocks, this new and alternative approach opens appealing and general perspectives in the preparation of new multifunctional supramolecular objects embedded with luminescence properties.

The study of the luminescent properties of 1 and D reveals that the balance between the differences in energy between excited states, the importance of SOC and the crystal packing is crucial to evaluate and understand the evolution of luminescence with the temperature.

Important perspectives are offered by these studies since a large number of compounds are accessible by the CDS procedure, as exemplified in Figure 4. The introduction of magnetic ions will provide multifunctionality to the systems. Computational studies will be of particular interest and are currently under investigation. We also started to study polymer type systems based on the A pre-assembled precursor, where phase change due to an irreversible and nondestructive thermal transition is observed; the study is not sufficiently advanced to be included in this manuscript (Figure 9). Molecular electronics can be defined as the science domain dealing with the study and development of transport properties of electronic components constituted of molecular building blocks and of the circuits resulting of the assembly of these components. 1,2 The concept of "molecular engineering" was first proposed by the physicist Hippel in 1956. 3 This field of research is an interdisciplinary field that gathers expertise in physics, chemistry, electrical engineering, and biology (Figure 1). The main feature of molecular circuits is their nanometer range or even below. Molecular electronics can be regarded as a sub-category of nanoscience.

Developments in molecular electronics often follow a bottom-up process which consists in creating a device from the association of basic fragments (atoms, ions, small molecules). Among the appealing features that motivate the scientific community to explore molecular electronics, the main factors are: 4,5 • Size -The characteristic size of an electronic component is an important feature in electronics in general. Reducing the size allows to conceive less energy-demanding systems and the possibility to increase the component density in a device.

• Self-Assembly -In many cases, molecules can naturally create highly ordered nanoscale assembly such as self-assembled monolayers (SAMs) which are of high interest in many engineering processes, and in particular as active layer in the domain of molecular junctions. 6 • Chemical diversity -The molecules can be chemically modified, which provides a high tunability of their intrinsic properties for their incorporation into functional devices. These properties can be externally and reversibly controlled in some cases, with optical, 7,8,9,10 magnetic, 11 or electrical inputs 12 or with redox reactions 13,14,15,16 allowing the switching of the molecular systems between at least two states.

• Speed -Even though most molecules are not good electronic conductors (nor thermal either), the time to perform operations (down to 10 -14 s) is importantly reduced compared to silicon-based systems. 4 Nevertheless, the use of molecules as electronic components also has disadvantages:

• Temperature limitation -The stability of most molecular electronic systems decreases upon temperature increase. It is highly difficult to measure current-voltage curves above room temperature or at high applied voltages (> 2 V) for most of the systems.

• Reversibility -The molecular junction lifetimes and current intensities are generally still insufficient for industrial applications.

• Reproducibility -The low reproducibility of measurements often encountered is problematic and requires at this stage a deep statistical treatment. 17

II -Most common experimental molecular junction setups II.1 Scanning Tunneling Microscope (STM)

The scanning tunneling microscope was invented by Binnig and Rohrer in 1981 at IBM. 18 This is the first type of microscope allowing for the imaging of surfaces at the atomic level. An STM measurement consists in approaching a sharp conducting tip (few angstroms wide) very close to a conductive (modified) surface. By applying a potential difference between the tip and the surface, electrons are extracted from the tip and create a current by tunneling effect down to the substrate. This current varies exponentially with the distance (making the approach really sensitive) and thus, allows the surface to be mapped with a great precision, around 0.1 nm in lateral precision and 10 pm (0.01 nm) in depth. 19 One of the most famous experiments performed in the early stage of molecular junction studies was done in 1996 by the group of Weiss. In this study, they prepared a mixed self-assembled monolayer (SAM) containing 4,4′di(phenylene-ethynylene) benzenethiolate derivatives dispersed among saturated alkanethiol chains on an Au (111) surface and probed their electrical properties by STM (Figure 2). 20 From this experiment, they observed that the tunneling current intensity was higher when the tip was above a conjugated molecule compared to the non-conjugated alkyl chains covering the gold substrate.

Until now, in the field of molecular junctions, STM is still an experimental setup used, allowing for the current-voltage measurements of molecules grafted on a substrate (referred to as Scanning Tunneling Spectroscopy -STS). In another famous experiment, the group of Kubiak in 1997 achieved the first single molecule current-voltage measurement of an ,'-xylyldithiol also on (111) gold substrate. 21,22 They found that single molecules can exhibit exotic I-V curve with non-ohmic characteristics (Figure 3) as the voltage increases, features never observed in conventional electronics. They also showed that the interface between the molecule and the electrodes has to be critically considered as well as the Fermi level positioning.

II.2 Conductive Probe -Atomic Force Microscopy (CP-AFM)

The second important experiment to conduct I-V curve measurements is conductive probe atomic force microscope, CP-AFM (Figure 4). 23,24,25 The atomic force microscope (AFM) was invented by Binnig et al. in 1985, 26 and was first modified with a conductive probe by O'Shea and coworkers in 1993. 27 One of the first uses of CP-AFM dates back to 1996 with the imaging of local electrical features of metallic surfaces 28 and in 1997 for the study of electronic transport in thin doped sexithiophene crystals. 29,31 In this technique, the landscape of the surface is captured by measuring the deflection of an AFM cantilever with a laser and the current is measured with a current-to-voltage amplifier. 30 It has the advantage of having two different detection systems, an optical one for the topography and a preamplifier for the current, instead of one as in STM measurements, which allows to decorrelate the current from the topography of the surface (in STM the tunneling current is used to build the image of the surface). The main mode of CP-AFM is "contact mode" in which the voltage is kept constant and the current measured. One disadvantage compared to the STM is the tip size. It is generally several nanometers wide which induces a lateral resolution of about 10 nm (vs 0.1 nm for STM). The number of molecules coupled with the tip is thus not precisely known but can be estimated.

Typically, the number of contacted molecules lies between 50 to 100 depending on the packing density and their tilt angle with the surface. Part

This method can be used for both physisorbed and chemisorbed molecules on the tip. In general, the tip is gold-coated to allow thiol bonding for example. The contact interaction between the tip and the sample is primordial especially when saturated molecules are involved; the current can vary by 3 orders of magnitude if the molecule is chemisorbed or not, and by 2 orders of magnitude for conjugated molecules compared to saturated ones. 32

II.3 Mechanically Controllable Break Junctions (MCBJ)

The third major technique in the field of molecular electronics is MCBJ, which appeared at about the same period as the STM. In a MCBJ setup, two metallic wires are gradually separated by a gap controlled using a piezo-actuator (Figure 5), which allows to close and open the junction. The gap can be as small as one atom and can be monitored with picometer precision. 33 A wire is initially created inside the gap either by chemical etching or electromigration. When the substrate is precisely curved by the piezo-actuator, the wire breaks and molecules can spontaneously create a molecular junction inside the aperture (Figure 5). -dithiol molecule between gold electrodes using the MCBJ method (Figure 6). 34 This has the advantage compared to STM and CP-AFM to eliminate intermolecular interactions that render difficult the rationalization of the intrinsic properties of the molecules within a junction.

III -Theoretical background

This present section aims at introducing the general theoretical background for molecular junctions and the introduction and technicalities of thermoelectricity.

III.1 -Ohm's law

Ohm's law, applicable in the case of conducting bulk materials, stipulates that through a conductor, the transient current between two points is proportional to the electric potential across the two points. The material resistance is linked to its electrical conductivity σ as follows: 35,5 𝑅 = 𝑙 𝜎. 𝐴 Eq. 1

with l the length of the wire (e.g. copper, iron) and A the cross-section area of the wire.

The origin of this resistivity is due to the scattering of electrons by: 36,37 • Phononscollective vibrations (or lattice vibrations) of a bulk material around the equilibrium position of the nuclei.

• Lattice defectsmisplaced or substituted atoms or ions which alter the periodicity of a crystalline solid. They can modify or suppress the delocalized electronic states involved in the conducting process.

These scattering events occur erratically. The mean distance that electrons travel in the material between scattering events is named the mean free path. For a metal, the mean free path is generally comprised between 10 to 50 nm at room temperature. 35 According to the Bose-Einstein statistic, the number of phonons increases with temperature until a certain threshold is reached. The latter is material-dependent; it is 170 K for gold using the Debye model. 38 The electron-phonon coupling leads to an increase in resistance and to a concomitant heating in the device. Thus, the heating of the material intensifies as the current increases.

The description of the physics linked to electrical flow in molecular-based systems does not follow the same rules. The ohmic regime does not apply mainly because:

• The resistance increases exponentially with length.

• Temperature independence of the resistance is generally measured.

• Current appears to increase non-linearly with voltage, revealing a voltage-dependence of the resistance.

• The representative mean free path for a bulk metal is higher than the length of standard molecules, which lies more in the 0.1 nm to 5 nm range (compared to the mean free path of 10 to 50 nm in a metal).

The physics in molecular junctions cannot thus be rationalized by Ohm's law. New theoretical insights were developed in the emerging community of molecular electronics to rationalize the resistivity measured in molecular junctions.

III.2 -Premise about the transport regime

The scattering events can be divided into two groups: 39,5 • Inelastic diffusionthe scatterer is not static or the electron has the possibility to interact with the internal degrees of freedom of the scatterer, which will alter the energy of the incident electron. Such inelastic scattering sources can be due to magnetic impurities, other electrons or phonons. 40 • Elastic diffusiona stationary scatterer without internal degrees of freedom will conserve both the electron energy and phase. Only an alteration of its trajectory is permitted.

Both scattering events, elastic and inelastic, can be categorized with their own mean free paths.

The inelastic mean free path is in the order of 20 nm for a conductor (38 nm for gold at room temperature). 37,41 while the elastic mean free path is typically one order of magnitude smaller. 42,43 A molecular junction is composed of a molecule physisorbed or chemisorbed on two bulk electrodes (or more in presence of gating effects). For such a short system, inelastic scattering events are unexpected since the length of the system is in most of the cases shorter than the inelastic mean free path. In that case, the main transport regime is the phase-coherent tunneling regime. 44,45,46 If both the elastic mean free path and the phase coherence length are larger than the molecule length, the transport is said to be ballistic, which means that the electron can cross the device without any loss of momentum. Generally, for molecular lengths greater than 5 nm, the incoherent hopping regime takes hover. This regime is thermally activated (Figure 7). 47,40 Part

The transport properties in molecular junctions are usually rationalized based on these assumptions. Consequently, the main hypothesis on the transport regime and its coherence is generally valid when the molecular junction is tightly coupled to the metal electrodes, with a length that does not exceed 5 nm. 49,50 Nevertheless, incoherent processes can still occur below this limit if the molecule is weakly coupled to the electrode (e.g. physisorption). This induces a longer crossing time of the incident electron because of the interaction with the molecule. This phenomenon is called Coulomb blockade regime. 51 Indeed, the electron inside the device creates a Coulomb repulsion preventing other electrons to flow. The current-voltage shape of a Coulomb blockade resembles a staircase. 52 Part II

The coherent regime has some interesting particularities: 45,5 • Ballistic electrons are not slowed by scattering events, their speed is thus constant. They can travel at the Fermi speed (v Fermi = √2E F /m e ; 1.4.10 6 m/s in gold). It is one order of magnitude higher compared to the actual speed in Metal Oxide Semiconductor Field Effect Transistor (MOSFET) 53 of about 2.10 5 m/s even though the resistance of molecules is high (see Figure 1: 10 5 to 10 10 ).

•

The molecular resistance occurs without heating of the molecule (no inelastic interactions), thus implying no energy loss inside the molecule. Indeed, the dissipation of the energy and the Joule heating appears to occur only in the contacting bulk electrodes. 45,54

III.3 -Conductance is transmission

We present below the first models used to explain and rationalize the electrical properties of molecular junctions.

III.3.1 -One dimensional square barrier model

The molecule in a molecular junction can be seen as an obstacle for the current flow, where the incident electrons can be transmitted or reflected. This can be interpreted at first sight by the tunneling barrier problem in quantum mechanics. The single barrier model considers a 1-D square potential barrier of width L and of height V0 as exemplified in Figure 8. Solving the time-independent Schrödinger's equation, the transmission probability can be determined as: 55,5 τ(E) = J III,transmitted J I,incident Eq. 2

With J the current probability. I and III correspond to the regions before and after the single barrier, respectively. This implies that when an electron encounters the potential barrier, it can be either reflected or transmitted through the barrier. The transmission probability τ(E) can be expressed as a function of the energy as follows: 40,55 τ

(E) = 4E(V 0 -E) 4E(V 0 -E) + V 0 2 sinh 2 (ρ 2 L) Eq. 3 with ρ 2 = 1 ℏ √2m e (V 0 -E),
and m e the electron mass. If ρ 2 L ≫ 1 the problem can be seen as:

τ(E) ∝ exp -2ρ 2 L
Eq. 4

The simplified Eq. 4 leads to a description in which an exponential decrease of the transmission is calculated with the increase of the barrier length L. This reproduces the experimental observations of an exponential increase of resistance as the molecule grows in size.

III.3.2 -One dimensional double square barrier model

A double square shaped barrier was also proposed in the literature. 37,40,55 The resulting transmission function for this second model is displayed Figure 9. Three resonant transmission peaks are obtained echoing the transparency of the system at these energies.

III.3.3 -Single resonant tunneling level

A more refined model based on a single resonant level tunneling was proposed to better reproduce the complexity of molecular junction transmission features. 40,55 It is summarized in

𝜏(𝐸) = 𝜏 1 𝜏 2 1 -2√ℛ 1 ℛ 2 𝑐𝑜𝑠(𝛥𝜙) + ℛ 1 ℛ 2 Eq. 5
where τ 1 , τ 2 , ℛ 1 and ℛ 2 are the energy dependent transmission and reflection coefficients of the first and second barrier respectively. The energy dependent phase shift is noted Δϕ. When we have: ϕ  2nπ and ℛ 1 , ℛ 2  1, this leads to a resonance peak energy ϵ 0 , at which the barriers become transparent. Such singularity would not be observed if an inelastic scatterer was introduced because it would randomize the phase, meaning that the transmission across N successive barriers would goes as 1/N instead, re-inducing an ohmic length dependency. 5 Developing Eq. 5 around ϵ 0 (i.e. 𝜙  2𝑛𝜋 and ℛ 1 , ℛ 2  1), using a Taylor expansion to the first order and using the fact that √ℛ 1 = √1 -𝜏 1 ≈ 1 -𝜏 1 /2, one can write: 40

𝜏(𝐸) ≈ 𝜏 1 𝜏 2 ( 𝜏 1 + 𝜏 2 2 ) 2 + (2 -𝜏 1 -𝜏 2 )(1 -𝑐𝑜𝑠(𝛥𝜙))
Eq. 6

Applying a second order cosine expansion and a first order development of ∆ϕ around the resonance energy ϵ 0 , we obtain: 5

𝑐𝑜𝑠(𝛥𝜙(𝐸)) ≈ 1 - 1 2 ( 𝑑𝛥𝜙 𝑑𝐸 | 𝐸=𝜖 0 ) 2 (𝐸 -𝜖 0 ) 2 Eq. 7
Assuming that τ i multiplied by the second order term is negligible and defining the coupling Γ as follows: 5

𝛤 𝑖 = 1 2 ( 𝑑𝛥𝜙 𝑑𝐸 ) -1
𝜏 𝑖 Eq. 8

the expression of the transmission τ(E) can be simplified as a Lorentzian function centered at the resonance energy, ϵ 0 , with a full width at half maximum (FWHM) of 4Γ:

𝜏(𝐸) = 4𝛤 1 𝛤 2 (𝐸 -𝜖 0 ) 2 + (𝛤 1 + 𝛤 2 ) 2 = 4𝛤 2 (𝐸 -𝜖 0 ) 2 + 4𝛤 2 | 𝛤 1 = 𝛤 2
Eq. 9

This expression is a really important equation which will be used in the next chapters of this thesis to calculate τ(E). Within this framework of the single tunneling resonant level, the coupling Γ i is directly related to the time needed for an electron to escape the scattering region delimited by the molecular system, and is often called the "scattering rates". It is an important quantity which is intrinsic to each molecular junction. It will be a key parameter to define the transmission properties of a molecular junction during this study. This model is one way to recover the Lorentzian shape transmission around a resonant level, but the same results can be derived using the Green's function formalism, 40 see Figure 11 for a representation of the single level resonant using Eq. 9. This model is a crude simplification since a molecule is not as simple as 1-D energy barrier. Its electronic structure is composed of several occupied and unoccupied molecular orbitals (MOs).

In an electronic transmission spectrum, the resonant peaks, shaped as Lorentzian functions, are Part II Chapter I 98 intrinsically connected to the energy position of the MOs. Each of these peaks can have different scattering rates, contingent with the ability of the electrons to flow from one end of the device to the other. This ability is mostly determined by the chemical bonding nature, and the delocalization of the orbitals between the molecule and the metal electrode which is highly sensitive to changes in the contact geometry. Undeniably, if the coupling between the molecule and the leads is strong, the passage of the electrons will be facilitated. This is why Γ i is also called the coupling strength. Now, even if we can argue that the electronic transmission through a barrier is related to the current emerging out of a molecular junction, the exact relation between the two is not straightforward.

In 1957, Landauer demonstrated that the conductance can be perceived as electronic transmission by introducing the notion of the quantization of the conductance. 57,58,59 In a molecular junction in which the molecule is grafted at both ends to electrodes (leads), the electrodes are considered as electron baths in equilibrium at a temperature T and with a chemical potential . Upon application of a bias, the electrons are injected from the electrodes to the molecule. Considering that a tunneling process occurs, it is possible to show that a relation exists between the electronic transmission and the quantized conductance. 40 Basically, considering the electrodes as one-dimensional electron gas, the directional current density J can be expressed as:

𝐽 𝐿→𝑅 = 2𝑒 ℎ ∫ 𝜏(𝐸) 𝑓 𝐿 (𝐸, 𝜇 𝐿 ) (1 -𝑓 𝑅 (𝐸, 𝜇 𝑅 )) 𝐽 𝑅→𝐿 = 2𝑒 ℎ ∫ 𝜏(𝐸) 𝑓 𝑅 (𝐸, 𝜇 𝑅 ) (1 -𝑓 𝐿 (𝐸, 𝜇 𝐿 ))
Eq. 10 with L and R representing respectively the left and right electrode and τ(E) the transmission.

The f L and f R are the Fermi-Dirac distributions, as the electron obeys the Pauli exclusion principle. The Fermi-Dirac functions write as:

𝑓 𝐿,𝑅 = 1 𝑒𝑥𝑝 ( 𝐸 -𝜇 𝐿,𝑅 𝑘 𝐵 𝑇 𝐿,𝑅 ) + 1 Eq. 11
with k B the Boltzmann constant. Finally, the total current can be expressed as:

𝐼(𝑉) = 𝐽 𝐿→𝑅 -𝐽 𝑅→𝐿 = 2𝑒 ℎ ∫ 𝜏(𝐸, 𝑉) (𝑓 𝐿 (𝐸, 𝜇 𝐿 ) -𝑓 𝑅 (𝐸, 𝜇 𝑅 )) 𝑑𝐸 ∞ -∞
Eq. 12 Part

Eq. 12 constitutes the Landauer formula. All theoretical calculated I-V curves shown in the next chapters are generated in this formalism. This equation which provides the current at each considered bias voltage is the integration of the transmission function multiplied by the difference between the left and right Fermi occupation of the electronic levels in the electrodes.

The transmission function can be evaluated for example with a single resonant level (lorentzian model 60,61 ) or calculated on the basis of quantum chemical calculations, applying methods like the Green's function formalism. In that case, since the transmission is possibly voltage dependent, it has to be calculated self-consistently at each considered bias.

In fact, when a voltage bias is applied between the left and right electrodes, the chemical potentials  𝐿 and  𝑅 , are shifted symmetrically below and above the equilibrium Fermi level.

This restricts the transmission spectrum to be considered in a specific bias window as shown in Considering 𝑇 = 0 K and 𝜏(𝐸) = 1, Eq. 12 which is yielding the current can be simplified as:

𝐼(𝑉) = 2𝑒 2 ℎ 𝛥𝑉 = 7.75.10 -5 𝛥𝑉 = 𝐺 0 𝛥𝑉 Eq. 13
with G 0 the conductance quantum. The reciprocal resistance quantum is 𝑅 0 = 12.9 kΩ. Thus, at 𝑉 = 1.0 𝑉, this equation establishes that the charge current cannot be higher than 77.5 μA in the hypothesis of a maximum of transmission around the Fermi energy. Obviously, this is the most favorable case; currents are expected to be comprised in the nA to μA range in realistic molecular systems.

Considering a coherent ballistic regime, from the Landauer's formula, the electronic transport is built from a difference of electron current being scattered at the electrodes. This means that the created electric field can be seen as an alteration of a potential barrier shape, which will in turn change the fundamental transmission probability of the scatterer (e.g. molecule). This is what we showed using different barrier models. Another famous barrier model is the Simmons model 62,63,64,65,66,67 Nevertheless, because the electronic structure of molecules yields a much complex electrostatic potential compared to a barrier model, the use of computational studies at the quantum level is required to access to the electrode-molecule-electrode transmission probability, which is, a priori, not known. This will be computed using the DFT coupled with the non-equilibrium Green's functions.

III.4 -Density functional theory (DFT) and non-equilibrium Green's functions (NEGF)

This part describes the methodologies which will be applied to calculate transmission spectra of organic and organometallic molecules employing ab initio type methods. A compromise has to be found between the precision of the results and the computational cost in order to deal with a large number of systems and additionally to provide sufficiently quantitative results to allow for the rationalization of experimental measurements; DFT appears here to be the best compromise. Contrarily to wave-function calculations which intend to solve the Schrödinger many-body equation in 3N dimensions (plus N spin degrees of freedom), with N the number of electrons, DFT essentially reduces the problem to a resolution based on the electronic density that depends only on 3 spatial coordinates to determine the properties of a many-electron system (Hohenberg and Kohn's theorems). 68 In the implementation that we use, the electrodes are considered as semi-infinite. In the directions which are perpendicular to the transport, the Part II Chapter I 101 periodicity is conserved, whereas along the transport direction, the scattering region (which contains the molecule) disrupts the periodicity of the metallic electrodes (Figure 13). Because this periodicity is broken, the system is referred to as "open". Furthermore, from the Landauer's picture, the molecule included in the scattering region is coordinated to two infinite electrodes, with each having a different Fermi energy (or chemical potential). The system is thus out of equilibrium. The problem which rises is thus how to calculate the electronic structure of such a system and its transmission considering an open system with infinite electrodes! The solution is to employ Green's functions. 69 Introduced by Green in 1828, these functions are used to solve linear differential equations with constant coefficients. The transmission probability can be obtained from the resolution of the Green's functions. 70,71,72,73,74,[START_REF] Stokbro | Ab-initio Non-Equilibrium Green's Function Formalism for Calculating Electron Transport in Molecular Devices[END_REF][START_REF] Cohen | Green's Function Methods for Single Molecule Junctions[END_REF] The resolution can be applied to non-equilibrium systems, which allows to calculate the transmission at different bias voltages using the Landauer's formula.

This NEGF formalism was developed by Kadanoff and Baym [START_REF] Kadanoff | Quantum Statistical Mechanics[END_REF] and independently by Keldysh in the early 60s. [START_REF] Keldysh | Diagram technique for non-equilibrium processes[END_REF] In the Keldysh formalism, the electronic transmission is defined in terms of The DFT coupled to the Non-Equilibrium Green's Function (DFT/NEGF) protocol is the best method to date which can resolve molecular junction systems in reasonable computational times without too much approximations. Because this method is based on DFT, it also encompasses its inherent approximations and limitations. Nevertheless, considering the I-V curves, the DFT-NEGF method can be semi-quantitative with experimental data, within one order of magnitude or less for the conductance [START_REF] Koentopp | Density functional calculations of nanoscale conductance[END_REF][START_REF] Mera | Are Kohn-Sham Conductances Accurate?[END_REF] , and even closer for the position of the MOs compared to the Fermi level. 112,115 Conclusions can thus be drawn based on qualitative features of the calculated electronic transmission spectra.

III.4.1 -Computational procedure

The geometries and electronic structures of the isolated molecules were studied using the Amsterdam Density Functional package (ADF), version 2019 (vacuum, 0 K). [START_REF] Te Velde | Chemistry with ADF[END_REF] For the periodic bulk-like systems involved in the transmission calculations (using the NGEF formalism), the Synopsys Atomistic ToolKit package (Quantum ATK 2019, 2020 and 2021 versions) was used. [START_REF] Markussen | An integrated platform of electronic and atomic-scale modelling tools[END_REF] The ADF calculations were performed using a TZ2P atomic basis set 84 (triple- basis

set with 2 polarization functions, small frozen core) with the rev-PBE [START_REF] Zhang | Comment on Generalized Gradient Approximation Made Simple[END_REF] exchange-correlation functional (GGA). Relativistic scalar ZORA corrections were included for all molecules. The numerical accuracy of the Becke integral was set to good (1.1.10 -4 eV). The SCF convergence and the geometry optimization tolerance were both set to 10 -6 a.u.

Device calculation procedure

The level of theory used was the following unless specified in the text:

Exchange correlation: GGA -revPBE (same functional as for the molecular calculations)

Basis set: Troullier-Martins type pseudopotentials (FHI); DZP (Double- and 1 polarization function) for all atoms expect for gold: SZP (Single- and 1 polarization function).

Density-mesh cut-off: 100-125 Hartree depending on the system.

Occupation method: Fermi-Dirac with a broadening of 300 or 1000 K.

Device k-points: generally, from 5 to 10 along the A and B directions (perpendicular to the current) and between 50-150 in the C direction (transport direction).

SCF tolerance convergence: 10 -5 a.u.

NEGF contour integrals:

• Integration method: Weighted

Transmission spectrum:

• Cover bias: generally, between -2 to 2 eV relative to the Fermi energy of the electrodes.

• Infinitesimal: 10 -6 eV

• Self-energy calculator: recursion

• k-point: between 4-10 along both A and B directions All the k-points distributions were realized using the Monkhorst Pack grid. [START_REF] Monkhorst | Special points for Brillouin-zone integrations[END_REF] Part II Chapter I 104

IV -Thermoelectric properties

Thermoelectricity is the terminology assigned to describe the electricity generated by a thermoelectric effect, i.e. the difference in temperature between two sides of a material This induces an electronic current that flows across the material and creates a voltage difference.

Reversely, applying a voltage between two sides of a material generates a heat flow that creates a difference in temperature. These thermoelectric effects are at the heart of many applications like refrigeration, thermometry, or generation of electricity. Thermoelectric effects can be described by three correlated phenomena: the Seebeck, Peltier, and Thompson effects.

Seebeck effect

The first effect discovered was by Seebeck in 1821. [START_REF] Seebeck | Über den Magnetismus der galvanischen Kette. Abhadlungen der Physikalischen Klasse der Königlisch-Preußsischen[END_REF] He showed that an electrical potential difference 𝑉 can appear when one part of a junction with two different conductors (iron and copper in Figure 1) is heated. Basically, the Seebeck effect emerges when two wires of different nature, iron and copper for example, are connected forming a thermocouple. A basic thermoelectric system is shown in Figure 15. Upon application of the temperature difference 𝑇, an electrical potential difference 𝑉 appears and can be measured by a voltmeter. Peltier effect Peltier, a part-time physicist, observed in 1834 the second thermoelectric effect which occurs when an electrical current flow through a thermocouple. [START_REF] Peltier | Investigation of the heat developed by electric currents in homogeneous materials and at the junction of two different conductors[END_REF] It generates heating of one of the two junctions, and cooling on the other (Figure 15). These temperature changes can be reversed depending on the current direction. At that time, the correlation between the Seebeck and the Peltier effect was not established. The heating on one side of the system and cooling on the other that is generated upon application of 𝑉 can be explained by the conversion the energy carried by electrons into heat upon flowing across different materials.

Thomson effect

Thomson discovered in 1855 that a correlation exists between the Seebeck and Peltier effects using thermodynamic theories. From these relations, he proposed that there must exist a third thermoelectric effect, which was later called the Thomson effect; this corresponds to reversible heating or cooling when an electrical current 𝐼 and a temperature gradient 𝑇 are applied simultaneously. Contrarily to the Seebeck and Peltier effect, the Thomson effect can be observed in a bulk material but, as of today, not in a junction configuration. Generation, emission or absorption of heat 𝑄 occur in some parts of the material.

After these discoveries, the possibilities of building thermoelectric devices for current generation, heat generation and refrigeration emerged in the scientific community. [START_REF] Altenkirch | Über den Nutzeffekt der Thermosäule[END_REF][START_REF] Altenkirch | Elektrothermische Kälteerzeugung und reversible elektrische Heizung[END_REF] He demonstrated that the performance of energy conversion can be increased by enhancing the electrical conductivities 𝜎 and reducing the thermal conductivities 𝜆 of the material. Nevertheless, at that time, the use of only metallic conductors did not allow to access to efficient thermoelectric systems. Actually, in 1949, a breakthrough was made in the development of a modern theory of thermoelectricity by Ioffe using the concept of figure of merit 𝑍𝑇. [START_REF] Vedernikov | Ioffe and origin of modern semiconductor thermoelectric energy conversion[END_REF] Ioffe endorsed and popularized the use of the semiconductors (silicon, germanium, gallium arsenide…) in thermoelectric systems to increase their performance. Nowadays, bulk materials with high thermoelectric figures of merit are doped semiconductors, some of the most famous ones are the tellurides of antimony, bismuth and lead. [START_REF] Goldsmid | The use of semiconductors in thermoelectric refrigeration[END_REF] The research institute of Ioffe, the laboratory of semiconductors of Academy of Sciences of the USSR, has been a pioneer institution in thermoelectric research and developments leading to some of the first thermoelectric power generators and cooling devices commercially available. These scientific developments are at the origin of the democratization of the use of alloys to decrease thermal conductivity of bulk systems.

Nowadays, the research strategies to obtain efficient thermoelectric bulk materials can be synthetized as follows:

• Lowering of the thermal conductivity 𝜆 which often results in incorporating heavy elements in the materials.

• Increase of complexity in the composition and structure such as disorder and nanostructuration of the materials.

• Narrowing of the band-gap (semiconductors)

• Raising of the electrical conductivity 𝜎 to improve efficiency

IV.1 -Definitions and relationships between the thermoelectric coefficients

This section intends to provide the necessary definitions and equations related to the Seebeck coefficient, the Peltier coefficient and the figure of merit 𝑍𝑇 and reveal their interdependence.

The Seebeck coefficient, 𝑆 is defined by:

Part II Chapter I 107 𝑆 = - 𝛥𝑉 𝛥𝑇 Eq. 15
The Peltier coefficient of a thermocouple, π, is equal to the ratio between the rate of heating or cooling 𝑑𝑄/𝑑𝑡 = 𝑄 ̇ (amount of heat transferred per unit of time) at each junction and the electrical current 𝐼:

𝜋 = 𝑄 İ Eq. 16
The second Thomson's relation expresses the Peltier coefficient in terms of the Seebeck coefficient as: [START_REF] Rowe | Thermoelectrics Handbook: Macro to Nano[END_REF] 𝜋 = 𝑆. 𝑇 Eq. 17

This relation correlates the Seebeck effect and the Peltier effect by the factor T which is the mean temperature of the system. This was fully demonstrated in the 1950s with the support of the Onsager reciprocal relations. [START_REF] Onsager | Reciprocal Relations in Irreversible Processes[END_REF] Finally, the figure of merit 𝑍𝑇 which determines the efficiency of a thermoelectric system is established as follows:

𝑍𝑇 = 𝑆 2 𝜎𝑇 𝜆 Eq. 18
With σ the electrical conductivity and λ the thermal conductivity. The thermal conductivity refers to the ability of a material to conduct heat. It can be divided in different contributions, the electronic thermal conductivity 𝜆 𝑒𝑙 and the phononic thermal conductivity 𝜆 𝑝ℎ .

In bulk materials, and more specifically metal, an empirical relationship between the electronic thermal conductivity and the electrical conductivity multiplied by the temperature exists and is called the Wiedemann-Franz law:

V -Thermoelectricity in molecular junctions

The ongoing research in nanotechnology is facing similar energy control issues than at the micro and macroscale. Among the different issues to be solved, one important problem concerns the heat generated by the resistance of the components of a nano-unit against the electrical current flow. It leads not only to a waste of energy but can also cause an important lowering of performance and even destruction of the nano-devices. Light irradiation, often used as external input or energy source in nano-devices, is also a cause of deterioration notably by inducing localized over-heating. [START_REF] Ceroni | Light to investigate (read) and operate (write) molecular devices and machines[END_REF] A straightforward approach that naturally comes to mind to prevent such deleterious effect is to exploit thermoelectric effects at the nanoscale. Besides solving these energy control problems, the development of such systems could provide thermal nano-detectors, nano-coolers or more generally nano-regulators by using their ability to transfer heat. The development of efficient thermoelectric nano-devices could also provide the conception of body implementable biosensors or monitors that need to be energetically autonomous.

The thermoelectric phenomena observed in bulk materials have been imagined to also occur at the nanoscale. This assumption came from the pioneering work of Hicks and Dresselhaus [START_REF] Hicks | Effect of quantum-well structures on the thermoelectric figure of merit[END_REF]100 in 1993 which predicted that atomic wires should drastically improve the efficiency of energy conversion. This was at the origin of the intensive theoretical and experimental research developments in molecular junctions (Figure 18) built of atoms, metallic wires, molecules, quantum dots, and even biomolecules like DNA. 101,102,103,104,105,106,107,108, 109,110,111,112,113,114,115 The main feature of molecular junctions is the combination and overlap between the discrete electronic levels of the molecule energy with the continuous spectrum of energy states of the electrodes. Numerous applications using nanoscale thermoelectric systems can be envisaged, for instance: heat to voltage converters, 116,117,118,119 local refrigerators, 120 thermal transistors, 121,122 thermal rectifiers, 123,124 logic gates 125,126 and in the future, even the concept of biocompatible implementable sensors or monitors which require integrated energy sources. On the fundamental point of view, a deeper understanding of thermoelectric properties of molecular junctions could certainly provide a deeper insight into the nature and characteristics of electron and thermal transport mechanisms in these systems that will surely allow to better drive the future developments in this area. 127,128,129,130 Several factors play a role in the charge transport across a molecular junction and thus in its thermoelectric ability:

• The intrinsic electronic characteristics of the inserted molecule.

• The coulombic interaction between electrons.

• The constructive and destructive quantum interferences (QI) that can occur across the junction.

• The electrode vibrational properties (phonons) and their coupling with the vibrational modes of the molecule.

V.1 -Linear regime response formulations

In this part, the coherent thermoelectric coefficients are described within the Landauer framework. We also assume that the chemical potentials of the electrodes are equivalent to the Fermi energy 𝜇 ≈ 𝐸 𝐹 . In addition to these approximations, the following developments are only valid in the linear response regime i.e., by considering that the system is under a slightly outof-equilibrium condition due to the perturbation. The linear response theory was introduced in 1957 by Kubo. 131 From Eq. 12 the expression for the current 𝐼(𝑉) and the heat current in the Landauer framework are:

𝐼(𝑉, 𝑇) = 2𝑒 ℎ ∫ 𝜏(𝐸, 𝑉) (𝑓 𝐿 (𝐸, 𝜇 𝐿 ) -𝑓 𝑅 (𝐸, 𝜇 𝑅 )) 𝑑𝐸 ∞ -∞
Eq. 21 Here, 𝐿 0 refers to the linear conductance of the system divided by the quantum of conductance 𝐺 0 :

𝐿 0 = 𝐺/𝐺 0 𝑤𝑖𝑡ℎ 𝐺 0 = 2𝑒 2 ℎ ≈ 77.46 𝜇𝑆 Eq. 25
The 𝐿 𝑛 coefficient (with n = 0, 1, 2) in Eq. 23 and Eq. 24 is expressed as:

𝐿 𝑛 = 2𝑒 ℎ ∫ 𝜏(𝐸)(𝐸 -𝜇) 𝑛 (- 𝑑𝑓 𝑑𝐸 ) 𝑑𝐸 ∞ -∞
Eq. 26

In Eq. 26, the chemical potential of the left and right electrodes, 𝜇 𝐿 and 𝜇 𝑅 respectively, are considered equal 𝜇 = 𝜇 𝐿 = 𝜇 𝑅 ; this corresponds to the system at equilibrium (V = 0).

V.2 -The Seebeck coefficient in the linear response regime

In this framework, considering an open circuit (𝐼 = 0), Eq. 23 becomes:

𝐿 0 𝛥𝑉 = - 𝐿 1 𝛥𝑇 𝑇 Eq. 27
and by identification:

𝑆 = - 1 𝑇 𝐿 1 𝐿 0 ≡ - 𝛥𝑉 𝛥𝑇 Eq. 28
This Eq. 28 will be applied to calculate the Seebeck coefficient for each studied system. The linear regime implies that ∆𝑇 ≪ 𝑇.

When the system is studied at low temperature, a Sommerfeld expansion 36 Eq. 29

From Eq. 29, it appears that the slope of the transmission at the Fermi level is essential to obtain high Seebeck coefficient. Nevertheless, this observation is not so valid upon increase of the temperature as shown in Figure 20, which displays the discrepancy between the Sommerfeld expansion (Eq. 29) and the integral form (Eq. 23) occurring between 100-150 K. In this example, at RT, the difference is about 20 μV/K (20 % error). This error is intrinsically linked to the electronic transmission's shape; thus, it can be different than the calculated 20 % in Figure 20.

V.3 -Calculation of the thermal conductance

For molecular systems, the use of thermal conductance is favored over thermal conductivity.

The total thermal conductance can be divided into two main contributions:

• electronic thermal conductance  𝑒𝑙

• phononic thermal conductance  𝑝ℎ

If only the electronic thermal conductance  𝑒𝑙 is considered ( 𝑝ℎ = 0), using Fourier's Law of thermal conduction one can expressed 𝑄 𝑒 as follows:

𝑄 𝑒 = 𝜅 𝑒𝑙 . 𝛥𝑇 Eq. 30

Using Eq. 24, Eq. 27 and Eq. 30, 𝜅 𝑒𝑙 can be expressed as:

𝜅 𝑒𝑙 = 1 𝑇 (𝐿 2 - 𝐿 1 2 𝐿 0 ) Eq. 31
From Eq. 31, one can deduce that the electronic thermal conductance is dependent on the electric conductance 𝐺 (because 𝐿 0 = 𝐺/𝐺 0 ). Therefore, it can be related to the aforementioned Wiedemann-Franz Law (Eq. 19) employing conductances instead of conductivities:

ℒ = 𝜅 𝑒𝑙 𝐺. 𝑇 → 𝜅 𝑒𝑙 = ℒ. 𝐺. 𝑇 Eq. 32
It should be noted that at the nanoscale the validity of the Wiedemann-Franz law remains an open question. 134,135 For the calculation of 𝜅 𝑝ℎ , omitting the electron-phonon interactions, the phonon contribution to the heat flux Q can be expressed as: 104,136

𝑄 𝑝ℎ = ℏ 2𝜋 ∫ 𝜔 𝜏(𝜔) ∞ 0
[𝑛 𝐵 (𝑇 𝐿 ) -𝑛 𝐵 (𝑇 𝑅 )] 𝑑𝜔 Eq. 33

with 𝑛 𝐵 the Bose-Einstein distribution:

𝑛 𝐵 (𝑇) = 1 𝑒𝑥𝑝 ( ℏ𝜔 𝑘 𝑏 𝑇 𝐿,𝑅
) -1 Eq. 34

where 𝜏(𝜔) is the transmission function for a phonon with an energy ℏ𝜔. Using the same approach (Fourier's law of heat conduction) than for 𝜅 𝑒𝑙 , one can state that 𝑄 𝑝ℎ = 𝜅 𝑝ℎ ∆𝑇, consequently, 𝜅 𝑝ℎ can take the form:

𝜅 𝑝ℎ = ℏ 2 2𝜋𝑘 𝑏 𝑇 2 ∫ 𝜔 2 𝜏(𝜔) ∞ 0 𝑒𝑥𝑝(ℏ𝜔/𝑘 𝑏 𝑇) (𝑒𝑥𝑝(ℏ𝜔/𝑘 𝑏 𝑇 ) -1) 2 𝑑𝜔
Eq. 35

Part

The phononic contribution to the total thermal conductance is present and can be substantial;

in our case, we will only take a constant value of 𝜅 𝑝ℎ = 20 pW/K, estimated from the literature (see next chapter for more details), to perform the figure of merit calculations.

V.4 -Expression of the Figure of merit (ZT)

As mentioned in the previous sections, the efficiency of the energy conversion is characterized by the dimensionless thermoelectric figure of merit ZT. 137 By introducing the electrical conductance 𝐺 and the total thermal conductance 𝜅 instead of the electrical conductivity  and thermal conductivity 𝜆, Eq. 18 can be expressed as:

𝑍𝑇 = 𝑆 2 𝐺𝑇 𝜅 Eq. 36
Finally, if only the electronic thermal conductance is considered, an "electronic" figure of merit 𝑍𝑇 𝑒𝑙 can be extracted using Eq. 27, Eq. 28 and Eq. 31:

𝑍𝑇 𝑒𝑙 = 𝐿 1 2 𝐿 0 𝐿 2 -𝐿 1 2
Eq. 37

With all those keys in hand, in the next chapter we will analyze in more details the intrinsic characteristics of molecular junctions.

Chapter II -Analysis of molecular junction characteristics

I -Introduction

From an experimental point of view, the main measurements that are performed for the characterization of the transport properties of a molecular junction are the I-V curve and the conductance. In order to better evaluate and compare molecular systems, parameters are extracted from these data by applying several models and hypotheses. The two main parameters that are usually used to characterize a junction are the interfacial electronic coupling Γ, and the energy position of the first resonant peak 𝜖 0 . This first part describes the different models, their approximations and limitations.

II -Methods to extract the key molecular parameters 𝝐 𝟎 and 𝚪

II.1 -Ultraviolet photo-electron spectroscopy

Ultraviolet photo-electron spectroscopy (UPS) consists in measuring the kinetic energy of electrons extracted from a compound upon absorption of UV photons. This is used to define the molecular orbital energies located in the valence region of the electronic structure.

According to the photoelectric equation of Einstein, the kinetic energy 𝐸 𝑐 of the extracted electron is given by: 1

𝐸 𝑐 = ℎ𝜈 -𝐸 𝑏 Eq. 1

with ℎ the Planck's constant, 𝜈 the frequency of the incoming photon, and 𝐸 𝑏 the binding energy of the electron that represents the level of energy from which it has been extracted. This technique is applied in the molecular junction field to access intrinsic characteristics (Figure 1).

In that case, the molecules to be tested are deposited on a substrate (e.g., SAMs on gold) and the energy of the highest occupied molecular level, lying closest to the Fermi level of the metallic substrate can be retrieved by UPS. However, this technique does not allow for a precise estimation of the value of Γ, but it could be theoretically deduced from the broadening of the experimental peak.

This experimental technique has some drawbacks which are:

• the 𝜖 0 measured is not the one characteristic of a molecular junction because the molecule is only contacted by one electrode instead of two; the actual energetic position

of 𝜖 0 is expected to depend on the number of chemisorbed contacts. Nevertheless, it was theoretically 2,3 shown that the addition of the second electrode (to create a junction)

does not have a large influence on 𝜖 0 , so that 𝜖 0-𝑠𝑢𝑏 ≈ 𝜖 0-𝑗𝑢𝑛𝑐 , as evidenced for purely organic systems. This is mainly due to the fact that the dipole bonds are mostly independent (there is no overlap) for a long enough molecule, so that the only difference is coming from an image effect between the molecule and the metallic surface. 18 • it allows to estimate the value of the HOMO of the system but not the LUMO; this implies that if the current is driven by the LUMO, the estimated 𝜖 0 does not correspond to the level participating in the transport properties.

• even if we can get the HOMO's energy, there is no warranty that this level is the transporting level; indeed, it is possible that this level would be localized (on one part of the molecule) and therefore would not be a conducting level and that the delocalized levels contributing to the current would be the HOMO-1 or HOMO-2. The resolution of UPS spectra typically prevents estimating accurately the deeper-lying levels.

Inverse photoemission spectroscopy (IPES) should be carried out to estimate the LUMO energy

II.2 -Transition Voltage Spectroscopy

The transition voltage spectroscopy (TVS) method relies on the assumption that the offset energy of the HOMO or LUMO, 𝜖 0 , can be derived from a current-voltage measurement at low bias when the resonance with the molecular level is not yet reached. To apply the TVS theory, which is based on field emission physics (Figure 2), 4,5,6 the I-V data are plotted as:

ln ( 𝐼 𝑉 2 ) = 1 𝑉
Eq. 2

The minimum appearing in this curve is defined as the transition voltage Vt. This method was first based on the tunneling barrier of the Simmons' model when the barrier shape was changed by the applied bias from trapezoidal to triangular. 7,8 To be applicable, the transport needs to match two types of regime (Table 1):

• at low voltage, a direct tunneling mechanism, with the current 𝐼 ∝ 𝑉.

• as the voltage increases, a Fowler-Nordheim (F-N) tunneling-type mechanism, leading to 𝐼 ∝ 𝑉 2 . Later, the TVS was constructed considering the coherent transport model based on a single level. 4 According to this model, and considering Γ 𝐿 = Γ 𝑅 , the relation between the transition voltage 𝑉 𝑡 and 𝜖 0 is determined as: 2,3,5,6 TVS was also studied to describe asymmetric F-N plots depending on the applied bias polarity.

𝜖 0 = √3 2 𝑉 𝑡 Eq. 3
This induces different values for the transition voltages (Vt+  Vt-) needed to determine the energy of 𝜖 0 and to quantify the asymmetric voltage division 𝛾 of the molecular junction. 6,10 Nevertheless, this method relies on approximations. 11 If the transport properties can be described by a single level model, TVS yields a quite correct estimation of the energy level position 𝜖 0 if the parameter does not evolve significantly with the bias voltage. If several levels are involved in promoting the current, the applicability of TVS can be rather restricted.

Part II

Chapter II 129

II.3 -Single level model fitting

The most widely-used model to fit I-V curves is the single level model (SLM). Since the transmission as a function of energy is a priori not known, the hypothesis is made is that the level closest to the Fermi level of the electrodes is the only level responsible for the evolution of the current as a function of the applied voltage. In that case, the transmission has a Lorentzian shape (see Part II, Chapter I). Based on this hypothesis, the current is expressed as:

𝐼(𝑉) = 2𝑒 ℎ ∫ 𝜏(𝐸, 𝑉) [𝑓 𝐿 (𝐸 - 𝑒𝑉 2 ) -𝑓 𝑅 (𝐸 + 𝑒𝑉 2 )] 𝑑𝐸 ∞ -∞
Eq. 4

with the transmission adopting a Lorentzian shape:

𝜏(𝐸) = 4Γ 2 (𝐸 -𝜖 0 ) 2 + 4Γ 2 | Γ L =Γ 𝑅 Eq. 5
Assuming that the voltage drop is symmetrical between the two electrode interfaces of the molecular junction, in the zero-temperature limit, Eq. 4 can be solved analytically leading to:

𝐼(𝑉) = 4 Γ 𝐺 0 [tan -1 ( 𝑒𝑉 -2𝜖 0 2Γ ) + tan -1 ( 𝑒𝑉 + 2𝜖 0 2Γ )]
Eq. 6

If a system is measured at low temperature (typically T < 50 K), Eq. 6 is precise enough.

However, at RT, it is advised to use Eq. 4 as the difference in the estimations of 𝜖 0 and Γ can be in the order of 0.1 eV and few meVs respectively.

Importantly, if the transmission is not Lorentzian or if the transport peak is voltage dependent, the fit can be altered or voltage dependency has to be considered (Figure 3). When considering experimental or theoretical I-V curves with transmission shape more complex than the single level model, the applied voltage range used in the fitting is definitively a critical parameter. This is illustrated in Figure 3

II.4 -Modified single level model 𝝐 𝟎 (𝑽)

The single level model with 𝜖 0 depending on the bias voltage accounts for a Stark effect linked to the polarizability of the system. It can be described by a single level model:

𝜏(𝐸) = 4Γ 2 (𝐸 -𝜖 0 (𝑉)) 2 + 4Γ 2
Eq. 7

In Eq. 7, the 𝜖 0 is voltage dependent: 𝜖 0 (𝑉) = 𝑎𝑉 2 + 𝑏𝑉 + 𝜖 0-𝑖𝑛𝑖𝑡 . 𝜖 0 (𝑉) shifts as the bias voltage increases.

II.5 -The G+S method

In this section, we propose another method which consists in adding in the fitting procedure to estimate the energy level alignment or interfacial electronic coupling (parameters 𝜖 0 and Γ, respectively), the use of the linear conductance and the Seebeck coefficient. 12 The field of molecular electronics aims at developing functional nano-electronic devices by inserting molecules between two metallic electrodes. The I-V curves of such devices can be measured by several techniques such as CP-AFM for SAMs or MCBJ for single molecules. 13,14,15,16 Fitting the I-V curves of molecular junctions by simple analytical models is often done to extract relevant molecular parameters, i. e. 𝜖 0 and Γ, to build up useful propertyrelationships. However, such models can suffer from severe limitations in some cases and hence provide unreliable molecular parameters. This is illustrated here by extracting key molecular parameters by fitting computed voltage-dependent transmission spectra and by comparing them to the values obtained by fitting the calculated I-V curves with a typical Lorentzian model used in the literature. Doing so, we observe a large discrepancy between the two sets of values which warns us about the risks of using simple fitting expressions. Interestingly, we demonstrate that the quality of the fit can be improved by imposing the low bias conductance and Seebeck coefficient of the junction to be recovered in the fitting procedure.

Although UPS is typically used to infer the position of the HOMO electronic level with respect to the Fermi level of a substrate in a SAM configuration, the relative position of the electronic levels is expected to be different in a molecular junction configuration (due to additional interfacial electronic coupling, built-in fields 17 and image charge effects when a second electrode is introduced). 18 Part II At the present stage, 𝜖 0 and Γ parameters are mostly deduced by fitting experimental I-V curves, considering that the electronic transmission at the origin of the current is of Breit-Wigner resonance type (Lorentzian-type shape); this model considers a single molecular level (generally the HOMO or LUMO of the molecule) at the energy 𝜖 0 coupled to the electronic states of the electrodes (Figure 4). 19 The full width at half maximum of this Lorentzian reflects the amplitude of the electronic coupling Γ. The Lorentzian-type expression of the electronic transmission 𝜏(𝐸) at a given energy E is expressed as:

𝜏(𝐸) = 4Γ 𝐿 Γ 𝑅 (𝐸 -𝜖 0 ) 2 + (Γ 𝐿 +Γ 𝑅 ) 2
Eq. 8

In the case of symmetric contacts between the molecule and the two electrodes, Γ 𝐿 and Γ 𝑅 are equal (Γ 𝐿 = Γ 𝑅 = Γ) leading to the simplified expression:

𝜏(𝐸) = 4Γ 2 (𝐸 -𝜖 0 ) 2 + 4Γ 2
Eq. 9

It is important to stress that this model suffers from severe limitations in considering that only one molecular transporting level is present close to the Fermi level, as evidenced theoretically by Neaton and co-workers. 20 Using the DFT +  (i.e., static self-energy correction) formalism, some junctions show electronic transmissions which deviate importantly from a Lorentzianshape curve. The single-level model is indeed expected to break down for molecules displaying several close-lying energy levels in the transmission window. A second important hypothesis of this fitting model is to consider that the molecular parameters Γ and 𝜖 0 are bias voltage independent. In reality, and depending on how the applied voltage drops across the molecule, the level 19-g) energy may be bias-dependent. Recent calculations clearly demonstrated that Γ is a bias dependent parameter. 21 Moreover, this model does not account for the typical decrease in the intensity of the transmission peak with increasing bias. Finally, if there are several molecules in the junctions, the total transmission is not simply the sum of the transmission spectrum of the individual molecules so that a prefactor accounting for cooperative effects should be introduced. Accordingly, we will only focus on single-molecule junctions in the following. Despite its significant restrictions, this model has been widely exploited by

Part II

Chapter II 133 experimentalists to shed light on the electronic structure of molecular junctions and to establish relationships between the I-V characteristics and the electronic structure of the molecular junction. In this context, we aim at addressing the reliability of this analytical procedure using quantum chemistry (QC) as a tool able to provide in parallel I-V curves that can be fitted using the same models as previously described in literature and the actual calculated electronic parameters of a single molecule junction based on calculated transmission functions. This allows us to evaluate the accuracy of the fitting models in recovering a correct amplitude for the two parameters using I-V curves in comparison to the values calculated independently. It is important to emphasize at this stage that the scientific evidences given below are not dependent on the level of theory used. The present work highlights that much care has to be taken while using the single-level fitting procedure to extract the physical characteristics of a molecular junction.

In order to exemplify the problems that can arise from I-V curve fittings using this single-level model, we choose to evaluate the transmission properties of the trans-Ru(TMA)(CC-C6H4-SH)2 molecule (TMA = 1,5,9,13-tetramethyl-1,5,9,13-tetraaza-cyclo-hexadecane) grafted to gold electrodes. This system has been chosen as study case for several reasons. First, it is a symmetric molecular junction in which the metallic center Ru(II) is coordinated in a trans arrangement to conjugated (thiol)aryl-acetylide organic linkers. The rest of the central metal coordination sphere is completed by a tetra-chelating ligand via nitrogen atoms (Figure 5). (electrode-molecule-electrode) built with QuantumATK. 26 The sulfur atoms are anchored in the hollow position on the (111) surface of gold. We use ghost atoms on top of the gold surface (represented by the yellow spheres) to obtain a better electronic description of the surface. 22 This pseudo-octahedral environment stabilizes the metal center to fulfill the 18-electron rule conferring a high thermodynamic stability to the system. As lots of organometallic complexes, it displays a weak oxidation potential in comparison to most organic conjugated molecules, 23 a feature which promotes molecular levels close to the Fermi energy of the electrodes. Since this system is known in its molecular form 24 and closely-related molecular junctions have been successfully investigated experimentally by us and others 25 , it is highly suitable to experimental investigations. It has been chosen as case study among a large series of candidates that we have been screening since its calculated transmission properties show a high transmission peak close to the Fermi level with a quasi-Lorentzian shape, as required for this study.

The molecular junction is built by grafting the molecule to two gold electrodes via sulfur-gold bonds on the hollow site of the gold (111) surface (Figure 5). The transmission spectra and I-V curves have been simulated in the coherent regime at the DFT/NEGF level using the ATK package software from Synopsys. 26 The calculated transmission spectrum given in Figure 7 (right) at zero bias (V = 0.0 V) displays a Lorentzian peak centered at 𝜖 0 = -0.21 eV below the Fermi level. A very narrow peak (hence not contributing to the current) is also found at 𝜖 1

Part II

Chapter II 135 = -0.40 eV and is associated to other states involving the molecular backbone (Figure 6). 27 Nonetheless, the main transmission peak can be accurately fitted by a single level as shown below. Eq. 10

In the following, we will also refer to the Seebeck coefficient S and the conductance G. S corresponds to the bias that can be generated by a difference of temperature between the two

electrodes (S = -V/T) 19a),30,31,32,33 S has been measured for many molecular junctions. 34,35 This parameter can be calculated using the transmission spectrum at zero bias (in the linear regime approximation) and is formally defined at zero current I. A simplification of the current equation (Eq. 10) can thus be done leading to Eq. 11 (see Part II, Chapter I, section 2). In this expression, the Ln terms are incorporating integrals of the Fermi-Dirac function at equilibrium (feq) (Eq. 12 and Eq. 13) which describes the distribution of electrons/holes over energy states in the electrodes. 30,31,32,33 𝑆 = -1 𝑇 𝐿 1 𝐿 0 Eq. 11

𝐿 𝑛 = - 2𝑒 ℎ ∫ 𝜏(𝐸, 𝑉)(𝐸 -𝜇 𝑒𝑞 ) 𝑛 𝜕𝑓 𝑒𝑞 𝜕𝐸 𝑑𝐸 Eq.
12

𝑓 𝑒𝑞 = 1 exp(𝐸 -𝜇 𝑒𝑞 ) 𝑘 𝐵 𝑇 + 1
Eq. 13

The conductance G is calculated within the linear regime approximation as: 𝐺 = 𝐿 0 𝐺 0 Eq. 14 with G0 the quantum of conductance (77.48 mS). The coupling (Γ) and the energy (𝜖 0 ) can be deduced in the single-level approximation directly from each calculated transmission spectrum by fitting the peak closest to the Fermi level by a Breit-Wigner function. The obtained values can be considered as benchmark reference values for this study since they are directly issued from the DFT/NEGF transmission calculations (labelled later as Γ t and 𝜖 0-𝑡 ). One couple of values Γ t and 𝜖 0-𝑡 is thus obtained at each voltage (Table 2).

Table 2 -Γ  (eV), 𝜖 0- (eV), S (V/K) and G (G/G0) obtained by application of the singlelevel method to fit (E) at each applied voltage (V). The average values of the different parameters in the bias interval between 0.0 and 0.2 eV and between 0.0 and 1.0 V are the values reported in Table 3. The Γ t and 𝜖 0-𝑡 values given in Table 3 for several ranges of bias are the averaged values over the considered voltage range. The calculated I/V curve of the Au-[Ru(TMA)(CC-C6H4-S)2]-Au junction has been generated from calculated transmission spectra from 0.0 to 1.0 V by voltage steps of 0.1 V. It is represented in Figure 7a. The physical characteristics of the molecular junction can be now deduced by a fitting procedure of the I-V curve, using the singlelevel model to obtain Γ 𝐼-𝑉 and ϵ 0 𝐼-𝑉 (Eq. 9).

V Γ  𝝐 𝟎-𝝉 S  G  /G 0 0.0 0.
Table 3 -Γ (eV), 0 (eV), S (V/K) and G (G/G0) obtained by: a) application of the single-level method to fit (E) at each applied voltage (see Table 2); we report here the values averaged over the voltage range; b) fitting of the I/V curve, c) by adding G and S to the fitting procedure. The percentage errors are given in Table S2. -0.189 --G"G+S"/G0 0.113 --S"G+S" 88 --This was done using several voltage ranges in the I-V curve: (i) from 0 to 100 meV, a range of bias in which the current is proportional to the bias (linear regime); the transmission properties are slightly changed within this small potential window (no noticeable energy shift of ϵ 0 or Γ);

Bias voltage range (

(ii) in a larger voltage window ranging from 0 to 200 meV; (iii) for the total calculated I-V curve (from 0 to 1 V). The results of these fits are given in Table 3 together with the reference values Γ t and 𝜖 0-𝑡 previously obtained. Although the three I-V curves fits are of very good accuracy (R 2  1), the associated values ϵ 0 𝐼-𝑉 strongly deviate from ϵ 0-t , whatever the bias range considered.

Table 4 -Percentage error of Γ (eV), 𝜖 0 (eV), S (V/K) and G/G0 obtained by a) the fitting of the I/V curve (I/V) and b) by adding G and S to the fitting procedure ("G+S") (see Γ"G+S" % error 𝜖0-"G+S" % error S"G+S" % error G"G+S"/G0 % error 8. 6 13.2 0.9 0.9

The Γ 𝐼-𝑉 values are at least doubled compared to Γ 𝑡 . As expected, the conductance Γ 𝐼-𝑉 is close to the reference value since it is the fitted quantity. Interestingly, there is an order of magnitude difference between the Seebeck coefficient estimated from the different fitting ranges (𝑆 𝐼-𝑉 ) and the reference one (𝑆 𝑡 ) (Table 3). The high quality of the I-V curve fits highly contrasts with the poor description of some of the molecular junction properties (𝜖 0 , Γ and S). In the framework of the transition voltage spectroscopy, 𝜖 0 can also be estimated in principle as the minimum value in an I/V versus ln(I/V 2 ) plot. 5,6 This approach is not expected to provide reliable trends when the transmission peak is noticeably shifted with the bias voltage (Figure 8). However, in the case of weakly polarizable organic systems, such as oligophenylene derivatives (Figure 9), the procedure based on the I/V versus ln(I/V 2 ) plot to evaluate ϵ 0 is applicable. 2,3 at biases ranging from 0 to 0.9 V by steps of 0.1 V, as obtained using the same computational approach.

Deviations are expected for a large integration voltage window due to the fact that ϵ 0 and Γ are bias dependent properties (Figure 8).

A major improvement can be obtained by imposing both the low bias conductance and the Seebeck coefficient of the junction to be recovered with a chosen accuracy (here above 95 %).

The procedure adopted is explained in Figure 11. This procedure is referred as the "G+S" method in Table 3. Doing so, the differences against the reference values of ϵ 0 and Γ are strongly reduced, with the right order of magnitude recovered at low bias (Table 3 and Figure 7a). This improvement can be rationalized by highlighting that in the I-V fit procedure, a perfect Lorentzian-type shape of the transmission is assumed. Thus, the more the system deviates from it, the more ϵ 0 and Γ have probabilities to be erroneous even though R 2  1. Imposing the constraint of also recovering S introduces a tolerance to the deviation from a perfect Lorentzianshape that significantly improves the extracted parameters. Importantly, although the physical meaning of the parameters is improved, the corresponding I-V curve is apparently less satisfactory, with an R 2 that significantly deviates from 1. This relates to the fact that deviations are appearing when the linear regime approximation becomes irrelevant, i.e. at higher bias (Figure 12). The "G+S" procedure can be applied to experimental I-V curves (see Table 5 for applications to data reported in reference 36). and bipyridine (N-gold coordination). The value of 𝜖0-theo is taken from the same article and corresponds to calculations performed at the DFT+ level. The values "I/V" and "G+S" are obtained by applying the procedures described in the main text to the experimental data (I/V curves, Gexp/G0 and Sexp). Interestingly, the "G+S" method in green gives results closer to the experimental data (in blue) and closer to 0-theo for the tri-phenyl and bi-pyridine systems. For the bi-phenyl case, the system can be well described with the fitting of the I/V curve only, and give similar results as the "G+S" method, pointing to a quasi-lorentzian shape of the transmission, in agreement with computational studies published in the same article. Restricting the fitting to small voltage segments of the I-V curve could be seen as an alternative since the ϵ 0 and Γ are not supposed to significantly change in each bias window. As shown in Figure 13, this procedure fails in reproducing the right bias dependence of the transmission spectra.

In conclusion, we have demonstrated that using a Breit-Wigner resonance type model to fit experimental or calculated I-V curves to get an insight into the electronic and transmission properties of molecular junctions can lead to significant errors in the evaluation of the molecular parameters 𝛜 𝟎 and 𝚪. We have shown that the correct position and width of the Lorentzian resonance cannot be reconstructed from the fitted values, even for a case study which presents a quasi-ideal transmission peak shape. The level of accuracy of the fitting procedure in the description of the physics has been strongly improved when considering simultaneously the conductance and the Seebeck coefficient of the junction that are both accessible experimental parameters. 34 Part The "G+S" method allows to determine the 𝜖 0-𝑖𝑛𝑖𝑡 with quite good precision compared to the single level model for fits done at low voltage. Applying this modified single level model, we can retrieve the displacement of 𝜖 0 (𝑉). The reliability of the traditionally done Lorentzian fitting procedures can be improved by measuring the Seebeck coefficient in combination with the conductance.

III -Transmission and thermoelectric junctions

One of the main questions to be addressed in this thesis is to determine the type of transmission desirable to get better thermoelectric properties. To do so, we have tested different types of theoretical transmission that could be accessible with molecular junctions and evaluate the corresponding thermoelectric properties.

III.1 -Phononic thermal conductance 𝜿 𝒑𝒉

The phononic thermal conductance 𝜅 𝑝ℎ has been both experimentally and theoretically obtained 36,37,38,39,40,41,42,43 for different types of molecular junctions. Different ranges of values that have been published in the literature for 𝜅 𝑝ℎ are summarized in Table 6. The common feature between all these studies is that 𝜅 𝑝ℎ decreases as the size of the molecule increases. In this thesis, we will consider quite rigid organic and organometallic molecules ranging from 1.5 to 3.0 nm for which a reduction of 𝜅 𝑝ℎ is expected compared to small molecules (< 1.2 nm). Since the denominator in the equation to calculate 𝑍𝑇 depends both on the electronic and phononic thermal conductance, we will consider 𝜅 𝑝ℎ = 20 pW/K to simulate our ZT in the next chapters.

Indeed, if only the electronic thermal conductance without the phonon contribution is taken to calculate ZT, i.e. the electronic where as ZT = 3.10 -4 using 𝜅 𝑝ℎ = 20 pW/K, leading to a ratio between both Figures of Merit of 2.10 4 . In fact, in all the systems where 𝜅 el ≪ 𝜅 𝑝ℎ and 𝜅 𝑝ℎ is neglected, 𝑍𝑇 𝑒𝑙 ≫ 𝑍𝑇.

III.2 -Lorentzian shape transmission

Considering that the molecule electronic transmission is driven by a single level chemisorbed to the electrodes, different 𝜖 0 and Γ couples could be tentatively tried to evaluate which thermoelectric properties would prevail at RT, as shown Figure 14. Part

In the case of a single Lorentzian shape transmission, the possible (𝜖 0 , Γ) couples giving a Figure of Merit ZT > 1 and a thermopower S 2 G > 5 pW/mK 2 are contained between -0.11 and -0.04 eV for 𝜖 0 and between 1 meV and 10 meV for Γ (Figure 15), the best ZT being  1.6 (𝜅 𝑝ℎ = 20 pW/K). This simulation can conclude that an electronic transmission having a shape close to a single Lorentzian will not induce high thermoelectric efficiency.

III.3 -Lorentzian shape transmission and Fano resonance

The second type of theoretical transmission shape discussed here is a Lorentzian shape coupled with a Fano resonance. A Fano resonance is a type of quantum interference 44,45,46 (QI). The expression of the electronic transmission from a single level model with a QI reads: 47

𝜏(𝐸) = 4Γ 2 (𝐸 -𝜖 0 - 𝑉 2 𝐸 -𝜖 1 ) 2 + 4Γ 2 Eq. 15
With 𝑉 the coupling between the first transporting state 𝜖 0 and the localized state 𝜖 1 responsible for the quantum interference. Indeed, even with a Fano resonance quasi perfectly placed compared to the Fermi level, the influence on the thermoelectrics properties are barely noticeable. Changing reasonably the coupling value V or the position of the level 𝜖 1 hardly change this conclusion. A huge Fano resonance which has maximum and minimum sufficiently energetically spaced can moderately influence the thermoelectrics values but less than the QI effect presented below.

III.4 -Lorentzian shape transmission and anti-resonance

In this model, we also consider a two-level model having an anti-resonance as QI in the transmission function instead of a Fano resonance. This can be created also using equation Eq. Inspection of Figure 18 shows that ZT is enhanced in presence of anti-resonance, reaching up to 3 (left graph of Figure 18). In this example, 𝜖 0 validates the conditions between -0.62 and -0.10 eV (compared to -0.11 and -0.04 eV in the simple Lorentzian shape). Moreover, in Figure 18, we only modified the energetic position of the 𝜖 1 level, though changing the coupling V also changes the thermoelectric properties. Consequently, the possibilities to obtain interesting thermoelectric properties are enhanced if anti-resonances are present close to the Fermi level.

It should be noted that using Eq. 8 with high value of coupling V, the energy of the maximum of the transmission peak does not corresponds to the 𝜖 0 of the junction without QI (or when the coupling V is small) and is actually given by the following expression accounting for the coupling between the two levels.

𝜖 𝑏𝑜𝑛𝑑,𝑎𝑛𝑡𝑖 = 𝜖 0 + 𝜖 1 ± √(𝜖 0 -𝜖 1 ) 2 + 4𝑉 2 2
Eq. 17

IV -Transmission study: an overview on OPE molecules

In this part of the thesis, we intend to explore the nature of the factors potentially responsible for discrepancies between experimental and theoretical data based on DFT/NEGF calculations.

To do so, we investigate the change in transmission with modifications of the electrode shape and the geometry of the coordination at the anchoring site.

In order to compare the DFT/NEGF calculations to experimental results, the influence of the shape of the electrode and the contact geometry in the simulation on the transmission is an important factor to consider. The selection of benzenedithiol (BDT), biphenyldithiol (BPDT), the oligo-phenyl-ethynyl-dithiol (OPE2), and (phenyl-ethynyl-dithiol)-anthracene (OPE5) in Figure 19 as test molecules was motivated first by the fact that experimental and computational studies were available. 48,36,2,3 Secondly, the UPS measurements of OPE2 and OPE5 junctions shows that the deduced 𝜖 0 are shifted by 0.32 V, a noticeable value that simulation should recover (Table 7). Since the theoretical model is built to promote structural configurations maximizing the conductance, only the highest experimental conductance values were taken for comparison purpose; in the cases where the conductance was obtained by both the MCBJ and STM methods, a mean value of the highest conductance was chosen. We have first considered different shapes of electrode, and different anchoring geometries between the sulfur and the gold substrate. Their shapes are described in Table 8.

Part Part

The #1-top geometry coordination (first entry, Table 8) was the configuration chosen in the computational study in reference 48. The calculated transmissions are shown in Figure 13. For In their combined experimental and computational study, Kaliginedi and coworkers 48 did calculate the same (#1-top / #1-top) configuration and also the mixed arrangement #1-top on one side of the molecule and #1-hollow on the other side (see Table 8). In order to evaluate the influence of the adatom on the results, a third arrangement in which both ending sulfurs are bonded in a top configuration to one gold atom of each gold (111) surface (flat electrodes) has been considered, i.e., T/T configuration. Figure 21 5 and text for the labelling.

To overcome this apparent problem, a lot of computational studies are considering the sulfur atom also bonded to a hydrogen atom or a methyl group. 49,50 We thus investigated those configurations together with the enlargement of the tip shape of the electrodes as listed in Table 8. These results are reported in Table 9 for the case study molecules OPE2 and OPE5. For the OPE2 system in hollow coordination, the device with the gold tip configuration (#3-hollow) yields the best match with UPS measurements with 𝜖 0 (UPS) = -0.81 eV and 𝜖 0 (#3-hollow) = -0.80 eV. The conductance ratio between #3-hollow and the experimental value is 25 with G#3hollow/G0 = 3.83E-3 and Gexp/G0 = 1.55E-04. In all top configurations, the values of the calculated 𝜖 0 are far from the UPS values, but the conductance values are closer to the experimental data (the ratios are 8.5, 3.9 and 10.4 for #2-top, #3-top, #4-top respectively). For the OPE5 system the same conclusions can be reached; the #3-hollow coordination scheme is the closest to UPS experiment with 𝜖 0 (UPS) = -0.49 eV and 𝜖 0 (#3-hollow) = -0.49 eV, but with a conductance ratio about 18 times greater for the theoretical approach (G#3-hollow/G0 = 8.42E-3 and Gexp/G0 =

4.75E-04). In top position, the calculated 𝜖 0 are also far from UPS measurements, but the conductance are better matching the experimental values with ratios of 3.5 and 1.1.

Table 9 -Theoretical data extracted from the transmission spectra. the 𝜖 0 is in eV, G/G0 the conductance, "LUMO" corresponds to the LUMO energy position compared to the Fermi level in eV. The HOMO-LUMO gap (H-L gap) in eV, S the Seebeck coefficient in V/K. On the right are collected the different ratios compared to the experimental data listed in Table 7.

Position 𝝐 𝟎 G/G 0 LUMO H-L Gap S 𝝐 𝟎 Ratio G/G 0 Ratio Gap ratio OPE2 #1-top - - - - - - - - #2-top -1.
From this analysis, several observations can be done:

• it is possible that the molecular configurations in an UPS measurement (made on SAMs)

does not match the configuration in a junction (made by CP-AFM or MCBJ). This could be one explanation for the discrepancy between hollow configurations, giving 𝜖 0 values in agreement with UPS measurements and top configurations (with H or CH3), giving close values for the conductance but not for 𝜖 0 .

• another possible explanation of the discrepancy is that the experimental conductance values of molecular junctions are obtained from statistical data. Thus, the retrieved conductance is due to an average of several configurations for the anchoring group. A statistical study is thus necessary to fully rationalize the experimental data.

• the influence of the DFT functional could be an important point for the values obtained before. We tested this in #3-hollow configuration for OPE2. The results are shown in Figure 26.

I -Introduction

Among the different projects on molecular junctions involving our team, the group of Rigaut "organometallic and catalysis" at the ISCR in Rennes and the group of Scheer in the Physics Department in the University of Konstanz, a study on anchoring group has been devoted to (Au-C) covalent bonds in MCBJ's at low temperature. Indeed, in order to study in-depth, the electronic and magnetic transport properties and other physical properties of functional molecules, the ability to form single-molecule junctions which are stable and robust at low temperature and in vacuum are also of particular interest for choosing suitable end-groups.

Thiols (-SH), 1 amines (-NH2), 2,3 cyanides (-CN), 4 pyridines 5 are among the most commonly used end-groups for their coupling strengths and known contact geometries. Recently, there has been surge of interest in establishing direct gold carbon (Au-C) covalent bonds to achieve more robust and highly conducting molecular junctions. 6,7,8,9,10,11 Initial studies with Au-C bonded single-molecule junctions were observed by Venkataraman's group on methylene terminated oligophenyls and polymethylene chains which shows near resonant transport and enhanced conductance compared to other end-groups. 7 Hong et al. characterized the (oligo)-phenyleneethynelene (OPE) derivatives using Raman scattering 11 and Arasu et al. proposed bipodal platforms based on biphenylene with two Au-C bonds which shows very stable and interesting electronical properties of Au-C anchoring scheme. 12 The experimental fabrication of the junction from a solution is the critical step. For the creation of direct Au-C bond, one possibility is to functionalize the molecule with the highly toxic SnMe3 group which cleaves off on contact with Au electrodes. 6,7,13 The in situ cleavage of trimethylsilyl group attached to alkynyl moiety using tetrabutylammonium fluoride is an alternative. 11 Recently, another binding mechanism based on alkynyls forming direct Au-C bonds without the need for any precursor was reported. 14 So far, all these experiments have been performed in solution at room temperature. The characterization with photoemission spectroscopy has shown that it is also possible to form Au-C bonds in ultrahigh vacuum conditions. 15 Sheer's and Rigaut's groups have been collaborating to fabricate and characterize single-molecule junctions with direct Au-C covalent σ bonds at low temperature (4.2 K) under vacuum conditions. The computational studies were performed to give some insights for the interpretation of the experimental results.

II -Experimental data and characterization by Rigaut's and Scheer's groups

The group of Scheer prepared the OPAn molecule solutions with DCM as solvent under nitrogen atmosphere to make single molecule junctions with the MCBJ technique (see Part II, Chapter I). The yield of molecular junctions formed with low concentrations was very small, as seen in Figure 2. In contact with the gold electrodes, the hydrogen atoms at the outer ends of the molecules cleave off to form direct Au-C bonds. 19 This method avoids the use of any additional precursor which could leave non-desired groups in the solution. Electrical transport measurements were carried out at 4.2 K. First, they opened and closed the samples to form Au contacts and molecule junctions repeatedly. The conductance G = I/V was measured at fixed bias voltage across the junctions. The evolution of G is evaluated as a function of displacement between the gold electrodes. From these data, 1-D conductance histograms can be created. respectively. 14 It should be noted that the uncertainty in the determination of the plateau length in the MCBJ method is about 30 %. 20 In addition, they study the IET spectra of individual junctions by using a lock-in technique in a bias range from -300 mV to 300 mV and an AC modulation voltage of 3 mV.

III -Results and analysis

Figure 3 shows representative opening traces and corresponding 1-D histograms obtained from 300-500 traces measured at a bias voltage of 100 mV. In these histograms, the maxima are labeled HC for high conductance and LC for low conductance. The histogram of OPA2 has one conductance shoulder (HC) around 1.5.10 -4 G0 and another (LC) at around 2.10 -5 G0 (Figure 2b). The majority of OPA2 opening traces has flat plateaus near 10 -4 G0 with a length of 0.2-0.5 nm. Slanted plateaus of length up to 1.5 nm starting above 10 -4 G0 and decreasing upon

stretching are also observed. The latter can be attributed to the presence of dimerized OPA2 systems in the junctions (OPA2-OPA2) resulting of alkyne homo-coupling and the possibility of having a part of the molecule lying over the electrode surfaces, as sometimes observed for long molecules. 21,22,23 The OPA3 shows pronounced flat plateaus around 10 -4 G0 of length 0.6 to 1 nm as shown in Figure 2c, corresponding to a single OPA3 bridged molecule. The conductance at OPA3 molecular plateaus is slightly higher than that of OPA2, which agrees with the reported room temperature studies. 14 The OPA3 histogram has a HC close to 2.10 -4 G0 and the probability of stretched junctions extends until a LC of 10 -5 G0 (Figure 2d).

OPA4 exhibits five different kinds of molecular traces in contrast to OPA2 and OPA3. Figure 2e shows the frequently observed traces for OPA4 which are based on observed conductance, plateau length and probability of formation:

• Type Ilong plateaus up to 1.5 nm and conductance between 10 -2 G0 and 10 -3 G0.

• Type IIlonger plateaus up to 2 nm with a conductance around 10 -5 G0.

• Type IIIflat plateaus at higher (10 -3 G0) and lower conductance (10 -5 G0) with plateau length around 0.5 nm.

• Type: IV -Slanted plateaus starting at lower conductance (10 -4 G0). Very short lengths, less than 0.5 nm

• Type V -Slanted plateaus starting at higher conductance (10 -2 G0). Very short lengths, less than 0.5 nm Maxima of histograms are labelled as HC and LC for higher and lower conductance respectively. The probability of each type of traces observed from the OPA4 molecule is also shown.

To confirm the traces, OPA4 measurement on multiple samples and statistical evaluation were carried out by the group of E. Scheer. The molecular junction (OPA4) yield creation was 32 %.

The majority of traces falls into Type I with 38 % probability and Type II with 17 %, which agrees with the molecular junction conductance found in previous studies at ambient conditions. 14 The possibility of parallel π-π stacking of the molecules could give rise to Type I traces. 24 Type IV with 32 % can be associated with junctions chemically bonded to only one electrode. Type III 7 % and Type V 6 %, indicate that some molecular junctions could not be well connected to the electrodes and breaks apart easily upon stretching, resulting in short plateau lengths.

An increase in conductance (HC case) as the number of phenyl groups increases is found, which is in contradiction with the usual trend of exponential decay of conductance with molecular length for hydrocarbon molecules in off-resonant transport. 25 However, a non-exponential, but still decreasing conductance with length was previously found by another group for OPAn junctions in room temperature conditions. 14 The authors noted a pronounced variation of the most probable conductance based on the concentration of the solution, suggesting that the formation of molecular junctions connected in parallel might play a role. The strength of the experimental MCBJ setup at low temperature is to allow to provide further evidence for Au-C coordinated molecular junctions by analyzing the IET spectrum.

The IET spectrum is defined as d 2 I/dV 2 normalized with the differential conductance dI/dV to compare spectra gathered from junctions with different conductance. The spectra were measured over a fixed bias range of -300 mV to 300 mV and symmetrized using the formula 𝑦 = [𝑓(𝑥) -𝑓(-𝑥)]/2, to suppress contributions that might arise from conductance fluctuations or other phenomena since an IET spectrum is supposed to be symmetric. 26 The IETS for OPA3 and OPA4 junctions with conductance around 10 -3 G0 were performed. In the case of OPA2, the IET spectrum was numerically derived from the dI/dV curve of a HC junction due to the limited lock-in sensitivity in this conductance range. 169 in all molecules were also observed. 30,31,32 The strong signature of Au-C and C≡C vibration modes provides a direct evidence of stable Au-C coordinated molecular junction.

IV -Computational study IV.1 -Computational Details

Molecular geometry optimizations of the OPA systems were performed applying the density functional theory method (DFT with the revPBE functional using a Slater TZ2P atomic basis set with a small frozen core), as implemented in the SCM ADF suite of program. 33 Analytical vibrational frequency calculations were performed, including the calculation of the infrared and Raman intensities; the calculated frequencies ensure that all geometries correspond to a minimum on the potential energy surface. The densities of vibrational states were also plotted by using a Gaussian broadening for each vibrational state (with a FWHM = 3 meV).

The simulation of the vibrational properties of the Au-molecule-Au junctions and monocoordinated Au-molecule systems were simulated by an Au3 cluster model using the selfconsistent-charge density functional tight-binding method (SCC-DFTB) using the DFTB+, 20.1 code. 34 Additional calculations showed no sensible changes in the vibrational property results when increasing the cluster size up to Au13. The parameter sets "mio" and "auorg" were used. The geometry optimizations were converged imposing a force criterion of 1.10 -5 a.u. In all cases, no imaginary frequencies were found ensuring that all geometries correspond to a minimum of the potential energy surface (MODES procedure). The densities of vibrational states were obtained by using a Gaussian broadening for each vibrational state (FWHM = 3 meV).

The calculations of the transport properties of the molecular junctions were performed by coupling the DFT with the NEGF approach, as implemented in the QuantumATK 2020 package. 35 The geometries obtained for the isolated molecules were used to build the molecular devices. The distance between the terminal carbon and the flat gold surface was set at 1.

IV.2 -Computational Results

In order to get a theoretical insight into the understanding of the IETS spectra, quantumchemical calculations at the density functional theory (DFT level) for the isolated molecules and tight-binding density functional theory (DFTB) level for both isolated and gold-coordinated molecules were performed.

DFT phonon calculations of the entire molecular junctions could not be performed because of excessive memory requirements. In Figure 5 is represented the DFT simulated density of vibrational states for the OPA2, OPA3 and OPA4 bare molecules with their simulated infrared and Raman spectra. The vibrational mode frequencies are in good agreements with the experimental data. The calculated C≡C stretching mode is at 265 meV for the three compounds, and the IR C≡C experimental frequency is at 261 meV for OPA2 and OPA3. 40,36 Part II The DFTB calculations on the DFTB optimized structure overestimate by 18 meV this vibrational mode mainly because of the change in interatomic distances; nevertheless, the overall density of the vibrational state spectra is in really good agreement between DFT and DFTB results, as shown in Table 1. of the OPA2-A system reported in Figure 5 reveals that this peak is well-separated from the rest of the vibrational modes (same result for OPA3-A and OPA4-A). The IETS spectra show effectively this signature for the HC regime junctions. Interestingly, the IETS of OPA2 in this region show at least two peaks (Figure 4). Several scenarios were intended to explain this feature. First, only one end of the molecule was chemically anchored (case B and F). In the same energy region, their vibrational signatures are made of two distinct vibrational C≡C modes separated by less than 10 meV. The presence of the (C≡)C-H stretching above 400 meV as shown in Figure 6 would permit us to discriminate between the two configurations but this part of the IETS spectrum cannot be accessed because of experimental stability issues. Nevertheless, since the IETS were done on the HC regime junctions, the calculated substantial drop in the conductance for single-gold-coordinated systems rule out these possibilities (see below).

In a recent paper, van der Zant et al. reported ambient condition MCBJs measurements of the

OPAn series and suggested a sp 2 -hybridized carbon to explain the unusual ordering in conductance when increasing the number of phenyl groups such as the presence of two distinct high-count conducting behavior. 14 We performed the vibrational study of the arrangement C, D and E to study this possibility (Table 1). Doing so, the vibrational signature of a C≡C around 275 meV disappears and the C=C vibrational modes is lower in energy, 250 meV in case (C) and down to 220 meV for case (D) and (E) (in the same region as the phenyl vibrational region).

This result does not satisfactorily explain the IETS measurements.

Considering that some opening traces that show plateaus which are larger than OPA2 (Figure 3a) can suggest the presence of longer molecules, we investigated the possibility of the presence of a dimer OPA2-OPA2 resulting from C-C coupling. This reaction was observed experimentally for OPA3 by Klappenberger and co-workers by covalent homocoupling of terminal alkynes on Ag(111) metallic surfaces. 40 Gold is also able to induce the same reaction as recently reported for phenylacetylene homo-coupling. 37 The (OPA2)2 vibrational signature in the region around 275 meV is made of two distinct peaks separated by 15 meV and are due to the terminal vibrational motion of the C≡C and the C≡C-C≡C central units of the molecule.

This could explain the more structured IETS peaks for OPA2 measurements. 8). The corresponding transmission spectra are given in Figure 9. The calculated conductance are about 2 orders of magnitude higher than the experimental high conductance peak for the symmetric OPA2, OPA3 and OPA4 junctions as expected for this level of theory with this most stable coordination. 38 The calculated evolution of log(G/G0) with increasing number of phenyl rings is shown in Figure 10. The calculated exponential decay β = 1.05 is in line with the usual exponential decay reported in literature. This value is 1.58 for the parent oligophenyl dithiol junctions. 39 The OPA2-OPA2 junction was also added to the series and turns out to have a conductance value intermediate between OPA3 and OPA4.

Considering that some of the experimental opening traces associated with high conductance peak plateau over more than 2 nm, one plausible explanation is presence of dimerized OPA2 systems in the junctions (OPA2-OPA2) resulting of alkyne homocoupling. For the junctions in which OPA2-OPA2 molecules would be trapped, the possibility of having a part of the molecule lying over the electrode surface, as sometimes observed for long molecules, could explain opening traces in which a regular decrease of conductance is measured with the displacement.

The large panel of conductance measured for OPA2 reveals a large variety of molecular junction types in which OPA2-A is in competition with (OPA2)2. Such dimerization is surely occurring for OPA3, as already shown experimentally, 40 and for OPA4; however, the resulting length of the dimeric systems, especially for OPA4 diminishes their probability to form molecular junctions via the present MCBJ method. (see Table 1).

For the B scenario, only one side of the molecule is covalently coordinated to the electrode (the other side is at van der Waals distance with H-Au = 2.86 Å). The transmission difference between the A and B scenario is given in Figure 11. The 𝛜 𝟎 is quite similar between both cases, but the conductance diminishes by about 2 orders of magnitude (from 10 -2 to 10 -4 , see Table 2)

Part

V -Conclusions

This cooperative work allows us demonstrating that single molecular junctions presenting direct Au-C covalent bonding can be fabricated in situ. Interestingly, the study of the conventional conductance histograms reveals several peaks for all molecules corresponding to different binding configurations. The IETS and detailed phonon mode analysis accessed with DFT calculations revealed the presence of the Au-C vibration mode and the signature of the C≡C band, providing further evidence to our analysis. This shows that the Au-C linking scheme is suitable for the formation of molecular wires as platform for functional molecular devices also at low temperature. The study clearly establishes that although all three molecules belong to the same class of molecular wires with the same end-groups, the formed junctions can be quite different in geometry and in formation yield. Our analysis also demonstrates the importance of the combination of complementary measurement techniques to elucidate the actual nature of the molecular junctions.

Chapter IV -Organometallic systems to improve thermoelectric properties

I -Introduction

This second chapter deals with organometallic wires that have been considered to date mostly as conducting molecules, as rectifiers, or for spintronics applications. 1 A great advantage of organometallic complexes is that they are easily adaptable to target desired electronic properties and that they can handle concomitantly several properties (photophysical, redox, magnetic...).

The combination of metal atom(s), ancillary ligands, organic conjugated linkers, and anchoring groups offers a vast range of possibilities which can be further modulated by adding functional groups. Moreover, electronic levels associated with the main backbone and pending ligands can The field of molecular thermoelectrics has been incredibly productive over the last ten years. 2 The feasibility and potentiality of molecular modules to achieve thermoelectric effects were demonstrated in 2007. 3 For example, Solomon et al. reported recently the largest thermopower measured to date for a bicyclo-octosilane junction (S = 970 V/K), 5 i.e., thirty times higher than the C60 dimer representing the best molecular system ever measured before. 6 Unfortunately, its ZT is very weak (~10 -2 ) because of really low conductance values. Large enough G is undeniably necessary to allow for a substantial quantity of converted energy, thus to improve the power factor (S 2 G). Nevertheless, 𝜅 𝑒𝑙 increases in parallel to G at the detriment of ZT. A subtle balancing must be found in the search of ideal electronic features and necessitates robust predictive protocols. The present efforts to improve TE properties of molecular devices have been focused on the chemical engineering of the molecular backbone to directly impact the conducting properties G and 𝜏(𝐸).

The choice of organometallic wires which is to date almost unexplored in thermoelectrics 7 is based on the previous experimental results obtained by OMC and those of other groups that demonstrate that organometallic MJ can be formed. They are sufficiently robust and show enhanced conducting properties compared to organic wires of similar length and rigidity. 8,9 This is notably the case for electron-rich systems incorporating transition metal (TM) atoms of group VIII (such as Fe and Ru) in the conjugated backbone. In such systems, the participation of the metal atom in the highest occupied MOs induces a lowering of the oxidation potentials and thus a closest proximity to the Fermi energy. Additionally, MOs similar to the conjugated MOs in the electronic structures of pure organic conjugated wires, and responsible for charge transport are also present in these systems. They are fully delocalized over the molecule and are coupled to the electrodes near the Fermi energy. They present an antibonding character between the metal d orbitals and the p orbitals of the neighboring carbon atoms. 11,12 The main strategy proposed to date to enhance thermoelectric properties has been to add pendant units connected to a conjugated organic wire though with limited possibilities of modifications. This increases also typically to a large extent the steric hindrance, which is detrimental to device fabrication. 3a On the positive side, this induces QI by a partial electronic confinement over the lateral substituent which creates an abrupt decrease in the electronic transmission near the Fermi level that can, in certain conditions, boost the thermoelectric

properties. Here, instead of introducing additional groups to induce QI, we took advantages of molecular levels already present in the organometallic MJs, i.e. metal-centered and ligandcentered levels. A second strategy, to be combined with the first one, is to conceive multiplepath wires that yield constructive or destructive QI effects due to the phase differences between the multiple conduction pathways. 5,10 This theoretical concept has been fully demonstrated experimentally and can be transposed to organometallic wires. 5,11 Remote control of the thermoelectric properties to alter the electronic conductance is also achievable. Indeed, if the electronic transmission is significantly lowered, the thermoelectric properties will concomitantly decrease. In that respect, external stimuli, such as light, 12,13 mechanical force, 14 magnetic field, 15 electrical field, 16 and electrochemical potential, 17 have already been used to change the conductance of molecular devices. This is thus directly transferable to TE organometallic molecular devices.

Introducing metal atoms in the conjugated path or in its vicinity allows to:

• Expand the possibilities of structural and electronic modulation compared to organic systems in view of the huge diversity of metal and ligand combinations.

• Benefit from the inherent properties of the transition metals (redox, magnetic, optical).

• Increase easily the number of functionalities by ligand modifications or incorporation of additional chemical functions. We have concentrated our efforts on organo-ruthenium and organo-iron systems. They have been shown to engender much better conducting properties than the rest of the transition metals. 9,18 The good overlap and energetic alignment between Ru and Fe orbitals and carbon atomic orbitals allows to keep the electronic delocalization when the metal centers are incorporated in organic conjugated wires and to decrease the redox potentials. The long-term plan would to be investigate the different systems shown in Figure 2. This family of platforms already exists and exhibits various ligand substitution schemes. In this thesis, we will concentrate on case studies in order to better target the factors that drive the properties for future

developments. The quantum-chemical calculations are done at the DFT level using a GGA functional, triple- atomic basis sets with two polarization functions. The same functional is used for the DFT/NEGF calculations for sake of coherence.

Figure 2 -Structures of candidate organometallic wires

Many possibilities exist for the choice of the conjugated spacers. Some are generating competition among the conducting paths that are creating QI in the transmission, and hence possibilities of TE enhancement. A lot of organic units can be combined to create conjugated linkers of varying nature and length, see Figure 3. The anchoring group also plays an important role notably in defining the strength of the interfacial coupling (Γ), and has thus an impact on the transmission 𝜏(𝐸). In this study, we deliberately concentrate our efforts on thiol-gold anchoring which provides a warranty of success on the experimental point of view. Calculations in which both the atoms of the organometallic compound and those of the electrodes are treated explicitly at a quantum level can constitute a heavy computational cost, even for DFT. To circumvent this issue, we study the possibility of using a Density Functional Tight-Binding (DFTB) method that consists in an approximate and parameterized DFT scheme (see Part II, Chapter VI). DFTB methods allow to run calculations with costs reduced by two orders of magnitude compared to DFT.

II -Functionalization of the Fe II (cyclam)(C 2 Ph-S) 2 system

Our first case study system is a Fe II (cyclam)(C2Ph)2 molecule (with cyclam = 1,4,8,11tetraazacyclotetradecane) which could be described as a three-phenyl oligo(phenyleneethynylene), OPE3, in which the central phenyl is replaced by an iron center chelated by a cyclam ligand (see Figure 4). We applied different types of chemical modifications on the cyclam and the phenyl groups of this initial system, Fe-bare, to evaluate the degree of tuning of the electronic transmission depending on the added/modified chemical groups. The studied systems, that have been chosen to screen different possible modifications, are named Femethoxy, Fe-fluorine, Fe-benzene and Fe-CH2COOH and are described in Figure 4. In this first part, two levels of methodology were used: molecular DFT calculations on cluster models and DFT/NEGF in junction configurations. Indeed, since one of the goals of this work is to perform at the end a chemical design to target molecular junctions presenting enhanced thermoelectric properties, numerous simulations need to be done. We thus investigated the reliability of DFT clusters models which are much less resource-demanding to account for some electronic features that could be used to determine the systems to be investigated more deeply.

II.1 -Description of the transport properties using Au 13 cluster-type molecular model compared with junction DFT/NEGF calculations

The cluster-model used to mimic a junction configuration consists in replacing the gold semiinfinite electrodes by clusters of several gold atoms. After a preliminary study in which different sizes and coordination modes were investigated, Au3 clusters have been chosen since this allows recovering geometric and electronic features of the reference DFT junction-type simulations. Indeed, if the cluster size is smaller the ordering of the last occupied moleculecentered and gold-centered levels is reversed compared with the reference DFT junction-type simulations. The odd number of gold atoms allows to form a thiol contact with the molecule a)

Part II

Chapter IV 189 without charging the total system. Finally, the triangular faces of the Au13 gold cluster give access to hollow coordination for the anchoring of the sulfur atoms. 19 The electronic structures of the five compounds sketched in Figure 4 in a junction-like cluster configuration are given in Figure 5. In these molecular calculations, the Fermi level is approximated to the energy of the last occupied gold-centered levels (in yellow, Figure 5). The molecular levels in blue (Figure 5) are those containing a high contribution of iron d orbitals (see Table 1). The analysis of the MOd levels in terms of atomic orbital participation given in Table 1 reveals in all cases that two of these three MOs are strongly metal centered and poorly delocalized over the conjugated path. We will show later that these orbital shapes are at the origin of Fano resonances. The difference in energy between the MOd and the last Au-centered MOs which are associated to the Fermi energy of the model junctions can vary importantly depending on the chemical modification performed (Table 1). Considering Fe-bare as the reference system, the fluorination of the methyl groups of the cyclam importantly stabilizes these levels (by up to 0.54 eV). This substitution by electron-withdrawing atoms is detrimental to the conductance.

The substitution of the hydrogen atoms on the amine is limited from a synthetic point of view to few possibilities (phenyl, carboxylate). The effect on the electronic structure is more limited but the MOd Fe-Phenyl get slightly closer to the MOAu levels. Another possibility to reach this is to modify the organic linker notably by adding electron donating groups (OMe here). It has effectively the same effect.

To evaluate the reliability of this cluster-type model, DFT/NEGF calculations were performed on the molecular junction configurations incorporating the same molecules. The computed transmissions as well as the main transmission eigenstates for the Fe-bare case are shown in Chapter IV 192 periodic potential described as plane waves). 20 The transmission eigenstates are obtained by propagating the linear combination of Bloch states:

∑ 𝑒 𝛼,𝑛 𝜓 𝑛 𝑛 Eq. 2
And by finding the 𝑒 𝛼,𝑛 that diagonalize the transmission matrix  𝑛,𝑚 , (i.e. the eigenstates) leading to:

∑  𝑛,𝑚 𝑒 𝛼,𝑛 𝑚 = 𝜆 𝛼 𝑒 𝛼,𝑛 Eq. 3
with 𝜆 𝛼 the transmission eigenvalue.

These eigenstates are highly comparable to the Au13-Fe-bare-Au13 MOd shown in Figure 4. The comparison between the relative energies of the MOd (Table 1) and the energy of the main transmission states given in Figure 6 allows us to conclude that the cluster model is reproducing qualitatively the main features:

• The ordering is globally the same. For the systems in which the difference between the energies of the MOd was small, an inversion can be found (Fe-methoxy and Febenzene).

• The eigenvalues have the same order of magnitude but the differences are much reduced in the DFT/NEGF scheme (range of energy: 0.04 to 0.66 eV vs 0.15 to 0.30 eV).

• A noticeable discrepancy is the ordering of the highest two MOd corresponding to a transport level and a localized one.

The device calculation confirmed that Fe-fluorine junction has a transmission main peak too far from the Fermi energy to promote a current at moderate bias voltage. For all the other systems, all transport levels which have a substantial metal character and are strongly coupled to the electrodes are really close to one another in each system (with a variation by less than 50 meV). This could be interpreted as a pinning effect in the system, as shown previously in the literature, notably for organic systems. We can thus conclude that the Au13 cluster-type model applied to a series of compounds provides electronic structure data that give qualitative trends on their transport properties. Even though not quantitative, it allows to exclude systems in a prospective study and thus limit the number of DFT/NEGF calculations to be performed. It is also interesting for large molecular systems for which the cost of DFT/NEGF calculations can become prohibitive.

II.2 -Thermoelectric properties

The key electronic and thermoelectric parameters extracted from the calculated transmission at zero bias are collected in Table 2. The 𝑍𝑇 was calculated considering the thermal phononic 𝜅 𝑝ℎ contribution at 20 pW/K. The thermoelectric properties are moderately affected by the chemical modifications performed except for Fe-fluorine whose conductance is really weak compared to

the others. For instance, excluding Fe-fluorine from the comparison, the ZT ranges between 0.256 for Fe-bare to 0.337 for Fe-benzene while the Seebeck coefficient S ranges from 102

V/K for Fe-bare to 108 V/K for Fe-methoxy.

Table 2 -Electronic and thermoelectric parameters for the five systems presented in Figure 4 extracted from the computed transmission at zero bias (DFT/NEGF level). S is the Seebeck coefficient in V/K, S 2 G the thermopower in pW/mK 2 , G/G0 the ratio of conductance over the quantum of conductance G0, 𝜅 𝑒𝑙 the electronic thermal conductance in pW/K, Qcal the maximum power that can be extracted from one electrode ( These results clearly reveal that substitutions that only alter the energy of the conducting levels and their atomic contributions will not provide sufficient changes to really modify the thermoelectric properties. Alternative routes need to be evaluated, notably the presence of quantum interferences in the transmission features as mentioned in the introduction.

III -Influence of meta/para thiol coordination on transmission properties

Among the different possibilities to generate QIs, multiple-path transmission is a possibility that has fully been demonstrated. 11a) We have investigated role of the meta and para coordination of the phenyl groups of the linkers in the organometallic framework, to assess whether QI can be created and what is the impact on the thermoelectric properties. The Fe-bare system was used as case study. Three coordination modes have been tested:

• para-para: both thiols are in para coordination with respect to the external phenyl rings (pp).

• para-meta: one thiol is in meta configuration and the other one in para (pm).

• meta-meta: both thiols are in meta coordination with respect to the phenyl rings (mm).

Part

The electronic transmissions and the transmission eigenstates of the pp-Fe-bare, pm-Fe-bare and mm-Fe-bare are given in Figure 7. The main impact of changing the para-para linkage to the meta-meta coordination mm-Fe-bare is a decrease of the coupling strength (Γ) between the molecular orbitals (notably 𝜖 0 ) and the gold surface, resulting into an important depletion of the device conductance from 0.1021 to 0.0026 G/G0. The pm-Fe-bare system shows a mixed behavior with the principal peak of transmission exhibiting a larger coupling than for the mm-Fe-bare system but smaller than for the pp-Fe-bare. We do retrieve the same eigenstate shapes in the three molecular junctions. The study of the thermoelectric properties depicted in Table 3 reveals that the mm-Fe-bare system yields a high Seebeck coefficient with S = 346 V/K, and ZTel = 4.5 which is high. Nevertheless, as stated before, it is important to include as well the thermal phonon conductance to have a complete picture. As shown earlier, 𝜅 𝑝ℎ is typically on the order of a few pW/K. The global ZT would be 0.357 by considering 𝜅 𝑝ℎ = 20 pW/K which is an estimation based on literature (see Part II, Chapter I and Chapter II). The system would be quite interesting with a non-negligible thermopower 𝑆 2 𝐺 = 26 𝑝𝑊/𝑚𝐾 2 if the real 𝜅 𝑝ℎ turns out to be smaller.

The conjugated organic linkers play thus a crucial role in inducing QI that importantly modifies the thermoelectric properties. Nevertheless, if the conductance is too small the ZT and the 𝑆 2 𝐺 thermopower are not improved even though the Seebeck coefficient is importantly increased (102 to 346 V/K) in that study. The nature of the metal center(s) and its coordination to the linkers are also of high importance in the molecular design.

IV -Screening of organometallic systems and their transmission

We have thus screened different coordination metal structure, with different ligands, one anchoring group (thiol), and different coordination modes (meta and para). The results are detailed below as a function of the nature of the metal ligands. For each junction containing thiol anchoring, the sulfur is in hollow coordination with respect to the gold electrodes.

IV.1 -Metallocene molecular systems

Ferrocene-containing molecular junctions: coordination to the electrodes via the same ligand Since its discovery in 1952, 24 the Ferrocene (Fc) has been used in a really large variety of domains because of its interesting redox, donor, chelating properties notably. It has been studied

in covalent molecular junctions both theoretically and experimentally with different linkers and anchoring groups. 25,26,27,28,29,30 These studies highlight the important role of the conformation and of the path of transmissions. To the best of our knowledge the phenyl-ethynyl linker with a thiol anchoring has never been investigated. The organometallic configurations that have been studied up to now present covalent ethynyl-metal bonds with the phenyl-ethynyl linker. The ferrocene incorporation is different since the coordination to the linkers is done via its cyclopentadienyl (Cp) groups. Conformational considerations were investigated by Camarasa-Gómez and coworkers. 28 The role of meta and para configurations of the linker-anchoring groups was also investigated.

First, the configurations in which the two linkers are substituting the same Cp group of the ferrocene were considered (Figure 8). In that configuration, the metallic center is lying out of the conjugated path. The associated transmission spectrum is displayed in Figure 9. The features of the transmission spectrum of the para systems arSe really far from the single-level model lorentzian-shape transmission. A really similar transmission feature was indeed calculated by Camaras-Gómez and coworkers for a phenyl-thiol linker. 28 It consists in two parts:

• at E-EF = -0.54 eV: Fano resonance. The associated transmission eigenstates n°3 and n°4 in Figure 9 is mainly due confinement on the ferrocene itself. The transmission pathways are also represented Figure 9, bottom image. The transmission pathways correspond to the casting of the transmission coefficients into local bond contributions.

The local bond contribution can be positive and negative. A negative value means that the electron is back scattered along the local bond. The direction of the arrows shows the flow of the electrons. 31 We can see here a lot of scattering and back scattering around Considering that the calculated Fermi level is not perfectly accurate in DFT, we have shifted the energy of the Fermi level to evaluate the range of thermoelectric values that could be expected. A similar study was performed by Lambert et al. recently. 32 The thermoelectric properties calculated as a function of this energy shift are represented in Figure 10. This procedure can also be seen as a simplified simulation of the impact upon gating the junction. Considering that the Fermi energy lies at the optimal value for ZT, we computed next the ZT values for only a single resonant level with a Lorentzian shape to understand the role of the anti-resonance in the thermoelectric properties of Ferrocene-1p. We made the hypothesis that ϵ 0-model = ϵ 0-1p-optimized = -0.082 eV (corresponding to a shift E = -0.23 eV). Because the anti-resonance is expected to modify the coupling strength of the transporting level, we tested different  values ranging from 1 meV to 50 meV. The resulting evolution of ZT is shown Figure 11. The best ZT that is obtained by this process in which no anti-resonance is takin into account is 1.4. The value obtained with the computed transmission of Ferrocene-1p is ZT = 2.74. This demonstrates that anti-resonance can be beneficial to the enhancement of the thermoelectric properties compared to a single Lorentzian shape transmission.

Part II

Chapter IV 202 Ferrocene-containing molecular junctions: coordination of electrodes to each ligand The Fc can be incorporated into the molecular junction via its Cp ligands. In the precedent study, the phenyl-ethynyl linkers were coordinated to the same Cp ligand of the Fe. In this part, we investigate the case in which each electrode is coordinated to a different Cp, as represented in Figure 12. The calculated transmissions are given in Figure 12 for the para and meta coordination of the thiol group on the phenyl groups. In these configurations, the conjugated pathway encompasses the metal center; the systems are referred to as Ferrocene-2p and Ferrocene-2m for the para-para and meta-meta coordination, respectively. The transmission spectra are similar to their counterpart Ferrocene-1p and Ferrocene-1m. The extraction of the thermoelectric values leads to ZT of 0.028 and 4.10 -4 respectively (Figure 12).

System

The inclusion of the Fe metal in the conjugated path moderately increases the thermoelectric features compared to the previous compounds in which the conjugated path was settled through the same Cp ligand. The main differences are:

• For the Ferrocene-2p, the main resonant peak is broader, and the anti-resonance lies above the Fermi level compared to Ferrocene-1p.

• For the Ferrocene-2m, the overall conductance and transmission maxima are much smaller compared to the Ferrocene-1m (G1m/G0 = 1.27.10 -5 and G2m/G0 = 1.37.10 -6 ), and the transmission maximum of the main transport level drops from 𝜏 1𝑚 = 0.360 to 𝜏 2𝑚 = 0.008).

As shown Figure 13, the variation of the Fermi level E to account for the limitations of DFT Part

The incorporation of the metal directly into the conjugated path does not appear as a breakthrough strategy to improve the thermoelectric properties. In the next section, the functionalization of the Cp ring will be investigated in the search of means to enhance thermoelectric properties.

Ferrocene-containing molecular junctions: functionalization of the Cp rings As seen earlier, the functionalization of the Fe(cyclam)(CC-Ph)2 systems were not conclusive for tuning the electronic transmission. Since the electronic scheme is quite different in Fcjunctions, chemical modifications are suspected to induce a different response. We choose to investigate this issue using the most promising system as reference, i.e. Ferrocene-1p. The hydrogen atoms of the Cp ligands have been substituted by methyl groups (Ferrocene-1pmethyl, representation in Figure 14). The expected consequence is to lead to a more electronrich metal center (thus more easily oxidizable) and thus to shift the main transport peak closer to the Fermi level while keeping the anti-resonance features in the transmission. The resulting transmission is shown in Figure 14.

Part Interestingly, even if the electronic structure of the isolated systems shows a destabilization of the HOMOs upon introduction of the methyl groups, the incorporation in the junction almost completely annihilates this molecular feature due to a pinning effect, as seen in Figure 14.

Another possibility to shift the transmission peaks closer to the Fermi level is to apply an external electrostatic field. This is investigated in the next section.

Electrostatic gating of the Ferrocene-1p-methyl system A dielectric gate, positioned to induce an electric field perpendicular to the transport direction through the molecular junction, was introduced in the computational model (Figure 16). Other gate positioning could be implemented but this one corresponds to experimental setups. 33,34 The expected result is to induce an energy shift the molecular-centered levels closer to the Fermi level to enhance the thermoelectric properties. The Ferrocene-1p-methyl which is the compound whose transmission peak lies closer to the Fermi level was used as test molecule.

Upon a negative increase in the gate voltage, the main transmission peak is effectively displaced toward the Fermi level, from -0.28 eV to -0.16 eV. Interestingly, this is associated with a decrease of transmission for this peak, from 𝜏 = 0.99 to 𝜏 = 0.62 upon a gating variation from 0.0 V to -2.0 V. In contrast, the second part of the anti-resonance which introduces a depletion close to the Fermi level is very moderately affected by the gating with an associated shift from -0.12 eV to -0.10 eV, with the gate set at 0.0 V and -2.0 V, respectively.

The inversion of the electric field orientation (positive gating) has a stronger effect on the main transmission peak which shifts from -0.28 eV to -1.06 eV for 0.0 V to +2.0 V gating, respectively; this is accompanied by a slight decrease in its transmission probability (𝜏 -2.0𝑉 = 0.62 and 𝜏 +2.0𝑉 = 0.66). Contrarily to the negative gating, the energy of the depletion of the anti-resonance moves significantly from -0.12 eV at a gating of 0.0 V to -0.88 eV at +2.0 V. Ruthenocene-containing molecular junctions: Ruthenocene-1p and Ruthenocene-2p

A similar study was performed on ruthenocene-containing systems (ruthenocene = RuCp2, 36 see Figure 18 for the representation of the complete molecule) to evaluate the impact of the nature of the metal. Ruthenocene is presenting an irreversible oxidation 0.41 V beyond the ferrocene oxidation. 37 It thus appears interesting to assess whether this change would also be

I -Introduction

In order to better predict the (thermo) electronic properties using computational approaches to perform molecular design and obtain molecular junctions with enhanced thermoelectric properties, the evaluation of the inconsistencies between computational and experimental results is compulsory. In collaboration with our experimental collaborators, i.e., Rigaut's group for the synthesis and the characterization of the molecules, Vuillaume's group for the SAMs fabrication and physical characterization (CP-AFM, UPS and XPS), and Scheer's group for the MCBJs study at low temperature, we have started a combined computational and experimental study of molecular junctions incorporating two different organometallic compounds.

The two molecules that are represented in Figure 1 have been chosen for a set of reasons. The Ru molecule which will be called Ru-dppe has been studied by the group of S. Rigaut over the last two decades for its conducting properties. Since their pioneering work with the group of Frisbie, 1,12 the Ru(dppe)2 unit has been integrated in several molecular junctions. 2,3,4,5,6,7,8 In most of the cases, the Ru(dppe)2 containing wires are giving higher conductance than organic conjugated wires of comparable length. It is one of the most studied organometallic molecular junctions to date and can thus been considered as a case study.

The prospective computational early work done in the group revealed that the Fe(cyclam)(CC-Ph-S-)2, noted Fe-cyclam, as a neutral molecule (iron formally in +II oxidation state) presents a really interesting Seebeck coefficient of 100 V/K at 300 K while having a theoretical conductance of about 0.1 G/G0. Interestingly, at the molecular level, the most stable species is the magnetic cationic one in which the iron is in the +III oxidation state. The synthesis of this organometallic platform is well described in the literature. The same linker (phenyl-ethynyl)

and the same anchoring (thiol) have been chosen because they are commonly used, wellmastered on an experimental point of view, and for sake of comparison.

Group of E. Scheer

The Ru-dppe substituted by EDMS protecting groups, as shown in Figure 1, was synthesized by the group of Rigaut (ISCR, Rennes). The MCBJs were fabricated by dropping a THF solution of Ru-dppe (1 mM) to which tetra-n-butyl ammonium fluoride (TBAF) was added to remove the EDMS protecting groups. The solvent was evaporated in a nitrogen atmosphere.

The electrical contacts were made using copper wire on gold electrodes with conducting silver paint. A full description of the setup and description is available in reference 9. 3. These curves are uncommon and not constituted of a linear part followed by a quadratic evolution, which is the typical profiles that are encountered in most molecular junctions. 10,11,12 The linear conductance extracted from these curves give a conductance GHC of about 4.9.10 -3 G0. The SLM procedure was tentatively applied to get the key parameters ϵ 0 and Γ but it failed even in the linear regime. The TVS method was also unsuccessful to estimate ϵ 0 since no minimum could be found as shown in Figure 4. The ϵ 0-final of -0.68 eV is more than twice the value at zero bias (ϵ 0-init = -0.30 eV). As said previously, this is certainly exaggerated by the fact that the possible bias voltage dependency of Γ and the reduction of transmission as the bias V increases were not considered.

Unfortunately, this could not be accounted for since it results in an increase of the quantity of parameters that lead to an overfitting of the data. close to GLC  2.10 -5 G0 whereas the high conductance, GHC  2.10 -3 G0, has the same order of magnitude as for the first batch (GHC  4.9.10 -3 G0). Selected I-V curves are extracted in Figure 8. Different contacts between the molecule and the gold electrodes can explain the important difference in the I-V curves between the two batches of measurements. This could be also due to a possible degradation of the solid-state powder. A new series of measurements is definitely needed to tackle these apparent discrepancies. UPS and XPS measurements were also performed as for Ru-dppe SAM. The XPS does not detect any feature which would indicate the presence of the CF3SO3 -counter ion (absence of F1s and oxidized sulfur SOx). The EDMS group was also not observed (no Si traces). For this new sample, the measured SAM thickness is d = 3 nm. It could be indicative of molecules lying on the molecular monolayer, thus not grafted to the gold substrate. The XPS and UPS data have to be taken with a grain of salt and further experiments would need to be done to confirm the present data. The UPS measurements point to 𝜖 0 = -0.73 eV (Figure 19).

IV -Comparison of the I-V characteristics between the two sets of experimental data

For Ru-dppe, the first batch of MCBJ measurements performed at low temperature by Scheer and coworkers are leading to quite different characteristics of the conducting properties compared to the SAM configuration studied by Vuillaume and collaborators. The conductance is also much lower for the SAM setup compared to the single molecule junction configuration, a feature that is often found but not fully understood to date. 24,25,26,27 In order to facilitate the comparison, current has been normalized in the next figures. Interestingly, this rescaling of the I-V curves shown in Figure 20 A good agreement between MCBJ and CP-AFM is also found for the Fe-cyclam system in Figure 20, right. Indeed, both SLM fit showed that 𝜖 0 (0 -1.0 𝑉)  -0.70 eV. V -Computational study of the Ru-dppe and Fe-cyclam junctions

Computational details

The geometries of the molecular systems were optimized in gas phase using the ADF program, • MCBJ: the mean value of ϵ 0 using the SLM up to V = 0.5 V when applicable gives ϵ 0-mean = -0.29 eV and Γ mean = 3.86 meV (see Table 1).

• CP-AFM (SAM): SLM with ϵ 0 = -0.32 ± 0.06 eV and Γ= 0.89 meV for a selected I-V curve.

• DFT/NEGF: the fit of the transmission gives ϵ 0 = -0.23 eV and Γ= 37 meV

Remarkably, even though the coupling and thus conductance in the computational study is one order of magnitude higher, the value of ϵ 0 is in reasonable agreement with both experimental series of measurements. This agreement is also found for junctions with organic conjugated molecules. Furthermore, the use of the #3-hollow binding configuration (i.e., adding a specific hollow tip) for the gold electrodes shape enhances the concordance between experiment and theory, as shown in Part II, Chapter II on the OPE-derivatives. It was not applied here due to the excessive size of the system. Nevertheless, a computational study reproducing quantitatively the experimental results would necessitate molecular dynamic calculations to gather a statistical bench of coordination geometries (different arrangements of the gold electrodes would also need to be considered).

The construction of a precise DFT/NEGF I-V curve requires to calculate the transmission at each bias voltage (considering that 𝜏(𝐸, 𝑉 = 0 𝑉) ≠ 𝜏(𝐸, 𝑉 > 0.1 𝑉)). Because of the important size of the system, this could not be performed for Ru-dppe. We only computed three additional transmission spectra at 0.2, 0.6 and 0.8 V, using the revPBE functional (Figure 23) in order to at least evaluate the degree of evolution of the transmission properties upon bias.

The normalized I-V curves with the computational results and the experimental ones are displayed in Figure 23, up to 0.8 V. Considering, three values for only one junction geometric configuration is clearly not sufficient to reproduce experimental I-V curves.

The calculated transmission is hardly modified between 0 and 0.2 V while higher voltages do shift the energy position of the main conducting peak, expand its FWHM and decrease its transmission probability. Part II Chapter V 240

V.2 -Fe-cyclam system

The Fe-cyclam molecules are stable as cations in which the formal oxidation degree of the iron is +III and are thus associated to a counter-ion (Fe III -cyclam). They therefore present an openshell electronic structure and are thus magnetic with one unpaired electron. Because the molecule is easily reduced, 28 we performed calculations for both oxidation states (+II and +III).

The DFT molecular study of Contrarily to the Ru-dppe system, the change from revPBE GGA functional to the HSE06 hybrid functional modifies the energetic position of ϵ 0 but only slightly the electronic coupling Γ. The ϵ 0 value obtained with the #3-hollow tip is equivalent to the value given by the HSE06 hybrid functional with the #1-hollow geometry, but Γ is different. Thus, by applying the #3hollow model, we have fashioned a pseudo-correction of the revPBE functional on ϵ 0 which has the consequence to over-corrects the coupling Γ and consequently the conductance. Part II Chapter V 241 Whatever the model, the calculated conductance is one order of magnitude higher than for Rudppe. In both sets of experimental data (MCBJ and CP-AFM), the conductance of the Ru-dppe junction is systematically more conductive than that of Fe-cyclam. The oxidation state of the Fe-cyclam molecular system certainly plays an important role when considering that the Fe IIIcyclam is more stable at the molecular level. The process of incorporation in the junction might partially modify this oxidation state. For instance, XPS measurements reveal that the counterion is not present after the immobilization of the molecule on gold and the electrodes can play the role of electron reservoirs. Thus, both oxidation states should be investigated.

The Fe III -cyclam was studied as isolated cationic radical molecule and substituted by Au13. The corresponding MO energy diagrams are given in Figure 25. The open-shell electronic structure of Fe III -cyclam presents some similarities with Fe II -cyclam, with the set of alpha and beta spinorbitals (SO) highly similar but with energies shifted upward with one electron less for the beta SOs, and on the contrary, stabilized for the alpha SOs. The highest conducting SO of beta type is much closer to the pseudo Fermi energy in the cluster-model calculations. The implementation of the DFT/NEGF protocol implies the actual charge distribution is the result of the electronic density of the central region reaching equilibrium with the gold electrodes, meaning that one cannot impose a charge to the system. Several computational strategies can be used to obtain an oxidized electronic configuration of the central molecule. A counter-ion can be added in the cell. We performed several simulations in which a chlorine counterion was positioned at different places in the central region of the junction to observe the impact of its position on the transmission shape. Indeed, the Cl -position has an influence on the transmission as shown in Figure 26 (red circles). These features are due to an undesired overlap of the electronic density between the chlorine anion and the Fe-cyclam molecule.

The retained Cl -position, avoiding interactions with Fe-cyclam, is distant from the closest molecule atom by 6 Å and the corresponding transmission and main transmission eigenstates are shown in Figure 27 (procedure B1 of Table 3). All systems containing a chlorine atom were treated with this configuration. In the B1 calculation, chlorine is negatively charged (-0.38 e, Table 3, B1 entry) but this is not sufficient to confer a cationic character to the Fe-cyclam.

Indeed, the Mulliken atomic charge is almost identical to the Fe II -cyclam calculation (A entry in Table 3), and the spin polarization is only of 0.5 electron between the spin up and spin down population (instead of 1). In this B1 scenario, the main transmission peak of the alpha spin is 𝜖 0-𝛼 = -0.67 eV and beta spin 𝜖 0-𝛽 = 0.01 eV which is basically pinned on the Fermi level (Figure 27). The alpha spin transmission is consistent with the experimental UPS and SLM values (𝜖 0 = -0.73 eV), but this is the beta spin main transmission peak which is responsible for the main transmission path and leads to a huge conductance of 0.66 G0. Both experimental setups are revealing that the Fe-cyclam conductance is much lower than that of Ru-dppe. The electronic configuration A (Fe II -cyclam) and B1 are thus not reproducing the experimental data.

Because the introduction a Cl -anion appeared insufficient to induce an oxidation of the molecular part, the application of an "atomic shift" appears necessary. We applied different values of atomic shift on the iron and/or chlorine atom to modify the initial electronic repartition. The results of the different tested protocols are summarized in Table 3 (B2, C, E1, E2, E3, and E4). The A, B1 and B2 systems were calculated with the #1-hollow flat electrodes, and all of the others with the #3-hollow tip electrodes. The atomic shift is introduced by adding a negative and/or a positive electronic potential term to the Hamiltonian of the central region.

Part

The use of these atomic shifts imposed to increase the size of the scattering region. The three gold slabs on each side of the junction ("+3 GS" in Table 3) were added to guarantee the charge neutrality and convergence of the global system.  corresponds to a scaled initial spin state (ranging from 0 to 1, with 1 the highest spin state and 0 the lowest) of the considered atom.

In Table 3, in the C and D protocols we try to only gate either the Fe (+1 eV) or the Cl -(-1 eV).

When only the Cl -is gated (C), the corresponding 𝜖 0-𝛼 and 𝜖 0-𝛽 were not modified compared to the B1 system, only the charge on the Cl atom was increased (-0.5 e). Interestingly, the gating of the Fe on the D system leads to modifications of both main transmission peaks when the system does not contain a chlorine ion. 𝜖 0-𝛽 is not pinned to the Fermi energy anymore.

When applying the E1, E2, E3 and E4 protocols, we tried different atomic gating (except in E3 in which only the initial scaled spin state  was increased from 0.20 to 0.75). Overall the E1 approach appears to be the best compromise.

The only difference between the B2 and the E1 system, is the electrodes shape (#1-hollow and #3-hollow, respectively) and the values of 𝜖 0-𝛼 and 𝜖 0-𝛽 are almost equivalent. The transmission features and the values of 𝜖 0-𝛼 and 𝜖 0-𝛽 that are provided by E1 are difficult to validate without further experimental inputs to determine the oxidation state(s) of the Fe atom for instance. Figure 28 represents the transmission spectrum generated with the E1 protocol.

The main problem with the addition of an "atomic gating" is that the chosen E1 protocol leads to a Fano resonance at the Fermi level, and a mixing of the transmission eigenstates compared to the non-atomic shifted B1 system (Figure 27).

We thus recently investigated an alternative approach which is to apply a gating perpendicularly Because of time constraints, we could not proceed to test further gating values, but it seems plausible that only a small perturbation could modify the oxidation state of the junctions. This would completely change their conductance values and I-V curves. This behavior could be traced back to the larger variation of 𝜖 0 and Γ from the SLM fits of Vuillaume's group compared to Ru-dppe.

II -Tight-binding methods

DFT is nowadays one of the most popular and versatile technique to calculate the physical and chemical features of various types of chemical systems (molecules, metallic clusters and periodic bulks and surfaces). It can be used as a cheaper alternative to highly accurate wave function-based methods (CASSCF, RASSCF), which are built from Hartree-Fock (with an exact exchange) and include electron correlation. Despite the constant increase in the computing power (memory/processors) available, there is still a need to calculate the properties of large systems composed of hundreds to thousands of atoms, for which even DFT may face computing resource limitations. In Scheme 1 is represented the different families of methods at the theoretician's disposal, depending on the time scale and the system size under study. The classical force field (FF) methods involved in molecular mechanics and dynamics (dark-yellow in Scheme 1), are several orders of magnitude faster than DFT, allowing to deal with millions of atoms and to trail their dynamics up to the nanosecond scale. It contains a large number of empirical parameters and can be very precise for well-parameterized systems but suffer from transferability when changing too much the chemical nature of the systems.

More importantly, molecular mechanics and dynamics dot not consider any quantum effects, preventing the access to the electronic structure of the systems.

II.1 -Density Functional Tight Binding (DFTB)

DFTB appears in the 90's under the impulsion of Seifert, Frauenheim, Eschrig and coworkers. 13 The principle is to use minimal a basis set ("tight") including valence orbitals only ("binding"). The idea is to pre-calculate DFT integrals (which represent the DFTB parametrizations) to accelerate the calculations compared to DFT. The resources needed to do a quantum-chemical calculation can be reduced by more than two orders of magnitude. Its parametrization is less empirical because it relies on DFT calculations. In contrast to the main SE methods, electron correlation is included in DFTB through DFT and is not parametrized. It also inherits all the DFT weaknesses, which are the delocalization error and the static correlation error. 14 In DFTB, to converge the energy of a system, the notion of self-consistent field (SCF) is replaced by the concept of Self-Consistent Charges (SCC). Indeed, the DFTB energy is obtained as a Taylor expansion of the total Kohn-Sham DFT energy around a reference density (sum of the density of neutral atoms in the system); the changes in density in the nonzeroth terms accounting for potential charge transfers in the real molecule, are replaced by changes in atomic Mulliken charges. The total energy is expressed as:

E tot = E orb + E SCC + E rep Eq. 1
With E orb , the electronic or orbital energy, E rep , the tabulated repulsion energy, and E SCC , the self-consistent charge term.

The three terms in Eq. 1 can be explained as follows:

• 𝐄 𝐨𝐫𝐛it represents the sum of the atomic densities between all pairs of atoms A and B present in the molecule. It is composed of two-center integrals. The integral calculations are based on the nature of the A and B atoms and their distance (related to the Hamiltonian H and the overlap matrix S, as in the extended Hückel theory). These are DFT pre-calculated integrals.

• 𝐄 𝐒𝐂𝐂the total density is not the simple sum of the atomic densities (E orb ). The charges (and the density) are self-coherently related to the total energy using an E SCC term which corresponds to a second-order Taylor expansion of the KS density functional for the total energy for the DFTB2 and up to the third order for DFTB3 with respect to the variation in the Mulliken atomic charges. The atomic Hubbard

parameters U a (or "chemical hardness") and its derivative U a d (calculated with DFT) are given as parameters to drive this SCC contribution for DFTB2 and DFTB3 respectively.

• 𝐄 𝐫𝐞𝐩it contains the effects overlooked by the various approximations (the most important being the missing core electrons). It is a two-body contribution and this parametrization process is the most time consuming. Typically, one has to determine for every pair of atoms in the targeted system the total energy E tot as a function of the bond distance R for specific bonds using high level DFT or ab initio methods. An example is given in Figure 2 for the repulsion function for the Au-S interaction.

In addition, it is possible to adjust the AOs confinement radii r 0 (usually, several times the covalent radius) to compensate the approximation of "atom-like" potential (otherwise, the AOs would be too diffuse) in order to further match the desired application. Noteworthy, none of the parameterization step necessitates experimental data, as they are all based on DFT reference calculations.

The resolution of the total energy E tot is done self-consistently by the resolution of the KS equation based on a variational principle, leading to the molecular orbitals of the system. The major drawback of this TB method is, like most parameter-based method, the need of explicit parameterization for each type of atom pair (E orb ), leading to numerous information to register. As a consequence, a fraction of the periodic table is covered by DFTB parametrization to date, and thus limits the molecules and applications to those including the parameters available for atoms and atom pairs (for example, no parametrization for a lot of transition metals). 16 (TD-DFTB) can be used to explore excitedstate properties. Solvent effects can also be described through a polarizable continuum model (PCM) 17 , as well as long-range interactions with empirical dispersion terms. 18 The methods are implemented in several codes like DFTB+, 19 Gaussian, 20 ADF 21 or QuantumATK. 22 In our study, we will use the DFTB+ program. Table 1 gathers the most used sets of parameters available in DFTB. The number of available parameters sets remains quite small compared to FF methods, and most of these parameters (for instance, the "trans3d" set for transition metals) have not been thoroughly tested. This definitely represents one of the main obstacles preventing more applications of the DFTB method to study materials of experimental interest.

Table 1 -Selection of parameter sets available in DFTB. 23 Set Elements Description mio 24,25 H -C -N -O -S -P Organic and bio molecules auorg 25 Au -(H, C, N, O, S) Gold-thiolate compounds halorg 26 (F, Cl, Br Biological systems interacting with surfaces or at interface with solid materials have been modeled in this framework. 30,31,32,33 The use of fast TB method is also interesting in terms of electron dynamics or TD-DFT calculations. 34,35 The Auorg set has also been extensively studied with organic molecules 36,37,38,39 and used to model gold electrodes in the case of charge transport. 40

II.2 -Extended tight-binding method (xTB)

The Geometry, Frequency, Noncovalent, eXtended TB (GFN-xTB) 41 reference data produced by hybrid DFT computations. 44 One advantage over DFTB2 is that all the transition metals are parametrized. It is also implemented in various software such as AMS, 45 CP2K, 46 Cuby4, 47 Entos, 48 ORCA, 49 or TeraChem. 50

this expansion after the third-order term. 252 It is important to note that this method is formulated in a spin-restricted way. The spatial molecular orbitals are identical for α (up) and β (down) electrons. This treatment, even for open-shell systems, is rooted in the fact that no spin-dependent terms are present in the GFN2-xTB Hamiltonian. This can lead to improper energetic splitting for states of different multiplicity, implying that GFN2-xTB Hamiltonian will always favor the low-spin configurations. 41 As for DFTB, xTB has been applied to biologic 53 and organometallic 54 systems and used for studies of electronic spectra of large systems, 55 conformer generation, 56 electron ionization, 57 , redox potential 58 or vibrational frequencies. 59,60

III -Application to transition metal complexes

In the following section, we aim at assessing the ability of the DFTB and xTB method to describe molecular junctions based on transition metal complexes. As a first step, this section is dedicated to the quantification of the accuracy of the TB description of isolated metal complexes at the molecular level. To do so, we selected a set of 51 representative complexes.

We used quite small molecules containing one transition metal center among the following: iron, nickel, cobalt, zinc, copper and titanium. For each metal, different types of ligand (Scheme 3) were tested.

Scheme 3 -Set of organic ligands used to build the set of complexes.

Part

The assessment of TB method accuracy is based on a comparison of metal-ligand distances and electronic structures, notably the HOMO-LUMO gaps and the shapes of the molecular orbitals with the corresponding DFT results and experimental data. The DFTB2 was tested using the parameter sets: "mio", "trans3d", "znorg", and "tiorg" (Table 1). We will also study the performance of the DFTB3 implementation, for the complexes for which the parameterization was available (i.e, Ni, 61 Zn 62,63 and Cu 64 ). The xTB results were obtained using the GFN2-xTB version of the formalism.

DFTB calculations were conducted using the DFTB+ software 65 and xTB calculations using the standalone version of the program. 66 The DFT results were obtained with the Amsterdam Density Functional program (ADF) using a TZ2P basis set, a small frozen core and the PBE GGA functional. This DFT scheme is selected as a relevant reference because it is analogous to the DFT framework used in the DFTB parameterization. 1 The metal-ligand distance references are either issued from experimental crystallographic information or from a high computational level method like coupled cluster (CC) taken from the literature.

In order to rationalize more easily the performance of the TB methods, we decomposed our benchmark study in two steps: i) an in-depth comparison between TB and DFT/CC/exp values for six relevant complexes encompassing various types of ligands and metallic centers and ii) a more general and statistical analysis of the TB accuracy for the set of 51 different complexes.

III.1 In-depth study of selected complexes

The different results for the handpicked complexes are presented in Table 1. We selected, The magnitude of the deviation appears to be metal-dependent. For the dM-L distance DFTB2 tends to systematically overestimate the dM-L distance compared to the reference and to DFT while the xTB method returns lower dM-L values except for [Co(phen)3] 3+ . Clearly, some of these deviations are dramatic and prevent the use of TB methods for such metallic complexes.

However, the magnitude of the errors is highly system-dependent and the small batch of complexes considered in this study does not allow to draw clear conclusions. In the following, we will then attempt at drawing a more general and statistical description of the TB methods accuracy.

IV -General statistics on the performance of the TB methods

The results obtained for the 51 studied complexes are gathered in Table S1 and S2, in the annexes. The calculated (DFT and DFTB2/DFTB3/xTB) metal-ligand distances (dM-L) are given in Table S1, and the HOMO-LUMO gap in Table S2. External reference values (experimental or CC calculations) are not considered in this step since they are not always available. We collect in Table 3 and Table 4 the mean, the absolute mean, the mean squared error Eq. 2

with xi-TB the value of the parameter for complex i for the tight binding method, xi-DFT the corresponding value for complex i for DFT, n is the size of the sample (i.e. the number of tested molecules) and m(xTB) the mean value of parameter for the TB method considered. Overall, obtaining a HL gap value close to DFT HL gap appears to be a real bottleneck for all TB methods, regardless of the ligand and the metal considered.

IV -Conclusions and perspectives

This chapter aimed at assessing the ability of several Tight-Binding DFT-based computational schemes to describe the geometry and electronic structure of transition metal complexes. In particular the metal-ligand distances, electronic gap and the nature of the molecular orbitals were thoroughly studied for a series of 51 complexes and compared to reference data. For an application targeting transmission calculations, and with the idea of improving the electric and/or thermoelectric properties using a fast sampling of various organometallic molecules, the output is rather negative. Indeed, TB approaches in their present forms are not consistently accurate and the errors returned are extremely system dependent (ligand and metal), which may impede further screening studies. Nevertheless, we present in Figure 10

Conclusion and perspectives

In the first part of the manuscript dealing with copper (I) luminescent complexes, a theoretical approach based on (TD-)DFT employing a relatively small atomic basis set and the PBE0 hybrid functional was used to rationalize the photophysical properties of large polynuclear systems, namely Cu6CN2 and Cu8CN4. The comparison between the computed and experimental data are in good agreement. The occurrence of a TADF process for Cu6CN2 is notably strongly supported by the computational study which reveals that considering the constraints imposed by the crystal packing reduces the energy between the first singlet and triplet excited states. The calculated emission wavelengths are also well reproduced. The calculations also point out the role of the Cu-Cu interactions which is enhanced upon excitation, hence favoring the spin-orbit coupling. A contrario, the computational results obtained for the smaller systems [Cu2(dppa)3] 2+ and [Cu2(dppm)3] 2+ are revealing a more complex energy ordering and coupling of the excited states that render difficult to draw firm conclusions. For [Cu2(dppa)3] 2+ , the hydrogen bonding between the [Cu2(dppa)3] 2+ and its counter-ion (either PF6 -or BF4 -) and more globally the crystal packing should be considered in our computational study.

Several points should be addressed in future studies to circumvent these problems:

• The crystallization of the [Cu2(dppa)3] 2+ with non-interacting counter-ion and a noncoordinating solvent will certainly lead to better agreement between the experimental and computational approach.

• As seen in Chapter II, several ISC routes are possible which indicates that the experimental study of the influence of the excitation wavelength used to realize the emission spectra can be interesting to perform to better understand these systems.

• The use of a more accurate computational approach would be necessary to access to more reliable energies and better evaluate the different competing processes.

Other multi-nuclear copper (I) systems are currently under investigation and were thus not presented in the manuscript, notably a polymeric version of the Cu4CN2 system using several types of organic linker like the 2,4,6-tri-4-pyridyl-1,3,5-triazine, bypiridine-33' or 1,3-di(4pyridyl)propane (Figure 1). These systems have interesting luminescent properties. Indeed, in the polymeric form, a yellow luminescence is created; by different mechanisms (e.g., mechanochromic or heating), the polymer can undergo structural changes leading to a blue • As shown in Chapter II, the presence of anti-resonance in the transmission peak close to the Fermi level enhances the thermoelectric properties compared to a classic Lorentzian-shaped transmission. Fano resonances have almost no effect at RT. The ferrocene-derivatives display such QI, leading to interesting thermoelectric properties if the main transport level can be adjusted to lie closer to the Fermi level, using an external gate for example.

• The nature of the metal is utterly important. Chromium, cobalt, platinum, molybdenum and manganese metal-containing systems were also studied without yielding interesting results. The coordination complexes incorporating iron and ruthenium were those for which the best properties were calculated.

• The coordination of the sulfur anchor in the meta position of the terminal phenyl group of the linker does not create Fano nor anti-resonance interferences contrarily to what is reported in literature for organic molecular junctions. The "T-shaped" molecular systems are promising alternatives.

Several limitations and possible solutions concerning the computational approach were also discussed:

• The Fermi level position accuracy compared to the first occupied and unoccupied molecular levels is clearly functional and basis set dependent. Using more sophisticated functionals like the screened hybrid-GGA HSE06 (recently available in QuantumATK 2021) can certainly enhance the quantitative agreement between experiments and computations. Modified DFT methods like DFT+ which includes static corrections to the Kohn-Sham energy levels built on many-electron calculations can also help to better reproduce the conductance of organic molecules, as reported by Quek et al. for systems with a nitrogen anchoring group. 1 • The geometries of the electrode greatly influence the position of the different molecular levels (even when using the same anchoring group and a similar hollow coordination).

The use of a statistical approach to consider the diversity of junctions observed in experimental conditions is highly desirable.

• Charged systems are complex to simulate using the DFT/NEGF tools.

• The electrical conductance is always overestimated compared to experimental data. This is certainly due to several factors, the dominant one being that only the best possible molecular junctions are typically computed. For a better comparison, the dispersal of the experimental junction types should be accounted for (see above).

• The use of semi-empirical tight-binding methods (DFTB or GFN2-xTB) was evaluated as way to give access to this statistical approach needed to better compare theory with experiment. Our results show that, to date, it is not reliable for transition metalcontaining systems.

• In this study, the lack of applicable methodology to evaluate the thermal phonon conductance at the theoretical level for the systems studied in the manuscript is damageable since ZT would be better evaluated. Indeed, we employed a value of 20 pW/K for the thermal phonon conductance on the basis of experimental evaluations.

The actual value might be smaller, especially when the rigidity and the length of the molecules increase.

The use of DFT as computational tool allowed us to rationalize at a semi-quantitative level the optical properties of copper (I) complexes of different sizes revealing the complexity of the multiple competitive processes that could occur after excitation. Even though more sophisticated tools would be need to access precisely to the de-excitation cascade, these calculations are giving valuable information on the nature of the excitation, the geometrical and electronic changes of the first excited states allowing to rationalize the TADF-like behavior measured experimentally.

DFT coupled with NEGF has also proven to be an interesting and comprehensive tool to perform a molecular design of organometallic junction in the aim of enhancing thermoelectric properties. The systems which has been identified in this manuscript by computational means are presenting rich electronic transmission features that give rise to appealing perspective in this emerging field. This study also reveals the need of faster computational tools that would allow to simulate the large diversity of geometries that are observed experimentally in order to simulate and therefore interpret and predict the properties of experimental molecular junctions. Keywords: coordination chemistry, molecular junction, computational chemistry Abstract: The computational studies developed in this thesis are divided in two research projects. The first part concerns a theoretical study of luminescent polynuclear copper (I) complexes. The second chapter reports computational results that are compared to experimental data obtained by C. Lescop and collaborators (ISCR -INSA Rennes) for copper dimers bridged by three diphosphines. Different levels of calculations are tested in order to quantify the accuracy of the results. The following chapter brings together the study of related metallacyclic compounds comprising respectively 6 and 8 copper atoms. These studies show the importance of the geometric reorganization of excited states and intermolecular interactions in the solid state. The second part of this thesis aims at providing a theoretical support to the development of molecular devices presenting high thermoelectric performance. The first chapter reports on the state of the art and the factors influencing the variation of thermoelectric properties. Chapters II and III present the theoretical tools available for the study of the transmission properties of molecular junctions and their thermoelectric characteristics. The advantages of organometallic compounds in inducing thermoelectric properties are treated in chapters IV and V. A significant increase in conductance and Seebeck coefficients is calculated. A computational molecular design implies the use of cheap computational methods. The density functional tightbinding methods are evaluated for this purpose in the last chapter.

1 Résumé L'équipe de Chimie Théorique Inorganique (CTI) de l'Institut des Sciences Chimiques de Rennes (ISCR) est un groupe de recherche regroupant des chercheurs du CNRS, de l'Université de Rennes 1 et de l'école supérieur des sciences chimique de Rennes (ENSCR). Le groupe de recherche utilise des outils de chimie computationnelle pour rationaliser les propriétés structurelles et physiques des molécules et du solide. Ses thèmes de recherche portent sur les : i) architectures complexes (i.e. gros clusters atomiques, verres) (axe 1), ii) matériaux solides pour l'énergie (axe 2), et iii) systèmes moléculaires multifonctionnels pour l'optoélectronique, la spintronique et le magnétisme (axe 3). Le présent projet s'inscrit dans l'axe de recherche 3 et vise à contribuer au développement des technologies en électronique moléculaire, optoélectronique ou spintronique utilisant une approche quantique computationnelle. L'intérêt principal de ces dispositifs est de fournir un rendement élevé à l'échelle nanométrique, ce qui est en accords avec les défis environnementaux d'aujourd'hui. Les systèmes moléculaires ont démontré leurs potentiels dans différents secteurs comme le stockage de données, les technologies de l'information, la médecine, ou encore l'énergie. Le projet de recherche de thèse présenté dans ce manuscrit est consacré à deux applications qui tirent parti des molécules contenant des métaux de transition : l'éclairage à bas coût et la récupération d'énergie (chaleur) ou d'électricité à l'échelle nanométrique (thermoélectricité). Pour la première application, les travaux se sont centrés sur des molécules contenant des atomes de cuivre. Ce sont des systèmes attrayants en raison de leurs propriétés physico-chimiques, disponibilité, faible toxicité et de leurs faibles coûts. L'objectif général est de proposer une alternative aux systèmes contenant des métaux lourds et des lanthanides qui sont les plus couramment développés pour la création de diodes électroluminescentes phosphorescente (PhOLED), ou pour des applications d'éclairage. Ce sont des composés plus toxiques, plus rares et donc chers. L'étude computationnelle qui est présentée dans ce manuscrit s'appuie sur des résultats expérimentaux obtenus dans l'équipe de Lescop de l'ISCR (Rennes) et permet de rationaliser les propriétés optiques de ces systèmes contenant du cuivre (I) en utilisant la théorie de la fonctionnelle de la densité (DFT). Pour la deuxième application, à savoir la récupération d'énergie à l'échelle nanométrique, les travaux se sont focalisés sur la conception de jonctions moléculaires in-silico pour des applications en nanothermoélectricité dans le but de récupérer et de convertir l'énergie dissipée dans des nano-dispositifs en électricité. Cette partie de la thèse a été développée en collaboration avec différents collaborateurs expérimentalistes : l'équipe de recherche de Rigaut à l'ISCR (Rennes), l'équipe de recherche de Scheer de l'université de Constance (Allemagne), et enfin l'équipe de recherche de Vuillaume (Lille).

La première partie porte sur l'étude computationnelle de complexes luminescents multi-nucléaires de cuivre La théorie de la fonctionnelle de la densité à liaison forte (DFTB, GFN2-xTB) qui a été utilisée avec succès dans des systèmes biologiques est évaluée pour des systèmes de coordinations. Cette étude montre que les paramètres disponibles pour les métaux de transition ne sont pas suffisamment fiables pour modéliser de nouveaux systèmes.

L'utilisation de la DFT comme outil de calcul nous a permis de rationaliser à un niveau semi-quantitatif les propriétés optiques de complexes de cuivre (I) de différentes tailles illustrant la complexité de plusieurs processus compétitifs qui peuvent se produire après excitation. Même si l'utilisation d'outils plus sophistiqués serait nécessaire pour accéder formellement au chemin de désexcitation, ces calculs donnent des informations sur la nature de l'excitation, la géométrie et les changements électroniques des premiers états excités permettant de rationaliser un comportement de type TADF mesuré expérimentalement. Enfin la DFT/NEGF s'est également avéré être un outil intéressant et complet pour cibler des jonctions organométalliques dont les propriétés thermoélectriques sont exacerbées. Les systèmes qui ont été identifiés in silico dans ce manuscrit présentent de bonnes caractéristiques de transmission qui donnent lieu à des perspectives attrayantes dans ce domaine émergeant. Cette étude révèle également le besoin d'outils de calcul plus rapides qui permettraient de simuler la grande diversité de géométries observées expérimentalement afin de simuler, d'interpréter et de prédire les propriétés des jonctions moléculaires obtenues expérimentalement.

  State of the art and methodology

Figure 1 -

 1 Figure 1 -Evolution of the number of articles by selected years containing the word "luminescent materials". Source: Google Scholar.

Figure 2 -

 2 Figure 2 -Jablonski-type schematic energy diagram of a closed-shell molecule with a singlet ground state 𝑆 0 . The singlet and triplet excited states, 𝑆 𝑛 and 𝑇 𝑛 , with n = 1, 2, ... are represented by bold lines. The thin lines represent the lower lying vibrational sub-levels of each state. The potential energy surfaces are given as a function of nuclear coordinate modifications.

Figure 3 -

 3 Figure 3 -Simplified mechanism of a phosphorescent/TADF emitter. The energy 𝛥𝐸 𝑆𝑇 is the barrier between the coupled triplet and singlet states. The ISC rate increases with SOC, which induces a competition between ISC and fluorescence (in green). For moderate SOC, RISC can be thermally driven leading to TADF (in orange). RISC is in competition with phosphorescence (red). 𝑆 0 # represents the transient state on the potential energy surface (PES) of 𝑆 0 at the triplet state 𝑇 𝐵 geometry. The system relaxes by non-radiative means from 𝑆 0 # → 𝑆 0 .

Figure 4 -

 4 Figure 4 -Left: molecular representation of the Cu(dppb)(pz2Bph2) complex. Right: emission lifetime 𝜏(obs) versus temperature. Adapted from reference 26.

Figure 5 -

 5 Figure 5 -Usual coordination modes for a Cu(I) complex. Here X corresponds to a halide atom and D, D' refer to poly-and mono-dentate ligand, respectively. A non-halide Cu(I) complex can either be cationic or neutral depending on the coordinating ligands. Adapted from reference 28.

Figure 6 -

 6 Figure 6 -From molecular chemistry to supramolecular chemistry with molecules, supermolecules, and (supra)molecular devices according to Lehn.38

  long range Corrected-Hybrid-GGA (LC-H-GGA), like the LC-wPBE similarly coupled with the 6-31+G(d) basis set. They are both providing the finest description of the complexes. It is important to highlight that the PBE density functional, which lies in the pure-GGA formalism is also giving interesting results. The same remark goes for the hybrid-GGA (H-GGA) B3LYP. With the dataset used, the DFs with added dispersion correction such as the DFT-D and DFT-D3 (from S. Grimme and co-workers 59 ) does not enhance the precision of the computational results. Here, we present some examples of copper-ligand distances compared to experimental data (mean signed error -MSE -, compared to the Cambridge database) as a function of the functional, as averaged over the whole basis set from reference 58: • Cu-ligand bond (average): o GGA-BLYP: +0.058 Å o H-GGA-B3LYP: +0.044 Å o HM-GGA-M06: +0.016 Å • Cu-N bond specifically: o GGA-BLYP: +0.052 Å o H-GGA-B3LYP: +0.040 Å o HM-GGA-M06: +0.029 the ligand-Cu-ligand angles (º) for copper(I) complexes are: • Ligand-Cu-ligand angle (average): o GGA-BLYP: +4.4° o H-GGA-B3LYP: +4.4° o HM-GGA-M06: +4.0°

II. 1 -

 1 This chapter is devoted to the theoretical investigations of [Cu(dppm)3] 2+ and [Cu(dppa)3] 2+ (dppa = diphenyl-diphosphino-amine) in which the CH2 unit is replaced by a NH group. The experimental characterizations of the solid-state compounds containing these cations are reported and discussed in light of our computational results. data obtained by Lescop and coworkers The permission of reporting the following unpublished results has been obtained from the research team which has obtained them: Lescop, Xu, Moutier and Calvez from the ISCR -INSA, Rennes. The synthetic protocol employed was to react a Cu + ionic source with a corresponding phosphine in a dichloromethane (DCM) solution under air. The TEF - ([Al(OC(CF3)3)4] -), PF6 -and BF4 -anions have been used as counter anions. Monocrystals were obtained upon solvent evaporation. X-ray crystallography was employed to characterize their solid-state structural arrangements. The [Cu 2 (dppm) 3 ] 2+ system II.1.1 -X-ray crystallographic geometries of the [Cu2(dppm)3] 2+ solid-state system The [Cu2(dppm)3] 2+ cations were crystallized with two TEF -counter ions giving rise to A TEF = [Cu2(dppm)3][TEF]2・DCM. This solid-state compound contains also DCM solvent molecules

Figure 1 -

 1 Figure 1 -Left: 2-D structures of [Cu(dppm)3] 2+ and [Cu(dppa)3] 2+ . Right: representation of the X-ray structural arrangement of A TEF . The orientation corresponds to the alignment of the Cu-Cu axis parallel to the sketch plane. Only one TEF -and DCM is shown for clarity. By the courtesy of Lescop and coworkers.

Scheme 1 -

 1 Scheme 1 -Numbering of the atoms for A (phenyl groups omitted for sake of clarity). The phenyl orientations of the dppm arrangements are determining the numbering, (similar for all geometries).

Figure 2

 2 Figure 2a) Experimental solid-state absorption spectrum of A TEF . b) Experimental excitation spectrum of A TEF for an emission at ex = 478 nm.

Figure 3 -

 3 Figure 3 -Temperature-dependent experimental normalized solid-state emission spectrum of A TEF in nm. Wavelength of excitation ex = 375 nm.

Figure 4 -

 4 Figure 4 -Luminescence decay time of the solid-state emission spectrum of A TEF from 80 K to 370 K.

Figure 5 -Scheme 2 -

 52 Figure 5 -Experimental X-ray structural arrangements of B BF4 and B PF6 . The chosen orientation corresponds to the alignment of the Cu-Cu axis normal to the sketch plane. Only one BF4 -and PF6 -counterion are shown for clarity. By the courtesy of Lescop and coworkers.

Figure 6 -

 6 Figure 6 -Experimental solid-state absorption spectra of a) B BF4 and b) B PF6 .

Figure 7 -

 7 Figure 7 -Experimental excitation spectra of a) B BF4 , ex = 488 nm and b) B PF6 , ex = 520 nm.

Figure 8 -

 8 Figure 8 -Temperature-dependent experimental emission spectra of a) B BF4 with ex = 375 nm and b) B PF6 with ex = 405 nm (80 K in blue, 300 K in red). See Figure 3 for the details of the temperature coloring scale.

Figure 9 -

 9 Figure 9 -Luminescence lifetime of the bi-nuclear copper systems a) B BF4 and b) B PF6 from

Figure 10 -

 10 Figure 10 -Representation of the DFT optimized ground state geometries of the complexes A theo and B theo complex.

Figure 11 -Figure 12 -

 1112 Figure 11 -A theo and B theo MOs energetic diagrams. The iso-values used for the representation of the orbitals is ± 0.03 (e/Bohr 3 ) 1/2 .

Figure 13 -

 13 Figure 13 -MOs energetic diagram of the B theo/-H+ . The iso-values used for the representation of the orbitals are ± 0.03 (e/Bohr 3 ) 1/2 .

Figure 14 -

 14 Figure 14 -Simulated UV-VIS absorption spectra of B PF6 with hydrogen atom positions optimized, B theo , B theo/-H+ , B theo/-2H+ , and A theo , as computed with a Gaussian smearing of 50

Figure 15 -

 15 Figure 15 -Representation of the Natural Transition Orbitals (NTO) associated to the S0 to T1vert (top) and S0 to S1-vert (bottom) excitations for A theo (level Gauss1 ).

Figure 16 -

 16 Figure 16 -Representation of the NTOs associated to the S0 to T1-vert (top) and S0 to S1-vert (bottom) excitations for B theo (level Gauss1 ).

Figure 17 -

 17 Figure 17 -One perpendicular (left) and one lateral (right) view of the isosurface description of the electronic charge-density difference between the first vertical triplet (T1-vert, top) and singlet (S1-vert, bottom) excited states, and the ground state (S0) of B theo (level Gauss1 ); red = density depletion; blue = density increase; the iso-density is 1.10 -4 e/bohr 3 .

Figure 18 -

 18 Figure 18 -For each table cell: (left) representations of optimized geometries of B theo superimposed onto the S0 B theo optimized geometry in light green (T1, T2, S1 or S2); (right) and iso-surface representations of the electronic charge-density difference between the considered

Figure 19 -

 19 Figure 19 -Relative energy diagram of several lowest optimized excited states as calculated in the TD-DFT paradigm relative to S0 in eV with their associated emission wavelengths in nm. Top: A theo . Bottom: B theo .

= 6 .

 6 25.10 -8 eV 2 . The other important factor to calculate the kRISC rate is the ρFCWD, the Frank-Condon-Weighted Density of States. Using the classical Marcus theory (CMT) to describe the thermo-kinetic barrier (Figure 20), the potential energy surface (PES) of both states around their respective minimum structures are described in terms of parabolic functions.

Figure 20 -

 20 Figure 20 -PES of the S1 and T1 states described as quadratic functions centered at their respective minimum.

1 -

 1 ) and computational studies of the luminescent hexametallic Cu(I) assembly 1 which results from the adaptive CDS reaction of A with CuCN, acting as an alternative cyano-based precursor, and of a Cu8(CN)4 metallacycle which results from the reaction of A at RT with NaN3 or KN(CN)2 are presented. The two systems are presenting interesting thermochromic luminescent properties. Summary of the experimental study -Group ofLescop

Crystalline samples of 1

 1 display an intense green luminescence upon excitation at 365 nm (Figure2) at 298 K. This corresponds to a broad emission band with an emission maximum em at 490 nm and an emissive lifetime  of 80 s. Upon cooling, the emission spectrum presents a gradual red-shift affording an intense yellowish luminescence at low temperature (Figure2: em = 522 nm at 80 K and  = 365 s). Upon heating up from RT to 363 K, very moderate thermal variations of both the emission spectrum and the lifetime are observed, and the luminescence maxima reach a plateau region at 330 K with em = 487 nm (em 80 K / 363 K = 0.132 eV / 1065 cm -1 ) and  = 57 s. Such emission behavior is typical of Cu(I) derivatives showing TADF behavior.5,6 By fitting the thermal evolution of  using Eq. 1 (chapter I), the values of E(S1-T1) = 1023 cm -1 ; (S1) = 325 ns and (T1) = 352 s could be obtained. Despite this set of values is significantly different from those obtained for the tetrametallic precursor A (E(S1-T1) = 1560 cm -1 , (S1) (T1) = 185 s) both account for TADF processes.3 In order to get more insights in the energetic and emission properties of 1, we performed a DFT/TD-DFT computational study.

Figure 1 -

 1 Figure 1 -a) Synthesis of 1 and comparative views of the X-ray crystal structures of A (top view) and 1 (top view and lateral simplified view; counter-anions, H atoms and solvent molecules have been omitted for clarity). b) View of the short contact inter-molecular interactions (shown with dotted lines) observed between neighboring imbricated metallacycles in the crystal structure.

Figure 2 -

 2 Figure 2 -a) Normalized solid-state emission spectra of 1 at temperatures between 80 K and 300 K upon excitation at 365 nm. The black arrow shows the direction of the thermochromic shift of the solid-state emission spectra observed upon cooling; b) plot of emission decay

74 Figure 3 -

 743 Figure 3 -a) Scheme of the TADF process and photophysical values experimentally obtained. b) Relative energy diagram of the first optimized excited states relative to S0 in eV; associated emission wavelengths in nm; see Table 1 for a complete summary of the emission energies. Top view and lateral views of the optimized geometries of 1-S0, 1-T1, 1-T1-const and 1-S1 (H atoms omitted); intermetallic Cu-Cu distances (in Å). Experimental T1 level (T1-exp) and associated experimental values in green for the sake of comparison.

  In each [Cu2(dppm)2] fragment, one of the metal centers has a distorted trigonal planar coordination sphere resulting from the coordination of two dppm ligands and a cyano ligand. The second Cu(I) metal center presents a distorted tetrahedral coordination sphere due to an additional coordination with the terminal atom of a DCM ligand. The two DCM ligands are located within the Cu8(CN)4 metallacycle system, connecting two-by-two, two Cu2(dppm)2 fragments (d(Cu-NDCM = 2.09 Å). Intermetallic Cu(I) distances in the [Cu2(dppm)2] fragments (d(Cu-Cu) = 3.43 Å) exceed those recorded in the precursor A (d(Cu-Cu) = 2.87 Å) and are too large for cuprophilic interactions.

77 =

 77 ) was conducted, revealing bi-exponential decay times all along the temperature range with a shortlived component having an almost linear and moderate progression upon cooling (at 80 K, 2 = Part I Chapter III 238 s) bearing a typical thermally activated delayed fluorescence (TADF) profile. By fitting the thermal variation of the 1 component to a Boltzmann-type equation, the values of E(S1-T1) = 978 cm -1 ,  (S1) = 229 ns and (T1) = 231 s were obtained. This set of values ranges among those determined for the other Cu(I) CDS assemblies based on the [Cu2(dppm)2]fragment (including the precursor A) that were identified as TADF luminophores.

Figure 4 -

 4 Figure 4 -Synthesis of several derivatives: B, C M , D, E and F M . Here, we focus only on the D system.

Figure 5 -

 5 Figure 5 -Views of the molecular X-ray structures of D, E and FCu derivatives; Selected metric data. Counter-anions, H atoms and solvent molecules have been omitted for clarity; location of the C and N atoms of the cyano ligands has been chosen arbitrarily, black dots represent the central points of the Cu-Cu segments of each [Cu2(dppm)2] fragments.

Figure 6 -

 6 Figure 6 -Solid-state temperature dependent emission spectra of D (ex = 365 nm), inset: plot of emission decay lifetime against temperature (80 K to 300 K); the red line represents the fit according to the TADF Boltzmann-type (Eq.1, chapter I).

Figure 7 -

 7 Figure 7 -Iso-surface (± 1.10 -4 e/bohr 3 ) representations of the electronic charge-density difference between the first vertical singlet (S1) or triplet (T1) excited state, and the ground state (S0) of D (violet = density depletion, cyan = density increase).

Figure 8 -

 8 Figure 8 -Left: optimized geometry of the Cu8(CN)4 complex at the DFT level. Right:

Figure 9 -

 9 Figure 9 -Luminescent coordination polymer based of the pre-assembled A system. The bridging ligand is a 2,4,6-tri(4-pyridyl)-1,3,5-triazine.

  It has emerged as a possible future alternative to silicon-based technology in the search of strategies for moving from microelectronics (bulk devices) to nanoelectronics by developing systems controllable at the molecular level. This domain offers the possibility to explore the electronic conductance, thermal conductance, thermoelectric effects, quantum interferences and more at the atomic scale. The involved chemistry and physics are completely controlled by quantum effects and thus can greatly diverge from bulk materials, in terms of optical, electronic or mechanical properties. Finally, it can provide tools to overcome Moore's law tied to the conventional silicon technology. This exciting perspective makes of molecular electronics a rapidly expanding domain of nowadays research.

Figure 1 -

 1 Figure 1 -Molecular electronics is an interdisciplinary field.

I. 2 -

 2 From molecular electronics to molecular junctionsThe idea of using a single molecule as a rectifier came from the paper of Aviram and Ratner in 1974. In this article entitled "Molecular Rectifiers", they present a theoretical study of the electronic transport through a charge-transfer molecule containing a donor and an acceptor endgroups through which the transport would be possible in a single direction. It was only at the 90's that the measurements of current-voltage curves (I-V) were accessible. This was made possible thanks to the developments of the Scanning Tunneling Microscopy (STM), Atomic Force Microscopy (AFM) and Mechanically Controllable Break Junctions (MCBJ) notably. In these experiments, the conventional electronic devices are replaced by molecules sandwiched between electrodes to form molecular junctions. Depending on the setup, the number of molecules varies between a few hundreds to a single one. The variety of molecular junctions and associated properties is almost unlimited considering the vast number of molecular structures that chemistry affords. The challenging interdisciplinary research in the domain is devoted to target molecular systems that could replace usual bulk electronic components or even offer new unique functionalities, and to build the experimental setups and rationalize their properties.

Figure 2 -

 2 Figure 2 -Representation of the 1996 experiment of Weiss et al. Adapted from reference 20.

Figure 3 -

 3 Figure 3 -Left: Calculated transmission function 𝝉(𝑬) through the molecule from the tip to the substrate. Right: I-V characteristics of a ,'-xylyldithiol molecule. The insert is the schematic representation of the experiment. Adapted from reference 22.

Figure 4 -

 4 Figure 4 -Scheme of a CP-AFM experiment in which an Au-coated AFM tip is used to probe the resistance of a thin crystal of the hexathienyl molecules (6T). During a measurement, the current is recorded as a function of the applied voltage between the conducting probe and a fixed electrode. Inset shows the orientation of the 6T molecules with respect to the probe and substrate. Graphic adapted from reference 31.

Figure 5 -

 5 Figure 5 -Left: Simplified scheme of a MCBJ experiment. Right: scheme of benzene-1,4dithiol molecules contacting two electrodes.

Figure 6 -

 6 Figure 6 -Scheme of the different measurement steps: (A) initial gold wire of the break junction before breaking and tip formation. (B) addition of the benzene-1,4-dithiol, SAMs formed on the gold wire surfaces. (C) mechanical breakage of the wire in solution that produces two opposite gold contacts that are SAM-covered. (D) after solvent evaporation, the gold contacts are slowly approached until the onset of conductance is achieved. Steps (C) and (D) can be repeated numerous times for reproducibility. Graphic adapted from reference 34.

Figure 7 -

 7 Figure 7 -Top: measurements of the resistance of different molecular wires with CP-AFM. A gold-coated tip is in contact with OPIn monolayer on a gold surface, with n = 1-10. The I-V traces were acquired over ± 1.5 V from OPI 3 to 10 and ± 1.0 V for OPI 1 and 2. Plot of the logarithm of the resistance (ohm) vs the molecular length (nm). Bottom: Schematic of an OPIn molecule with n ranging from 0 to 9. Adapted from reference 48.

Figure 8 -

 8 Figure 8 -Tunneling process over a 1-D barrier. An incident electron wave (I) scatter against the square barrier leading to an evanescent wave (II). If the barrier length is small enough, a certain quantity of the initial wave can be transmitted (III), implying that the associated particle has a high probability to tunnel across it.

Figure 9 -

 9 Figure 9 -Top: transmission plot of two 3 nm barriers (L = 3 nm) separated by 10 nm. The barrier height, V0 = 0.5 eV. The transmission is only calculated for energies below V0. Bottom: schematic of a double squared potential barrier. An incident wave, in red, is partly transmitted or reflected (black arrow) into the central region. The transmitted wave in the central region is in turn, transmitted or reflected by the second barrier.

Figure 10 .

 10 Figure 10. Considering two potential energy barriers, with arbitrary potential shapes (V0 and V0'), resolving the scattering matrices leads to the expression of the transmission τ(E) as a function of the energy:

Figure 10 -

 10 Figure 10 -The single resonant level can be represented by two separate potential barriers of arbitrary shape. V0 and V0' represent the corresponding potentials.

Figure 11 -

 11 Figure 11 -Representation of a single resonant level model. The presence of the molecule, in the center of the junction, is approximated by the presence of single energy level which is the frontier molecular orbital (FMO) lying closest to the Fermi energy of the electrodes (generally the HOMO or the LUMO). Left and right electrodes are represented by electron reservoirs. If a potential difference 𝛥𝑉 is applied, a symmetric splitting around the zero-bias Fermi energy of ± 𝛥𝑉/2 occurs (different heights in green). From reference 56.

Figure 12 .

 12 Figure 12. Only the transmission between the two shifted chemical potentials is integrated in the calculation of the current at the considered bias voltage.

Figure 12 -

 12 Figure 12 -The green line is the transmission probability as a function of 𝐸-𝐸 𝐹 from -1.0 eV to 1.0 eV. The 𝐸 𝐹 is the Fermi energy of the electrodes at zero bias. The blue curve represents the bias window defined by the electrodes, here 𝑉 = 1.0 𝑉. It corresponds to the difference between the Fermi level of the left electrode minus that of the right electrode (𝑓 𝐿 -𝑓 𝑅 ). The Fermi-Dirac function is accounted for considering T = 300 K. The whole green portion of the transmission within this window is integrated in Eq. 12 to calculate the current at this bias voltage.

Figure 13 -

 13 Figure 13 -Representation of a molecular junction, the directions perpendicular to the transport (A and B) are periodic, and the C direction (transport direction) is not periodic because of the scattering region (highlighted by the dotted lines).

Figure 14 -

 14 Figure 14 -Basic flowchart for the computation of transmission and current using the DFT/NEGF method. Adapted from reference 79.

• 40 •

 40 Type: Semi-circle • Integral lower bound: -50-80 eV • Circle points: 30-Real axis point density: 0.005-0.02 eV • Real axis infinitesimal: 0.005-0.02 eV Poisson solver: • Fast Fourier Transform 2D or multi-grid, with periodic conditions along A and B, and boundary conditions along C (Dirichlet or Neumann).

Figure 15 -

 15 Figure 15 -Representation of a basic thermoelectric circuit, with two different conductors (iron and copper) that are connected together to make a thermocouple. To measure the voltage, a voltmeter is placed at the ends of the copper wires.

Figure 16 -

 16 Figure 16 -Representation of the Seebeck effect, with observation of a thermovoltage with a voltmeter. With 𝑄 ̇= 𝑑𝑄/𝑑𝑡, the heat current.

  However, this was difficult to demonstrate experimentally because of the competing effects that are the Part II Chapter I 106 intrinsic Joule heating (heat generation induced by electrical current flow due to the collision between electrons and atoms of the conductor) and the thermal conduction (due to electrons and lattice vibrations) which greatly reduce the efficiency of such devices. The conversion effectiveness of thermoelectric generators and Peltier devices have been fully established by the work of the physicist Altenkirch in the early 1900s.

When

  heat conversion into work of a thermoelectric device is defined as the ratio between the energy provided to the external load over the heat energy absorbed at the hot junction. Indeed, in a perfect thermodynamic cycle, the efficiency  defined by the Carnot cycle (named after the French physicist Carnot in 1824) has an upper limit that a classic thermodynamic machine can achieve during the conversion of heat into work. The efficiency of a power generator, with the added figure of merit ZT, is defined as:[START_REF] Nolas | Thermoelectrics: Basic Principles and New Materials Developments[END_REF] 𝑍𝑇 tends to infinity (Figure17), one can retrieve the classic efficiency of the Carnot cycle 𝜂 = (𝑇 𝐻 -𝑇 𝐶 )/𝑇 𝐶 , with TH and TC the hot and cold temperature of each reservoir, respectively.

Figure 17 -

 17 Figure 17 -Evolution of the conversion efficiency  as a function of the temperature of the hot part of the junction 𝑇 ℎ𝑜𝑡 . The cold part of the junction is set at RT. The dashed line represents the theoretical Carnot cycle.

Figure 18 -Figure 19 -

 1819 Figure 18 -Schematic of a thermoelectric molecular junction.

  𝐸, 𝑉) 𝐸 (𝑓 𝐿 (𝐸, 𝜇 𝐿 ) -𝑓 𝑅 (𝐸, 𝜇 𝑅 )response regime and considering a small difference of temperature T = (T L -T R ) and a small difference of potential V = ( L - R )/e between the electrodes, one can simplify Eq. 21 and Eq. 22 into:132,133 𝐼(𝑉, 𝑇) = 𝐿 0 𝛥𝑉 + 𝐿 1 𝛥𝑇 𝑇 Eq. 23𝑄 𝑒 (𝑉, 𝑇) = 𝐿 1 𝛥𝑉 + 𝐿 2 𝛥𝑇 𝑇 Eq. 24

Figure 20 -

 20 Figure 20 -Evolution of the Seebeck coefficient for the Fe II (cyclam)(C2PhS)2 molecule (cyclam = 1,4,8,11-tetraazacyclotetradecane, see Part II, Chapter IV) as a function of the mean temperature applying Eq. 23 for the blue line and Eq. 29 for the dotted orange line.

  photons is collected and displayed as a function of the photon counts vs incident electron energy spectrum. This technique is thus fully complementary to UPS.

Figure 1 -

 1 Figure 1 -Schematic representation of an UPS spectrum.

Figure 2 -

 2 Figure 2 -F-N plot representation (Eq. 2) from the extracted data of an experimental I-V curve of a molecular junction with a biphenyl-dithiol (OPD2) molecule coupled to gold electrodes. 2 Inset: Zoom on the minimum of the F-N plot. The minimum of this plot is 𝑉 𝑡 , the transition voltage. Here, 𝑉 𝑡 ≈ 1.15 𝑉 -1 ≈ 0.87 𝑉 which, using Eq. 3, leads to |𝜖 0 | ≈ 0.75 𝑒𝑉.

  by fitting an OPD2 experimental I-V curve (Au-OPD2-Au) as in the F-N plot (Figure2), considering different voltage ranges. Both parameters evolve as a function of the selected voltage range Δ𝑉, which represents a maximum modification of 35 % for ϵ 0 and 19 % for Γ.

Figure 3 -

 3 Figure 3 -Left: evolution of the absolute value of 𝜖 0 and Γ in eV as a function of the integrated voltage range Δ𝑉 in volt used to fit the experimental I-V curve of the Au-OPD2-Au molecular junction. 2 Right: associated table that provides the R 2 (coefficient of determination) of the I-V fit (Eq. 6) and the parameters (𝜖 0 , Γ) values for each bias voltage range.

  𝜖 0-𝑖𝑛𝑖𝑡 corresponds to the maximum of transmission at zero bias. It can be estimated by UPS (if the conducting level is below the Fermi energy) or it can be calculated by quantum chemical calculations.Nevertheless, if the transmission is noticeably voltage-dependent, the use of the single level model considering that 𝜏(𝐸, 𝑉 = 0 𝑉) ≈ 𝜏(𝐸, 𝑉 > 0.1 𝑉) is not applicable as detailed in the a calculated I-V curve of Au-[Ru(TMA)(CC-Ph)2S2]-Au junction as study case.12

Figure 4 -

 4 Figure 4 -Simplified scheme featuring a single transporting molecular level of energy 𝜖 0 (here the HOMO) coupled by a coupling strength R to the right electrode and L with the left electrode. The Fermi energy (EF) is represented by the dashed line.

Figure 5 -

 5 Figure 5 -Top: chemical structure of trans-Ru(TMA)(C≡C-C6H4-SH)2 (TMA = 1,5,9,13tetramethyl-1,5,9,13-tetraazacyclohexadecane). Bottom: Representation of the junction

Figure 6 -

 6 Figure 6 -Left: Molecular orbital diagram of the isolated molecule trans-Ru(TMA)(C≡C-C6H5)2; Iso-contour plots of the highest occupied molecular orbitals (MO).28,29 Calculations and MO plots were obtained using ADF 2019; Slater basis set: TZ2P, small frozen core; Functional revPBE; Iso-contours: ± 0.01 (e/bohr 3 ) -1/2 Right: Transmission spectrum of the molecular junction at 0.0 V; representation of two transmission eigenstates.

  of (E)

Figure 7 -

 7 Figure 7 -a) Calculated transmission spectrum at zero bias; Lorentzian curves generated from the transmission curve, the I-V and the "G+S" fits are given in dotted green, red and black lines respectively; see Figure 8 with bias voltages up to 1.0 V. b) I-V curve obtained from transmission calculations (steps of 0.1 V) and fit of the I-V curve.

Figure 8 -

 8 Figure 8 -Calculated transmission spectra of the Au-trans-Ru(TMA)(C≡C-C6H4S)2-Au junction at biases ranging from 0.0 to 1.0 V by steps of 0.1 V.

Figure 9 -

 9 Figure 9 -Calculated transmission spectra of an Au-oligo(phenylene-ethynylene)-Au junction

Figure 10 -

 10 Figure 10 -Evolution of 𝜖 0-𝜏 , and Γ 𝜏 (in eV) as a function of the applied bias V (in mV) based on the fitting of the transmission peak. Left: 𝜖 0-𝜏 vs voltage. Right: Γ 𝜏 vs voltage.

Figure 11 -

 11 Figure 11 -Workflow for the calculation of G  /G0 and S  which yields the (Γ, 0) couple within the "G+S" method. Γ ranges from 0.01 to 0.1 eV and 0 range from -2 to 0 eV.

Figure 12 -Table 5 -

 125 Figure 12 -Left: plot of Γ"G+S" and 0-"G+S" values calculated when including in the fit the conductance G from the relation G = I/V and the Seebeck S at each voltage. Right: representation of the Lorentzian curves constructed using the previous results for three voltage ranges. The DFT/NEGF calculated transmissions at comparable voltages are shown in solid lines for comparison.

Figure 13 -

 13 Figure 13 -Top: visualization by color segment of the 100 mV parts of the I/V curve used for the integration to extract ΓI/V and 0-I/V. Bottom left: plot of the resulting ΓI/V, 0-I/V values as a function of the lower value of the bias voltage range. Bottom right: representation of the Lorentzian curves constructed using the previous results for three voltage ranges (the fit did not converge in the range V = [0.9 -1.0] V). DFT/NEGF calculated transmissions at comparable voltages are depicted as solid lines for comparison.

  Figure of Merit ZTel, non-realistic values can be obtained. For Part II Chapter II 146 example, taking S = 500 V/K, G/G0 = 10 -6 , 𝜅 𝑒𝑙 = 10 -15 pW/K, T = 300 K, gives ZTel = 5.8

Figure 14 -

 14 Figure 14 -Evolution of the Figure of Merit ZT and the power factor S 2 G as a function of 𝜖 0 and Γ. Left: evolution of 𝑍𝑇. Right: evolution of 𝑆 2 𝐺.

Figure 15 -

 15 Figure 15 -Colored area corresponding to (𝜖 0 , Γ) couples that lead to S 2 G > 5 pW/mK 2 and ZT > 1, the condition being not satisfied in the blue areas. The coloring is based on the ZT values.

Figure 16 -

 16 Figure 16 -Schematic model of a molecular junction with a side group, as considered in this study. A molecule consisting of two fragments is coupled between two electrodes. The energies of the transporting level of the two fragments are labelled 𝜖 0 and 𝜖 1 . The transfer integral or coupling between them is set to V. Γ 𝐿,𝑅 represents the coupling strength between fragment 0 and the electrodes while fragment 1 is not (weakly) coupled to them.

Figure 17 -

 17 Figure 17 -Left: representation of a transmission created with Eq. 15. The parameters are 𝜖 0 = -0.20 eV, V 2 = 10 -4 eV 2 and 𝜖 1 = -0.02 eV. Right: values of (𝜖 0 , Γ) couples for which the thermopower S 2 G > 5 pW/mK 2 and the Figure of Merit ZT > 1. The blue zone represents all the (𝜖 0 , Γ) couples which do not satisfy the above conditions.

Figure 18 -

 18 Figure 18 -Representation of the obtained values of (𝜖 0 , Γ) couples for which the thermopower S 2 G > 5 pW/mK 2 and the Figure of Merit ZT > 1. The blue zone represents all the (𝜖 0 , Γ) couples which do not satisfy the above conditions. Left: coupling V 2 = 3 eV 2 and 𝜖 1 = 0.1 eV. Right: coupling V 2 = 3 eV 2 and 𝜖 1 = 0.2 eV.

Figure 19 -

 19 Figure 19 -Chemical structures of OPE2 and OPE5 investigated from reference 48.

Table 8 -

 8 The different electrode shapes and anchoring geometries considered in this study. Flat gold electrode coordinated to a sulfur atom in hollow position. #2-hollow Gold tip (size 1) coordinated to a sulfur atom in hollow position. #3-hollow Gold tip (size 2) coordinated to a sulfur atom in hollow position. #4-hollow Gold tip (size 3) coordinated to a sulfur atom in hollow position.

  these molecular junctions in which the sulfur terminal atom is in top coordination with the gold electrodes (#1-top / #1-top), the HOMO is inevitably pinned close to the Fermi level. Its energy (compared to EF) is only marginally affected by the nature of the molecule. We illustrate this pinning effect in Figure20for BDT, BPDT and OPE2.

Figure 20 -

 20 Figure 20 -Calculated electronic transmission spectra of a benzenedithiol (BDT), biphenyldithiol (BPDT) and an oligo(phenylene ethynylene) (OPE2) with the sulfur atoms coordinated to the gold surface in (#1-top / #1-top) anchoring configuration. All these chainlike molecules are lying perpendicularly to the gold substrates.

  reports their results. The transmission for the (#1-top / #1-top) fully matches with our results (see Figure 20) and the calculation for which Part II Chapter II 153 only one contact is in #1-top position and the other in #1-hollow lead also to a pinning of the transmission peak close to the Fermi energy. The comparison between the transmission curves (#1-top / #1-top) and (T/T) allows to conclude that the effect of the gold adatom is mainly found in the strength of the coupling (weaker coupling for #1-top / #1-top than T/T) but the influence on the position of the peak is weak.

Figure 21 -

 21 Figure 21 -Effect of the coupling configuration on the transmission of an OPE2. Adapted from reference 48. See Table5and text for the labelling.

Figure 22 -

 22 Figure 22 -Transmission spectra of the OPE2 molecule with the sulfur coordinated in hollow position. #1-hollow in blue, #2-hollow in orange, #3-hollow in green and #4-hollow in red.

Figure 23 -Figure 24 -

 2324 Figure 23 -Transmission spectra of the OPE2 with the sulfur in top position with either an H or a CH3 bonded to it. #2-top in blue and #3-top in orange. The #4-top in green is with an additional gold tip and a CH3 bonded to the sulfur atoms.

Figure 25 -

 25 Figure 25 -Transmission spectra of the OPE5 with the sulfur in top position with either an H or a CH3 bonded to it. #2-top in blue and #3-top in orange.

Figure 26 -

 26 Figure 26 -Left: transmission spectra for the OPE2 in #3-hollow coordination for different types of GGA functional available in Quantum ATK. Right: table of 𝜖 0 and G/G0 for each tested GGA functional.

  thus indeed functional-dependent but the role of the diversity of configurations in the experimental results is surely also important. Considering that our goal is first to enhance the thermoelectric properties by chemical engineering, we intentionally restrict our calculations to the hollow-configuration which yields the best comparison with experiment (for 𝜖 0 at least).

Figure 27 -

 27 Figure 27 -Comparison of transmission spectra for the OPE2 molecule in #1-hollow configuration with revPBE functional and HSE06 and in #3-hollow configuration (revPBE functional).

  oligophenyl-diethynyl molecules with alkynyl end-groups, OPAn (n = 2,3, 4) where n represents the number of phenyl units, are shown in Figure1. They have been synthesized in the group of Rigaut. The MCBJs, which were fabricated by Scheer and coworkers are robust and the results are reproducible over several weeks at T = 4.2 K which allows to carry advanced physical characterization among which inelastic electron tunneling spectroscopy (IETS). Although IETS observations are known for several other anchoring groups,16,17,18 an experimental characterization of the vibrational modes from single-molecule junctions with Au-C linkers were still unknown before this work. This chapter presents the main experimental results obtained by our collaborators. We performed quantum chemical calculations at the density functional theory (DFT level) for the isolated molecules and tightbinding density functional theory (DFTB) level for both isolated and gold-coordinated molecules.

Figure 1 -

 1 Figure 1 -The bare oligophenyl-diethynyl molecules with alkynyl end-groups (OPAn) used in this study. Number of phenyl rings n = 2, 3, 4.

Figure 2 -

 2 Figure 2 -Example highlighting the difficulty to obtain molecular junctions with low concentration (left, 1 mM) compared to higher concentration (right, 5mM) for OPA4 molecules.

Figure 3

 3 Figure 3 shows representative traces obtained for each device. The noise floor of the conductance measurement is below 10 -6 G0. Au single-atom contacts form on every opening trace, as confirmed by the plateau formation at 1 G0. After the Au-Au contacts breaks, several conductance plateaus at different G for the different molecules are observed and are attributed to the formation of molecular junctions. The length of plateaus should increase with molecular length but exceptions are found in this study. Upon further stretching of the junctions, their conductance finally drops to noise level indicating the breaking of the junction. The theoretical lengths of fully stretched Au-OPAn-Au junctions are 1.2, 1.6 and 2.1 nm, for n = 2, 3, and 4,

Figure 3 -

 3 Figure 3 -Examples of opening traces with molecular plateaus and 1-D histograms observed for OPA2 (a, b), OPA3 (c, d) and OPA4 (e, f) respectively with an applied bias voltage of 100 mV. Displacement is measured as the inter-electrode distance upon stretching the junction.

Figure 4

 4 compares experimental and symmetrized IETS from 0 -300 mV for each OPAn molecule. Previous studies of surface enhanced Raman characterizations on molecular systems involving Au-C covalent bonds determined the Au-C stretch to lie around 46-52 mV. 11,27 Later, Giuseppe et al. using first principle simulations of IETS in metal-molecule junctions, found a signature of the Au-C stretch vibration mode near 60-70 mV. 28 The OPAn devices show bands around 60-75 mV, which we attribute to the vibrations of Au-C bond. The C≡C stretching mode is also observed in all molecular junctions around 250-270 mV as shown in Figure 4. 11 Typical Au phonon mode near 10-20 mV, 29 ring breathing near 140-160 mV and typical ring vibrations at 190-220 mV Part II Chapter III

Figure 4 -

 4 Figure 4 -Experimental inelastic electron tunneling spectra (IETS) for OPA2, OPA3 and OPA4 with red, green and blue curves respectively. Symmetrized IETS spectra calculated for each molecular junction with 𝑦 = [𝑓(𝑥) -𝑓(-𝑥)]/2 in black lines. The yellow regions indicate the typical energies of Au-C vibrations (60-75 mV) and C≡C stretching mode (250-275 mV).

  97 Å in a top position (optimized distance obtained with the DFTB calculations with the cluster model). The semi-infinite gold electrodes were cleaved in the (111) direction. Ghost atoms were added near the gold surface which is used to add basis sets to better depict the surfaces. The ATK computational details used are: GGA RevPBE exchange-correlation functional; FHI atomic basis set double-zeta plus polarization for C and H and single-zeta polarized basis for Au; SCF accuracy tolerance of 1.10 -5 Hartree; Density mesh cut-off 100 Hartree; 5 × 5 × 75 kpoints (along the a, b, and c directions, with c the transport direction) using the Monkhorst-Pack grid; Weighted semi-circle method with integral lower bound of 1.85 Hartree and 50 circle points; Fast Fourier transform 2-D Poisson solver. The zero-bias transmission spectra were simulated by calculating 801 points between -2.0 eV and 2.0 eV (Fermi energy located at 0.0 eV), with 5 x 5 k-points in the a and b direction.

Figure 5 -

 5 Figure 5 -Vibrational study for the OPA2 (top left), OPA3 (top right) and OPA4 (bottom) molecules at the DFT level. a) Calculated IR normalized intensity. b) Calculated normalized Raman intensity. c) Calculated vibrational density of states.

  scheme. It has to be noted that these low-energy vibrational modes are a collective motion of atoms involving gold atoms and most of the carbon atoms of the molecules. The results in Table1reveal that the triple C≡C bond vibrational modes are lowered by roughly 10 meV when the terminal carbon is substituted by Au3 (OPAn-A). This Au3-C≡C vibrational signature is found

Figure 6 -

 6 Figure 6 -DFTB vibrational density of states of OPA2-B (left) and (OPA2)2-A (right).

Figure 7 -

 7 Figure 7 -DFTB-calculated density of vibrational states for the Au3-OPA2-Au3 system.

Figure 8 -

 8 Figure 8 -Representation of the OPA2-A molecular device as simulated in Quantum ATK 2020. The yellow spheres above the gold surfaces represent ghost atoms.

Figure 9 -

 9 Figure 9 -Transmission spectra of OPA2, OPA3 and OPA4 at zero bias. ε0 given in eV is the energetic position of the main peak responsible for the conductance of the system with respect to the Fermi energy, G/G0 is the conductance.

Figure 10 -

 10 Figure 10 -Log (G/G0) as a function of the number of phenyl rings in the OPAn molecules with an exponential decay factor β = 1.05 (left) and for the scenarios OPAn-B and (OPA2)2-A

Table 2 -Figure 11 -

 211 Figure 11 -Transmission spectra for each OPAn-A and OPAn-B system. Top: transmission when both sides of the molecule are connected to the Au surface of the electrodes. Bottom: transmission when only one side is connected to the surface with the other side at the van der

  couple to generate QI effects, ideally in the vicinity of the Fermi level of the electrodes. A theoretical guidance is clearly necessary to discriminate the best systems among the really large number of chemical compositions. This research is done in collaboration with experimental physicists and chemists to have a comprehensive study that combines computational investigations, synthesis and measurements. The synthesis and characterization of target molecules are done by the group "organometallic and catalysis" in ISCR (OMC). The fabrication and characterization of the molecular devices are done in parallel on singlemolecule junctions in the group of Scheer at the University of Konstanz (UKON) and on SAMbased junctions in the group of Vuillaume at the "Institut d'électronique, de microélectronique et de nanotechnologie" (IEMN). The purpose of studying SAM configuration is to assess the possible impact of intermolecular interactions (cooperative effects) on the electrical and TE properties. The comparison between simulations and experimental results is expected to help identifying the factors to be improved in the subsequent molecular design protocol. A target goal would be to obtain thermopower-enhanced molecular units combining high conductance and high Seebeck coefficient. A power generation efficiency of 25 % is an optimum value for most applications.

  Several important milestones have been achieved since then, especially at the experimental level by the first measurements of phonon conductance and of Peltier Part II Chapter IV 184 cooling in molecular junctions. 2f A large part of the investigations has been carried out on main group molecular systems in line with the ongoing work in the closely related domains of molecular electronics. 4 So far, promising, carbon-, silicon-and more recently bio-based devices are all showing the same experimental drawback: high S but low G and consequently low ZT.

Figure 1 -

 1 Figure 1 -The general structure of the target organometallic wires.

Figure 3 -

 3 Figure 3 -Typical linkers, optical switches and anchoring groups involved in the candidate organometallic wires.

Figure 4 -

 4 Figure 4 -Chemical structures of the molecules used to test different functionalization schemes: a) Fe-bare. b) Fe-methoxy c) Fe-fluorine d) Fe-benzene and e) Fe-CH2COOH.

Figure 5 -

 5 Figure 5 -Left: DFT MO energetic diagram of the 5 systems with Au13 gold clusters coordinated by a thiol group in the hollow position. Right: Representation of the 3 orbitals with the largest iron d character for the Fe-bare cluster. Coloring of the levels: brown = virtual goldcentered MOs, yellow = occupied gold-or gold-thiol-based levels, blue = occupied MOs presenting a predominant d iron contribution (MOd), black = other occupied MOs.

Figure 6 .

 6 Figure 6. The latter are obtained as eigenstates of the transmission matrix defined by, as implemented in QuantumATK:

Figure 6 -

 6 Figure 6 -Top left: computed transmission of the five systems at the DFT/NGEF level. Top right: main transmission eigenstates of the Fe-bare device. Bottom: energy difference in eV between the Fermi level and the main molecular-based transmission states.

Figure 7 -

 7 Figure 7 -Top: computed electronic transmissions for the three different coordination modes: pp-Fe-bare, pm-Fe-bare and mm-Fe-bare. Bottom: transmission eigenstates of the three main peaks below the Fermi level of the pm-Fe-bare and mm-Fe-bare systems. The main transmission eigenstates of the pp-Fe-bare systems are shown Figure 6.

•Figure 8 -

 8 Figure 8 -Top: transmission spectra of both meta and para ferrocene systems with the metal lying out of the main conjugated path. Middle: chemical structure of the molecules, left: Ferrocene-1p and right: Ferrocene-1m. Bottom: Calculated thermo (electric) properties of the two junctions.

Figure 9 -

 9 Figure 9 -Transmission and selected transmission eigenstates of the Ferrocene-1p. Bottom: transmission pathways of the Fano resonance at E-EF = -0.54 eV (see text for more details).

Figure 10 -Figure 11 -

 1011 Figure 10 -Evolution of the figure of merit ZT as a function of the variation of the Fermi energy E for both systems with Emax = ±0.3 eV. The Seebeck coefficient is only reported for the best value of the respective ZT.

Figure 12 -

 12 Figure 12 -Top: Transmission spectra of both the meta and para ferrocene systems with the metal incorporated in the main conjugated path. Middle: chemical structure of the molecules,

  has less impact compared to Ferrocene-1p and Ferrocene-1m. The maximum of ZT is ZT2p = 1.74 compared to ZT1p = 2.74.

Figure 13 -

 13 Figure 13 -Evolution of the figure of merit ZT as a function of the variation of the Fermi level energy E for both systems with Emax = ±0.3 eV. The Seebeck coefficient is only reported for the best value of the respective ZT.

Figure 14 -

 14 Figure 14 -Comparison of the transmission spectrum upon methyl substitution of Ferrocene-1p. Top: transmission of Ferrocene-1p and Ferrocene-1p-methyl. Bottom: representation of the Ferrocene-1p substituted by 5 methyl group.

Figure 15 -Figure 16 -

 1516 Figure 15 -Electronic transmission shifts of the Ferrocene-1p-methyl as a function of the applied gate voltage, from -2.0 V to +2.0 V.

Figure 17 -

 17 Figure 17 -Evolution of ZT ( 𝑝ℎ = 20 pW/K) in red and of the Seebeck coefficient (V/K) in blue of Ferrocene-1p-methyl as a function of the applied gate voltage (V).

Figure 18 -IV. 2

 182 Figure 18 -Top: transmission spectra of both para ruthenocene systems with and without the metal in the main conjugated path. Middle: chemical structure of the molecules, left: Ruthenocene-1p and right: Ruthenocene-2p. Bottom: Thermoelectric properties of the two junctions.

Figure 19 -

 19 Figure 19 -Chemical structures of the tested molecules: [Ru(TMA)X2], with TMA = 1,5,9,13tetramethyl-1,5,9,13-tetraazacyclo-hexadecane and X corresponding to the different types of organic linker sketched.

Figure 20 -Figure 21 -

 2021 Figure 20 -Seebeck coefficient (V/K) as a function of the different linkers, in meta (m) or para (p) coordination. The error bar represents the variation of the Seebeck coefficient considering EF = ±0.2 eV.

Figure 1 -II. 1 -

 11 Figure 1 -Schematic representation of the Ru(dppe)2(CC-Ph-S(EDMS))2 and Fe(cyclam) (CC-Ph-S(EDMS))2 molecules -dppe = diphenyl-diphosphino-ethane; cyclam = 1,4,8,11tetraazacyclotetradecane; EDMS = ethyl(dimethyl)silane (protecting group).

Figure 2 -Figure 3 .

 23 Figure 2 -Left: 2-D conductance histograms of the Ru-dppe device. Right: two typical selected traces. The x axis "position" is in nm.

Figure 3 -

 3 Figure 3 -Left: typical HC current-voltage curve of the Au-Ru-dppe-Au device measured by Scheer's group at a sweeping rate of 0.003 V/s. Right: representation of 4 independent HC I-V curves at positive bias.

Figure 4 - 6 .

 46 Figure 4 -F-N plot issued from the I-V curves shown from the right of Figure 3.

Figure 5 -

 5 Figure 5 -Scheme of the molecule used to estimate 𝜖 0-𝑖𝑛𝑖𝑡 and 𝛤 from reference 13 for X = (CH2)2.

Figure 6 -II. 2 -

 62 Figure 6 -Modified SLM fit on a typical Ru-dppe I-V curve. The parameters are 𝜖 0-𝑖𝑛𝑖𝑡 = -0.30 eV, 𝛤 = 11 meV. The 𝜖 0-𝑓𝑖𝑛𝑎𝑙 (V = 1.0 V) = -0.68 eV.

Figure 18 -

 18 Figure 18 -Fe-cyclam SAM I-V curves and extracted data by Vuillaume's team. Statistical values of 𝜖 0 , using both the SLM and TVS method on a hundred of I-V curves leading to 𝜖 0 (SLM) = -0.76 ± 0.18 eV and 𝜖 0 (TVS) = -0.70 ± 0.16 eV.

Figure 19 -

 19 Figure 19 -UPS measurements (blue curve with binding energies relative to the Fermi energy of the gold substrate) for the Fe-cyclam molecule. 𝜖 0 is estimated at -0.73 eV (orange dotted lines).

  reveals really similar behaviors between the second batch of MCBJ measurements and SAM characterizations of the Ru-dppe system. The agreement is particularly good for the red IV-curve (Type-2 curve), whereas the deviations observed for the green and purple curve, associated to different values of 𝜖 0 , are certainly due to different junction geometries (Part II, Chapter II). The curvature of the I-V curves is indeed mainly governed by the position of 𝜖 0 (and with Γ predominant in the current intensity).

Figure 20 -

 20 Figure 20 -Normalized current I-V curves. Left: Ru-dppe system; CP-AFM in blue, first batch MCBJ in orange, second MCBJ batch in red, purple, and green. Right: Fe-cyclam system; CP-AFM in blue, MCBJ in orange.

V. 1 -Figure 21 -

 121 Figure 21 -A) DFT MO energy diagram of the bare Ru-dppe molecule. B) DFT MO energy diagram of the functionalized Ru-dppe molecule with two Au13 gold clusters (cluster model). Coloring of the levels: yellow = occupied gold levels, blue = occupied MOs presenting a predominant d iron contribution (MOd), black = other occupied MOs, grey = unoccupied levels.

Figure 22 -

 22 Figure 22 -Zero bias transmission of the Ru-dppe junction calculated at the DFT revPBE level in blue and DFT HSE06 level in orange. Main (thermo)electric characteristics.

Figure 23 -

 23 Figure 23 -Left: DFT/NEGF transmission as a function of the applied bias to the Ru-dppe junction (revPBE). Right: experimental and computational normalized I-V curves up to 0.8 V.

  Figure 24.

Figure 24 -

 24 Figure 24 -Zero bias DFT/NEGF transmissions of the Fe II -cyclam. Blue: #3-hollow electrode configuration, revPBE GGA functional; Green: #1-hollow electrode configuration, HSE06 hybrid functional; Orange: #1-hollow electrode configuration, revPBE GGA functional.

Figure 25 -

 25 Figure 25 -Left: DFT SO energy diagram of the bare Fe III -cyclam molecule. Right: DFT SO energy diagram of the functionalized Fe III -cyclam molecule with two Au13 gold cluster (cluster-model). Coloring of the levels: yellow = occupied gold levels, blue = occupied SOs presenting a predominant d iron contribution (SOd), black = other occupied SOs, grey = unoccupied levels.

(

  as already shown in Part II, Chapter III) to the transmission direction to gradually oxidize the molecule. The transmission spectra obtained using a positive voltage on the gate (2, 3, and 4 V) are displayed in Figure 29. The transmissions are not spin polarized anymore since we retrieve the Fe II -(cyclam) oxidation state.

Figure 26 -

 26 Figure 26 -DFT/NEGF zero-bias transmission of Fe-cyclam junction with the Cl -counter-ion inducing perturbations in the transmission shapes (red circles). Blue curve: spin down transmission; Orange curve: spin up transmission.

Figure 27 -

 27 Figure 27 -Left: DFT/NEGF zero-bias transmission of Fe-cyclam junction following the procedure B1 of Table 3. Blue curve: spin down () transmission; Orange curve: spin up () transmission. Right: main transmission eigenstates.

Figure 2 -

 2 Figure 2 -Representation of the Au-S repulsion function for the calculation of 𝐸 𝑟𝑒𝑝 .

  method is another type of DFT-based SE method designed to reproduce ground-state properties of molecules. GFN-xTB (see Scheme 2 for a listing of the xTB models) roughly uses the same approximations for the definition of the Hamiltonian and the electronic energy as DFTB42 but its parametrization does rely on pair of atoms. Indeed, the parametrization used in GFN-xTB is based on element specific and empirical fitting allowing covering most of the periodic table (up to Z = 86, Radon). The semi-empirical parameters are largely global and element-specific. As an example, for each of the second-row elements (from boron to fluorine), 11 different parameters are specified: two atomic energy levels (the 2s and 2p), their respective AO exponents, two chemical hardnesses (for the definition of chemical hardness in DFT see reference43), two electronic repulsion factors, two off-diagonal Hamiltonian polynomial factors, and one atomic-shell charge scaling parameter. The parameters are extracted using

Scheme 2 -

 2 Scheme 2 -A glance over the two GFN methods (GFN1 and GFN2) with their main characteristics. Green squares designate quantum description. Pale yellow ones describe a classical or semi-classical picture. The parts surrounded by arrows are treated selfconsistently. Adapted from reference 13. D3 (BJ) 51 and D4 52 correspond to Grimme et al. empirical dispersion.

among the 51

 51 systems, three Fe coordination complexes with different type of anchoring for the ligand: [Fe(bipy)3] 2+ (nitrogen coordination), [Fe(H2O)6] 2+ (oxygen coordination) and [Fe(CO)6] 2+ (carbon coordination). In parallel, we considered three tri-phenanthroline-based complexes encompassing different metallic centers: [M(phen)3] 2+ with M = Fe, Ni and Co.These six complexes are pseudo-octahedral compounds. For each system, we computed the metal-ligand distance (dM-L), the molecular orbitals (HOMO-1, HOMO, LUMO, LUMO+1), their energy level diagram and compared the results to DFT calculations and to an external reference.

Figure 4 and

 4 Figure 4 and Figure 5 present the energy level diagrams of the different complexes, with a comparison between DFT and TB frontier molecular orbitals. In most cases, the topology of the last occupied (HOMO and HOMO-1) and first unoccupied (LUMO and LUMO+1) are roughly retrieved by both DFTB2 and xTB. However, discrepancies appear in the relative weight of the metal and ligands in the orbital for some complexes, as well as inversions in the orbital ordering in some cases. For example, for the [Fe(phen3)] 2+ , both TB methods have MOs in good agreement with DFT. In contrast, the xTB MOs of [Ni(phen3)] 2+ are quite different from DFT MOs (no metal contribution of the virtual LUMO and LUMO+1 for

Figure 4 -Figure 5 -Figure 6

 456 Figure 4 -Molecular orbital diagram at different levels of theory for: [Fe(bipy)3] 2+ (top), [Fe(H2O)6] 2+ (middle), and [Fe(CO)6] 2+ (bottom). Occupied levels in black; Empty levels in orange. The HOMOs were aligned for sake of clarity. Iso-values is 0.05 e/bohr 3 .

Figure 6 -

 6 Figure 6 -Summary of the reference and computed metal-ligand distances and the HOMO-LUMO gaps obtained for three indicated Fe complexes.

Figure 7

 7 Figure 7 summarizes the values obtained for phenanthroline-based complexes coordinated to different metals. The electronic gaps obtained with DFTB2 and xTB methods are lower than those obtained in DFT in the case of the Fe and Ni complexes, but higher for the Co complex.

Figure 7 -

 7 Figure 7 -Summary of the reference and computed metal-ligand distances and the HOMO-LUMO gaps obtained for three indicated phenantroline complexes.

Figure 8

 8 Figure 8 displays the variations of dM-L (ΔdM-L) among all TB methods and DFT. Globally, the xTB distances are smaller than the DFT ones, and the magnitude of the error is not particularly metal-dependent. Only two systems, [Fe(C5H5)2] and [Zn(Ph)2], behave reversely with deviation of +0.117 Å and +0.103 Å, respectively. Nonetheless, the TB geometry optimizations of cobalt complexes (in orange) lead to the closest distances to the DFT values compared to the other metals.

Figure 8 -Figure 9 ,

 89 Figure 8 -dM-L, which is the difference between the metal-ligand distance obtained with the considered TB method (top: xTB, middle: DFTB2 and bottom: DFTB3) and DFT for each system. The graphs are grouped by metal and colored as following: blue is Fe, orange is Co, green is Ti, red is Ni, Zn is purple and brown is Cu.

Figure 9 -

 9 Figure 9 -Representation of HL, which is the difference between HOMO-LUMO gap obtained with the considered TB method (top: xTB, middle: DFTB2 and bottom: DFTB3) and DFT. The graphs are grouped by metal and colored as following: blue is Fe, orange is Ni, green is Zn, red is Co, Ti is purple and brown is Cu.

(

  MSE) and the mean absolute deviation around the mean value (MAD) of the dM-L and the HL gap values compared to DFT.The MSE and MAD are defined as:

  analysis increases the MSE and MAD values for each of the TB approach. The MSE is the Part II Chapter VI 270 lowest for the DFTB3 method with +0.002 Å 2 (only closed-shell systems) followed closely by DFTB2 with +0.006 Å 2 and xTB with +0.007 Å 2 . The TB methods are, overall, similarly accurate for closed-shell complexes. Adding of the open-shell systems in the analysis roughly doubles the MSE for DFTB2 and xTB with MSE of +0.011 Å 2 and +0.013 Å 2 , respectively. DFTB3 accuracy is six times lower when considering open-shell systems with a MSE value of +0.012 Å 2 . Table 4 details the error on the HL gaps. The xTB method is totally inaccurate upon addition of the open-shell systems in the set of molecules, with the MSE increasing from +0.506 eV 2 to +10.929 eV 2 . The DFTB2 and DFTB3 are more suited than xTB to describe open-shell systems. Upon inclusion of open-shell systems the MSE for the DFTB2 method is modified from +0.899 eV 2 to +0.865 eV 2 and for DFTB3 the MSE increases from +1.350 eV 2 to +1.618 eV 2 . The xTB is the closest to DFT for closed-shell systems only, and DFTB3 is the farthest.

  a first application of DFTB2 on two systems of interest for molecular junctions: Fe(cyclam)(CCPh2)2(SH)2 and Ferrocene(CCPh2)2(SH)2 (both tested in previous chapters). It can be seen from the molecular orbital diagram that some frontier orbitals are inverted compared to DFT for both molecules. This would change the transmission spectra and therefore the properties of the junction. The HL gaps are also different implying that the MOs alignment of the molecule with gold might be really different between the TB method and DFT. Finally, after making a single-point calculation on the Au13-Fe(cyclam)(CC-Ph-S)2-Au13 cluster geometry, we noticed that the d occupied MOs of the iron were located above the Part II Chapter VI 271 gold, (see Part II, Chapter III on the Au13 cluster for more details) so that an incorrect ordering of the gold orbitals and MOs is obtained. We thus tend to conclude that none of the tested TB methods can be used, even qualitatively, at the present stage to perform a screening on metallic-containing molecules. An essential improvement of the DFTB2 method encompasses necessarily the redefinition of the metallic parameter sets, specifically for targeted organometallics molecules. It is a time-consuming procedure but once finished, would permit a real breakthrough to alleviate the technological barrier that is limiting the development of molecular electronics by allowing the testing of larger molecules and polymetallic systems. Fast device optimization (i.e., concomitant optimization of the electrodes and the molecules) would allow us to consider statistically different anchoring geometries, and hence to provide a more realist comparison with experimental data in terms of conductance, thermoelectric properties and I-V curves.

Figure 10 -

 10 Figure 10 -Top) Fe(cyclam)(CC-Ph-SH)2 and bottom) Ferrocene(CC-Ph-SH)2 MOs energy diagram with the different methods. The HOMOs are aligned for sake of clarity. Right: frontier molecular orbitals obtained in DFT and DFTB2.

Titre:

  Élaboration in silico de systèmes moléculaires pour la fabrication de dispositifs en nano-et opto-électronique.Mots clés : chimie de coordination, jonction moléculaire, chimie computationnelleRésumé : Les travaux de chimie computationnelle développés dans cette thèse ont concerné deux axes de recherche. La première partie concerne l'étude théorique de complexes de cuivre(I) polynucléaires luminescents. Le deuxième chapitre s'attèle à rendre compte des résultats expérimentaux obtenus par C. Lescop et collaborateurs (ISCR -INSA Rennes) pour des dimères de cuivre pontés par trois diphosphines. Différents niveaux de calculs sont testés afin de quantifier la précision des résultats. Le chapitre suivant rassemble l'étude de composés métallacycliques apparentés comprenant respectivement 6 et 8 atomes de cuivre. Ces études montrent l'importance de la réorganisation géométrique des états excités et des interactions intermoléculaires à l'état solide. La deuxième partie de cette thèse vise à appor-ter un appui théorique au développement des dispositifs moléculaires à haute performances thermoélectriques. Le premier chapitre rend compte de l'état de l'art et des facteurs influençant la variation des propriétés thermoélectriques. Les chapitres II et III présentent les outils théoriques disponibles pour l'études des propriétés de transmission de jonctions moléculaires et de leurs caractéristiques thermoélectriques. Les atouts des composés organométalliques à induire des propriétés thermoélectriques sont traités dans le chapitre IV et V. Une augmentation sensible de la conductance et des coefficients de Seebeck est calculée. Un design moléculaire computationnel implique l'utilisation de méthodes peu coûteuses. Les méthodes "density functional tight-binding" sont évaluées à cette fin dans le dernier chapitre.Title:In silico elaboration of molecular systems for the fabrication of nano-and opto-electronic devices.

  (I) et est divisée en trois chapitres. Le premier développe les différents concepts de base de la luminescence théoriques permettant de rationaliser le transport thermoélectrique dans des jonctions moléculaires. Le deuxième chapitre introduit des paramètres permettant la comparaison entre les données expérimentales obtenues à partir des courbes courant-tension (courbes I-V)de jonctions moléculaires en utilisant différents modèles (modèle à niveau unique, « transition voltage spectroscopy », modèle à deux niveaux…) et des mesures additionnelles telles que la spectroscopie photoélectronique UV. Les différents termes de la conductance thermique phononique (ph)termes intervenant dans les grandeurs physiques thermoélectriquessont aussi expliqués et quantifiés. Enfin, la présence d'interférence quantique (IQ) sur des transmissions théoriques est abordée. L'existence de résonances Fano n'ont pas d'effet visible à température ambiante sur les propriétés thermoélectriques, en revanche la présence d'anti-résonance dans le pic de transmission le plus proche du niveau de Fermi peut améliorer les propriétés thermoélectriques par rapport à une transmission théorique de forme Lorentzienne. La présence d'anti-résonance est une caractéristique recherchée dans les chapitres suivants. Les calculs de transmission électronique de différentes jonctions moléculaires étudiées dans les chapitres III, IV et V sont réalisés en utilisant l'approche DFT couplé avec le formalisme des fonctions de Green hors équilibre (NEGF). Le chapitre III rassemble des données expérimentales obtenues par le groupe de Scheer sur les systèmes organiques, à savoir la jonction Au | oligophényl-diéthynyle | Au, dont les groupes d'ancrage sont des alcynyles (Au-C) -OPAn (n = 2, 3, 4), où n est le nombre d'unités phényles. Deux approches computationnelles (DFT/NEGF et DFTB) ont été utilisées pour rationaliser les propriétés électroniques, analyser le potentiel de la liaison Au-C à très basse température (4 K) et d'étudier les différents types jonctions possibles. L'objectif principal du chapitre IV est d'évaluer le potentiel de différents systèmes moléculaires comprenant des métaux de transition pour obtenir des propriétés thermoélectriques intéressantes. La nature du métal est très importante et l'étude montre que parmi les complexes de coordination étudiés ce sont ceux qui incorporent des atomes de fer et de ruthénium qui présentent les meilleures caractéristiques. Les dérivés du ferrocène présentent des antirésonances conduisant à des propriétés thermoélectriques intéressantes dans le cas où le niveau de transport principal peut être ajusté à l'aide d'une grille de tension pour se rapprocher du niveau de Fermi. Pour un espaceur comprenant un cycle phényle terminal auquel est coordiné le groupement d'ancrage (ici un atome de soufre) en position méta, les calculs montrent qu'il n'existe pas d'interférence Fano ni d'anti-résonance contrairement à ce qui est rapportés dans la littérature pour des jonctions moléculaires purement organiques. Les simulations montrent que les molécules en forme de « T » sont aussi une alternative prometteuse. Plusieurs limitations concernant l'approche computationnelle utilisée ont également été discutées : la différence d'énergie entre le niveau de Fermi et le premier niveau occupé ou inoccupé du système est clairement dépendant de la fonctionnelle. L'utilisation d'une fonctionnelle hybride tel que la GGA HSE06 (récemment disponible dans QuantumATK 2021) semble améliorer l'accord quantitatif entre expériences et calculs. Une méthode DFT modifiée tel que la DFT+ comprenant des corrections statiques pour les niveaux d'énergie Kohn-Sham peut également aider à mieux reproduire la conductance des molécules organiques, comme le rapportent Quek et al. pour des systèmes liés par un atome d'azote aux électrodes. La géométrie de l'électrode influence grandement la position des différents niveaux moléculaires par rapport au niveau de Fermi. L'utilisation d'une approche statistique pour considérer la diversité des jonctions observées dans les conditions expérimentales est hautement souhaitable. Les systèmes chargés sont complexes à simuler à l'aide des outils DFT/NEGF. La conductance est toujours surestimée par rapport aux données expérimentales ; ceci est certainement dû à plusieurs facteurs, le principal étant que seules les meilleures jonctions moléculaires possibles sont calculées. Pour une meilleure comparaison, la dispersion des types de jonctions expérimentales doit être prise en compte. Le chapitre V rassemble les données expérimentales obtenues par nos collaborateurs sur des jonctions organométalliques dans différentes configurations et conditions expérimentales. Sur cette base, une analyse des propriétés conductrices des jonctions et une comparaison avec les données expérimentales sont apportées. Elles révèlent un besoin de concevoir des méthodes de calcul plus rapides capables de mieux prendre en considération la diversité des géométries des jonctions révélées par la dispersion de la conduction observée dans les mesures expérimentales. L'utilisation de méthodes semi-empiriques a été évaluée dans la chapitre VI comme moyen de donner accès à cette approche statistique nécessaire pour mieux comparer la théorie avec l'expérience.

Table 1 -

 1 Typical applications and usual luminous efficiency (order of magnitude) of selected luminescence types.1

	Luminescence type	Typical application	Efficiency
	Black-body radiation	Tungsten filament lamp	≃ 5 %
	Photoluminescence	Fluorescence lamps and phosphorescent devices	≃ 20 %
	Cathodoluminescence Cathodic television screen	≃ 10 %
	Electroluminescence	Light-emitting diode (LED), flat TV screen panels	≃ 0.1 -50 %

Table 2 -

 2 Typical abundance, in mass percent, of transition metals available in the earth's

	crust. 10,16
	Fe	4.7 %
	Cr	0.01 %
	Ni	7.10 -3 %
	Zn	7.10 -3 %
	Ce	6.10 -3 %
	Cu	5.10 -3 %
	Co	2.10 -3 %
	Gd, Dy, Er, Yb, U, Eu, W, Mo, Tb, Tl, Lu, Tm	> 10 -3 ; < 10 -5 %
	Ag, Pd, Pt, Au, Os, Ru, Ir, Re	> 10 -5 %

Table 3 -

 3 Description of selected density functionals and basis sets involved in reference 58.

	Functional	Year	Type	Basis set	Type
	BLYP	1988	GGA	6-31G(d)	All electron
	BP86	1986	GGA	6-31G(d,p)	All electron
	PBE	1996	GGA	6-31+G(d)	All electron
	B3LYP	1994	M-GGA	6-31+G(d,p)	All electron
	B3LYP-D3	2010	M-GGA	SDD	Small core
	M06	2007	HM-GGA	TZV	Triple-ζ
	TPSSh	2003	HM-GGA	TZVP	Triple-ζ
	B2PLYP	2006	DH-GGA	cc-pVTZ	Triple-ζ
	mPW2PLYP	2006	DH-GGA	aug-cc-pVTZ	Triple-ζ
	CAM-B3LYP	2004	LC-H-GGA	6-311G(d,p)	Triple-ζ
	wB97XD	2008	LC-H-GGA	6-311+G(d,p)	Triple-ζ

In another work of Ansbacher et al. in reference 60, several copper (I) coordination complexes with thiophenyl ligands were calculated with several types of basis sets with the objective to establish a consistent level of prediction for Cu(I) complexes using PBE0, a hybrid-GGA functional.

From the work of Quintal et al. in reference 61, the performance and accuracy of a large selection of DFT functionals for transition metal reaction profiles were evaluated. From their point of view, there is not a single "best functional" for the transition-metal reactions, but there is a group of several hybrid-GGA functionals like PBE0, B1B95 or PW6B95 that can globally achieve the same precision. The optimal Hartree-Fock exact exchange percentages needed in Part I Chapter I 27 hybrid functionals depends on the class of systems considered. The use of hybrid-GGA density functionals appears to be suitable, notably PBE0.

Table 4 -

 4 Percentage difference of selected distances between experimental and theoretical results for Cu(I)-N bonds from reference 57.

	Functional	Basis set Cu-Nimine, guanine Cu-Nquinine	Cu-Namine
	GGA BP86		1.1 %	1.3 %	1.4 %
	GGA BLYP	def2-SVP	3.2 %	2.4 %	1.8 %
	H-GGA B3LYP		3.1 %	3.6 %	0.9 %

Table 1 -

 1 Main

experimental geometrical data of the [Cu2(dppm)3] 2+ A TEF and the DFT optimized structure of [Cu2(dppm)3] 2+ (A theo ) for sake of comparison. This will be discussed in the theoretical section. Distances are given in Å and angles in degree. For the numbering see Scheme 1.

Table 2 -

 2 Interestingly, the geometries of the [Cu2(dppa)3] 2+ ions are slightly different in the two compounds. Indeed, hydrogen interactions of different extent are found between the counterions and the protons of the bridging amine groups of the dppa ligands. The sum of the P-Cu-P angles around each copper atom is 359.4°, very close to a planar trigonal configuration,

	Compound	B BF4	B PF6	B theo	B theo/-H+	B theo/-2H+
	Cu1-Cu2	2.606	2.635	2.643	2.632	2.664
	Cu1-P1	2.296	2.300	2.339	2.332	2.331
	Cu2-P2	2.269	2.290	2.344	2.318	2.345
	Cu1-P3	2.285 *	2.286	2.344	2.347	2.320
	Cu2-P4	2.294 *	2.284	2.339	2.324	2.331
	Cu1-P5	2.322	2.279 *	2.372	2.373	2.323
	Cu2-P6	2.297	2.304 *	2.372	2.329	2.347

Main experimental geometrical data of [Cu2(dppa)3] 2+ . Distances are given in Å and angles in degree. Optimized structures of [Cu2(dppa)3] 2+ , B theo ; those for the mono and bisdeprotonated [Cu2(dppa)3] 2+ , B theo/-H+ and B theo/-2H+ , are given for sake of comparison. This will be discussed in the theoretical section. For numbering see Scheme 2. (*) dppa ligand whose amino group is interacting with a counteranion.

as already observed for the A TEF system. The Cu-Cu distance is shorter in B BF4 than in B PF6

Table 3

 3

and compared with the corresponding experimental data in Table 2. Comparing the A theo and B theo optimized structures, one of the Part I

Table 3 -

 3 Comparison between the optimized geometrical data of A

	Compound	A theo	B theo	B theo/-H+	B theo/-2H+
	Cu1-Cu2	2.799	2.643	2.632	2.664
	Cu1-P1	2.351	2.339	2.332	2.331
	Cu2-P2	2.348	2.344	2.318	2.345
	Cu1-P3	2.353	2.344	2.347	2.320
	Cu2-P4	2.348	2.339	2.324	2.331
	Cu1-P5	2.384	2.372	2.373	2.323
	Cu2-P6	2.377	2.372	2.329	2.347
	Compound	A theo	B theo B theo/-H+ B theo/-2H+
	P1-Cu1-P3	125.9	124.5	127.4	113.5
	P2-Cu2-P4	131.0	124.5	117.8	126.3
	P3-Cu1-P5	107.3	126.9	120.9	129.4
	P4-Cu2-P6	118.7	107.7	112.4	115.3
	P5-Cu1-P1	126.4	107.7	111.4	117.0
	P6-Cu2-P2	109.5	126.9	129.0	118.4

theo , B theo , B theo/-H+ and B theo/-2H+ . For numbering, see Scheme 1 and 2.

theo at the level Gauss1 and level ADF relatively

	at the
	same level of theory (basis set / functional). If unspecified, the geometry used is the optimized
	geometry at the level Gaus1 .
	Vertical excited state energies and absorption spectra
	The energies of the five first vertical singlet and triplet states were calculated for A to the energy of the ground state S0. The oscillator strength
	of the first two singlet excited states are really small (f < 0.01) so that a comparison with the
	experimental absorption spectra is not possible. It appears interesting to us to evaluate the level
	of precision of the Level Gauss1 which will be used in the next chapter by comparing it with
	alternatives, i.e. larger basis set and relativistic effect corrections. The vertical energies of the
	singlet excited states are 0.15 eV smaller on average at the Level ADF . Interestingly, the first two
	triplet excited states are a tiny less shifted (∼0.11 eV).
	A more complete study of the DFT level impact was performed for B theo . These results are
	summarized in

Table 5 . The deviation between Level Gauss1 and Level ADF is really similar to

 5

	A theo with a lowering by 0.16 eV on average of the energies of the first singlet states, and by
	0.10 eV for the first three vertical triplet states. To better discriminate between the effects of
	the augmentation of the atomic basis set, the change between Gaussian function type to Slater
	function type, and the effect of relativistic corrections, we performed the same calculations with
	different levels of theory. First, an extension of the Gaussian-type function basis set from SVP
	to 6-311G (3df, 3pd) (Level Gauss2 ) was performed (Table 5). The result on the excited state
	energies is really homogeneous with a lowering by 0.07 eV on average. Changing from

Level ADF-w/o-ZORA leads

  to the same trend, with a lowering by 0.11 eV for the singlet excited states and 0.05 eV for the triplet states. The effect of the basis set can explain the main part of the changes between Level Gauss1

	and Level ADF . Yet, the relativistic effects are also playing a part since introducing a scalar
	ZORA correction (Level ADF ) weakens by 0.05 eV more on average all excited state energies.

Table 4 -

 4 Relative energies in eV of the first five singlet and triplet vertical excited states for

A theo at the Level Gauss1 and level ADF

  . The energy of S0 is taken as reference energy.

		Level Gauss1	Level ADF
	x	Sx-vert	Tx-vert	Sx-vert	Tx-vert
	1	3.453	3.242	3.288	3.114
	2	3.655	3.385	3.492	3.299
	3	3.808	3.435	3.647	3.455
	4	3.876	3.453	3.726	3.496
	5	4.259	3.462	4.128	3.771

Table 5 -

 5 Relative energies in eV of the first five singlet and triplet vertical excited states for

the B theo at the level Gauss1 , level Gauss2 , Level ADF-w/o-ZORA , level ADF , Level ASF-SOC (see

	text). The
	energy of S0 is the reference energy.
	x

Level Gauss1 Level Gauss2 Level ADF-w/o-ZORA Level ADF Level ADF-SOC

	Sx-vert Tx-vert Sx-vert Tx-vert Sx-vert Tx-vert Sx-vert Tx-vert
	1 3.201 2.993 3.126 2.911 3.087 2.914 3.030 2.863 2.901 × 3 (∼T1)
	2 3.515 3.259 3.444 3.188 3.405 3.214 3.345 3.159 3.067 (∼S1)
	3 3.644 3.377 3.572 3.317 3.535 3.350 3.484 3.303 3.201 × 3 (∼T2)
	4 3.812 3.394 3.741 3.355 3.704 3.505 3.654 3.459
	5 3.978 3.458 3.907 3.430 3.895 3.551 3.854 3.478

theo/-H+ and B theo/-2H+ described

  meV. Inset: summary table of the wavelengths of the lowest (energy) absorption peaks in nm.

			B PF6 (H optimized)
		1st	2nd	3rd
		peak	peak	peak
	Atheo	359	339	326
	Btheo	387	352	340
	Btheo/-H+	462	381	357
	Btheo/-2H+	475	389	376
	BPF6	388	366	342
	The hydrogen interactions (partial hydrogen deprotonations) that are observed in the
	experimental X-Ray structures are detailed in the experimental section description. The studies
	of the deprotonated B previously show that the electronic
	structure is deeply modified, with the HOMO-LUMO gap getting smaller (3.57 eV (B theo/-H+ )
	vs 4.25 eV (B theo ))

theo and B theo at the level Gauss1 .Table 6 -

 6 Table 6 and Table 7 are gathering the description of the lowest singlet excited states for A Calculated electronic vertical excitation energies of A theo (level Gauss1 ). f corresponds to the dimensionless oscillator strength. The nature of the excitation in terms of the main MO transitions is given in the column "Main transition".

			A theo -level Gauss1
	State eV	nm	f	Main transition	%
	S1-vert 3.45 359 0.0038 HOMO → LUMO	91
	S2-vert 3.66 339 0.0218 HOMO-1 → LUMO 90
	S3-vert 3.81 325 0.0032 HOMO-2 → LUMO 86
	S4-vert 3.88 320 0.0489 HOMO-3 → LUMO 86
	S5-vert 4.26 291 0.1374 HOMO-4 → LUMO 72

Table 7 -

 7 Calculated electronic vertical excitation energies of B

theo , B theo/-H+ and B theo/-2H+ (level Gauss1 )

  . f corresponds to the dimensionless oscillator strength. The nature of the excitation in terms of MO transitions is given in the column "Main transition".

			B theo -level Gauss1
	State eV	nm	f	Main transition	%
	S1-vert 3.20 387 0.0019 HOMO → LUMO	93
	S2-vert 3.52 352 0.0073 HOMO-1 → LUMO 87
	S3-vert 3.65 340 0.0315 HOMO-2 → LUMO 89
	S4-vert 3.82 324 0.0742 HOMO-3 → LUMO 83
	S5-vert 3.98 311 0.2292 HOMO-4 → LUMO 84

B theo/-H+ -level Gauss1 State eV nm f Main transition %

	S1-vert 2.69 462 0.0163 HOMO → LUMO	98
	S2-vert 3.26 381 0.0137 HOMO-1 → LUMO 92
	S3-vert 3.47 357 0.0023 HOMO-2 → LUMO 68
	S4-vert 3.52 353 0.0026 HOMO-3 → LUMO 64
	S5-vert 3.68 337 0.0242 HOMO → LUMO+1 80

B theo/-2H+ -level Gauss1 State eV nm f Main transition %

	S1-vert 2.61 475 0.0144 HOMO → LUMO	96
	S2-vert 3.18 389 0.0224 HOMO-1 → LUMO 85
	S3-vert 3.29 376 0.0062 HOMO-2 → LUMO 76
	S4-vert 3.39 366 0.0090 HOMO → LUMO+1 88
	S5-vert 3.43 362 0.0221 HOMO → LUMO+2 81

Table 8 -

 8 Spin-orbit coupling matrix elements between the lowest vertical singlet and triplet excited states in cm -1 and difference in energy between selected states for A

theo and B theo (level ADF-SOPERT )

  in cm -1 and eV. The crossed-out table cells are corresponding to cases for which the triplet states are higher in energy than the considered singlet states.

A theo -level ADF-SOPERT <S|H so |T> T 1 T 2 T 3 T 4 T 5 ΔE / cm -1 (eV) S 1-vert

		7	84	197	54	25 S 1-vert -T 1-vert 1403 (0.174)
	S 2-vert	112	5	42	190	64 S 1-vert -T 2-vert -88 (-0.011)
	S 3-vert	189	52	13	86	44 S

2-vert -T 1-vert 3049 (0.378) B theo -level ADF-SOPERT <S|H so |T> T 1 T 2 T 3 T 4 T 5 ΔE / cm -1 (eV) S 1-vert

		2	122 165	27	27 S 1-vert -T 1-vert 1347 (0.167)
	S 2-vert	143	4	56	142	92 S 2-vert -T 1-vert 3904 (0.484)
	S 3-vert	147	57	6	125	75 S

3-vert -T 1-vert 5018 (0.622) III.3 -Excited state geometry relaxation, emission properties In order to study the emission properties, optimizations of the geometries of the first excited state of A theo and B theo were performed

	. Considering that the ISC should occur between S2-vert
	and T1-vert as established in the previous section, a relaxation of geometry at the T1 state is
	possible (the lifetime of triplet states is being much longer than that of singlet states). To have
	a full description of the process, other excited states should be considered. The optimized
	distances for each relevant state are given in Table for the B theo series. As expected from the
	description of the electronic density changes upon excitation, the Cu-Cu distances and Cu-P
	bond lengths are shorter in the first singlet and triplet excited states (-0.14 Å to -0.24 Å for Cu-
	Cu, and -0.02 Å to -0.03 Å for Cu-P), reorganization which occurs upon population of the
	LUMO. The metal electronic density is therefore involved in the excitation processes
	potentially reinforcing the SOC, and hence the RISC and the spontaneous emission. The
	geometries of S1, S2, T1, T2 of B theo are displayed in Figure

Table 9 -

 9 Optimized distances of A

theo and B theo in

				their different spin states S1, S2, T1, T2
	compared to S0. For numbering, refer to Scheme 1 and 2.
	A theo state	S 0	S 1	S 2	T 1	T 2
	Cu1-Cu2	2.799	2.574	2.658	2.556	2.634
	Cu1-P1	2.351	2.325	2.333	2.327	2.332
	Cu2-P2	2.348	2.351	2.382	2.363	2.375
	Cu1-P3	2.353	2.324	2.342	2.315	2.333
	Cu2-P4	2.348	2.316	2.366	2.319	2.376
	Cu1-P5	2.384	2.364	2.331	2.373	2.335
	Cu2-P6	2.377	2.335	2.311	2.340	2.314
	B theo state	S 0	S 1	S 2	T 1	T 2
	Cu1-Cu2	2.643	2.461	2.502	2.474	2.454
	Cu1-P1	2.339	2.305	2.306	2.301	2.310
	Cu2-P2	2.344	2.340	2.303	2.309	2.314
	Cu1-P3	2.344	2.338	2.371	2.379	2.314
	Cu2-P4	2.339	2.304	2.294	2.300	2.310
	Cu1-P5	2.372	2.336	2.345	2.342	2.360
	Cu2-P6	2.372	2.336	2.323	2.325	2.360

ADF-SOPERT and level ADF-SO at

	Part I	Chapter II
	the experimental result section. To be able to reproduce these features, a study of the excited
	states should be done in the solid-state. Considering that [Cu2(dppa)3] 2+ could be possibly co-
	crystallized with a non-interacting counter-ion, we performed a full study of the B theo which
	will be compared with these future results.
	The energies of the excited states and emission wavelengths of B theo states are collected in
	Figure 19 (bottom). Contrarily to A theo , S1 and S2 states are close in energy (415 cm -1 ) and only
	the triplet excited, T1, is found below. Once populated via ISC, from S2-vert to T1-vert, two main
	processes can occur: RISC that can then lead to TADF and spontaneous emission. Calculations
	at the level the T1 geometry were performed to evaluate their
	probabilities.
	Modeling the RISC from T1 to S1 implies to calculate |VSOC| 2 . Because T1 actually consists of
	three SO states, VSOC can be considered as the average of the SOC matrix element. Considering
	that the RISC process occurs in a shorter timescale than the geometrical reorganization of the
	systems during the T1 → S1 transition, the factor |VSOC| 2 is computed at the optimized geometry
	of the T1 state of B
		80 K to
		64

theo using level ADF-SOPERT and

	gives VSOC(T1-S1) = 2.5.10 -4 eV, thus, |VSOC| 2

Table 1 -

 1 Optimized

	States	E S0 (eV)	E S1 (cm -1 ) E em (eV)	E em (nm)
		First relaxed singlet excited states
	S1	3.055	0	2.502	496
	S2	3.100	364	2.492	497
	S3	3.119	524	2.511	494
		First relaxed triplet excited states
	T1	2.843	-1703	2.212	560
	T1-const	2.988	-533	2.412	514
	T2	3.090	290	2.598	477
			Experimental data
	S1	-	0	2.546	487
	T1	-	-1023	2.375	522

excited states of 1: relative energies compare to S0 and S1 and emission energies. Experimental data are given for sake of comparison.

Table 4 -

 4 Calculated vertical electronic triplet excited state energies of D (zero oscillator strength in the absence of SOC consideration). The description in terms of MO transition is not given since they are numerous for each state with participations by less than 10 %. The difference in the total density with S0 is given in Figure7for the description of T1.

	State	Energy (eV)	 (nm)
	T1	3.75	330
	T2	3.78	328
	T3	3.79	327
	T4	3.80	326

  Lescop, C. A Solid State Highly Emissive Cu-i Metallacycle: Promotion of Cuprophilic Interactions at the Excited States. Chem. Commun. 2016, 52, 11370-11373. doi:10.1039/C6CC06613E.

	Part II							Chapter I
	Chapter I -State of the art and theoretical background
	Organic I -Introduction Stacked Supramolecular	Assemblies.	Acc.	Chem.	Res.	2017,	50,	885-894.
								83

1

Lescop, C. Coordination-Driven Syntheses of Compact Supramolecular Metallacycles toward Extended Metallodoi:10.1021/acs.accounts.6b00624. 2 Lescop, C. Coordination-Driven Supramolecular Synthesis Based on Bimetallic Cu(I) Precursors: Adaptive Behavior and Luminescence. Chem. Rec. 2021, 21, 544-557. doi:10.1002/tcr.202000144. 3 El Sayed Moussa, M.; Evariste, S.; Wong, H.-L.; Le Bras, L.; Roiland, C.; Le Polles, L.; Le Guennic, B.; Costuas, K.; Yam, V. W.-W.; I.1 -Molecular electronics, general aspects

  can be used to simplify Eq. 23 by removing the integral, leading to:127

	Part II			Chapter I
	𝑆 𝑙𝑜𝑤𝑇 = -	𝜋 2 𝑘 𝐵 2 𝑇 3|𝑒|𝜏(𝐸)	𝑑𝜏(𝐸) | 𝑑𝐸	𝐸=𝐸 𝐹
				113

Table 1 -

 1 Different

	Conduction mechanism	Characteristic behavior	Temperature dependence	Voltage dependence
	Direct tunneling	𝐽 ~ 𝑉 exp (-	2𝑑 ℏ	√2𝑚𝜑 𝐵 )	None	𝐼 ~ 𝑉
	Fowler-Nordheim tunneling	𝐽 ~ 𝑉 2 exp (-	4𝑑√2𝑚𝜑 𝐵 3𝑞ℏ𝑉	3/2	)	None	ln (	𝐼 𝑉 2 )	~1 𝑉
	Thermionic emission	𝐽 ~ 𝑇 2 exp (-	𝜑 𝐵 -𝑞√𝑞𝑉/4𝜋𝜖𝑑 𝑘 𝑏 𝑇	)	ln (	𝐼 𝑇 2 )	~1 𝑇	ln(𝐼) ~ 𝑉 1/2
	Hopping conduction	𝐽 ~ 𝑉 exp (-	𝜑 𝐵 𝑘 𝑏 𝑇	)		ln (	𝐼 𝑉	)	~1 𝑇	𝐼 ~ 𝑉

models of coherent transport mechanisms for a molecular junction. Adapted from reference 9. With 𝐽 the current density, 𝑉 the bias voltage, 𝑑 the barrier length, ℏ the reduced Planck's constant, 𝑚 the mass of the electron, 𝜑 𝐵 the barrier height, and 𝑇 the temperature.

Table 3 )

 3 with respect to the reference values Γ  , 0- , S  , G  /G0 respectively.

	a)
	ΔV	R 2	ΓI/V % error	𝜖 0-𝐼/𝑉 % error	SI/V % error	GI/V/G0 % error
	0-1.0 0.997	340.0	177.1	77.8	5.1
	0-0.2 1.000	208.4	164.4	73.7	0.9
	0-0.1 1.000	398.6	335.6	84.2	0.0
	b)

Table 6 -

 6 Typical range of phononic thermal conductance 𝜅 𝑝ℎ in pW/K issued from different references, both theoretically and experimentally.

	Ref Type	Molecules	Coordination Electrode	𝛋 𝐩𝐡
	36	Experiment Alkane chains	S	Au	12 -30
	38	Theory	Alkane chains -(H and F)	S, N	Au	12 -45
	39	Theory	Alkane chains	-	-	20 -130
	40	Theory	Benzenediamine	N	Au	25
	41	Theory	Fullerene	C	Au	7.0 -46.3
	42	Theory	Oligoynes and Alkanes	S	Au	5.6 -33.4
	43	Experiment	Oligo (phenylene ethynylene)	S	Au	20  6

Table 7 -

 7 UPS experimental data for the extraction of 𝜖 0 , conductance, and UV-Vis gap for different OPE derivatives. Adapted from reference 48.

	Molecules	UPS 𝜖 0 (eV)	G/G0	UV-Vis gap (eV)
	OPE2	-0.81	1.55E-04	3.38
	OPE5	-0.49	4.75E-04	2.53

Table 1 -

 1 DFTB calculated energies of characteristic vibrational modes (meV) and their localization for OPA2, OPA2-OPA2, OPA3 and OPA4 molecules (DFT energies are given in

		Mol.
	Coordination	length	Au-C	C≡C	C=C-H
		(nm)
	OPA2	1.46	-	283 (265) -
	OPA2-A	1.24	38	275	-
	OPA2-B	1.35	26	275 283	-
	OPA2-C	1.39	43	249	-
	OPA2-D	1.16	43	-	220
	OPA2-E	1.21	48	275	219
	OPA2-F	1.25	27	274 267	-
	(OPA2)2	2.84	-	283	-

parentheses) and coordinated to two Au3 clusters at both ends (A), or coordinated to one Au3 cluster (B). For OPA2, different coordination scenarios were added (see text).

Table 1 -

 1 Description of the MOd energy levels: E(MOd-MOAu) is the energy difference in eV between the gold levels and the Fe-centered MOd; main atomic contributions are also given

	(> 3 %).
	E(MOd-MOAu)		Atomic character
	(eV)	Fe	Linker  C	 S	 Au
			Fe-bare

  Peltier effect) in pW and Vp the voltage in mV at which Qcal occurs.

	System	ZT	ZT el	S	S²G	G/G 0	 el	Q cal	V p
	Fe-bare	0.256	0.324	102	82	0.1021	76	4037	-35.0
	Fe-methoxy	0.321	0.387	108	126	0.1391	97	6207	-37.4
	Fe-fluorine	0.098	0.166	72	16	0.0397	29	742	-23.0
	Fe-benzene	0.337	0.398	107	147	0.1670	111	7245	-36.6
	Fe-CH2COOH	0.235	0.301	99	71	0.0940	71	3466	-33.6

Table 3 -

 3 Electronic and thermoelectrics data for pp-Fe-bare, pm-Fe-bare and mm-Fe-bare from their computed transmission spectrum at zero bias. Estimation of ZT with 𝜅 𝑝ℎ = 20 pW/K. Seebeck coefficient S in V/K, S 2 G the thermopower in pW/mK 2 , G/G0 is the reduced conductance. Qcal in pW, and Vp in mV.

	System	ZT	ZT el	S	S²G	G/G 0	 el	Q cal	V p
	pp-Fe-bare	0.256	0.324	102	82	0.1021	76	4037	-35
	pm-Fe-bare	0.059	0.458	133	5	0.0033	3	243	-52
	mm-Fe-bare	0.340	4.523	346	25	0.0026	2	3412	-191

Table 2 -

 2 Metal-ligand distance (dM-L), HOMO-LUMO (HL) gap and spin state of the molecules at different levels of theory. The Spin state column is the most stable spin state from the external reference. The last column is the most stable spin state in the considered level of theory. A positive E HS/LS means that the LS state is more stable.To perform the geometry optimizations at each level of theory, we used the most stable spin state either based on DFT or reference. Table1presents such computed spin states. For the selected Fe complexes, the LS state (S = 0) is the most stable for [Fe(bipy)3] 2+ at the reference level. The DFTB2 method results in an inversion of stabilization of the HS state (S = 2) over the LS state, by 0.082 eV. On the contrary, the xTB method favors the LS state by 3.129 eV. 2+ is found to be a low spin state (S = 0) in the reference calculations, and in the TB methods also by 0.762 eV and 4.979 eV, respectively. Overall, for these different Fe-based complexes, the TB methods are not consistently able to retrieve the correct spin state of the system. Especially, as enounced in the introduction, the xTB method always greatly over-stabilizes the LS state in all cases.

	Method	dM-L (Å)	HL gap (eV)	Spin state	Most stable spin state (E HS/LS in eV)
			[Fe(bipy)3] 2+
	Ref 67	1.980	-	-	-
	DFT	1.969 (-0.011)	1.63		S = 0 (+1.457)
	DFTB2	2.038 (+0.058)	1.46 (-0.17)	S = 0	S = 2 (-0.082)
	xTB	1.914 (-0.066)	1.47 (-0.16)		S = 0 (+3.129)
			[Fe(H2O)6] 2+
	Ref 68,69	2.165	-	-	-
	DFT	2.165 (+0.000)	0.40		S = 2 (-1.110)
	DFTB2	2.207 (+0.042)	0.07 (-0.34)	S = 2	S = 0 (+0.408)
	xTB	2.350 (+0.185)	8.48 (+8.08)		S = 0 (+1.605)
			[Fe(CO)6] 2+
	Ref 68,69	1.949	-		-
	DFT	1.889 (-0.060)	4.65		S = 0 (+3.767)
	DFTB2	2.146 (+0.197)	2.48 (-2.18)	S = 0	S = 0 (+0.762)
	xTB	1.858 (-0.091)	4.10 (-0.55)		S = 0 (+4.979)
			[Fe(phen)3] 2+
	Ref 67	1.975	-	-	-
	DFT	1.976 (+0.001)	1.69		S = 0 (+1.310)
	DFTB2	2.051 (+0.076)	1.54 (-0.15)	S = 0	S = 2 (-0.299)
	xTB	1.934 (-0.041)	1.50 (-0.19)		S = 0 (+3.075)
			[Ni(phen)3] 2+
	Ref 70,71,72	2.106	-		-
	DFT	2.114 (+0.008)	2.01		-
	DFTB2	2.177 (+0.071)	0.93 (-1.08)	S = 1	-
	DFTB3	2.097 (-0.009)	0.28 (-1.73)		-
	xTB	2.104 (-0.002)	1.30 (-0.71)		-
			[Co(phen)3] 3+
	Ref 73,74	1.942	-		-
	DFT	1.968 (+0.026)	2.31		HS not converged
	DFTB2	1.967 (+0.025)	2.98 (+0.67)	S = 0	S = 0 (+3.320)
	xTB	1.946 (+0.004)	2.51 (+0.20)		S = 0 (+3.510)

These tendencies are similar when changing the metal in the sub-series of phenanthrolinebased complexes. For [Fe(phen)3] 2+ , the DFTB2 calculation results in an energy inversion toward the HS state compared to references, with a ΔE HS/LS of -0.299 eV, while the xTB but the distances are either underestimated or overestimated depending on the complex. For xTB, the high spin system [Fe(H2O)6] 2+ is the less accurately reproduced, as expected.

III.1.3 -Electronic structures

In the following, the DFT electronic structures are used as reference. The HOMO-LUMO electronic gaps for the different complexes calculated with DFT and DFTB2/xTB methods are compared in

Table 2 .

 2 This value appears to be extremely sensitive to the approximations made in the TB models, and the deviations are also extremely dependent on the complex

	nature. For DFTB2, the largest deviations are found in the case of the [Fe(CO)6] 2+ complex
	(deviation of -2.177 eV) and [Ni(phen)3] 2+ (-1.076 eV) while the gap values for [Fe(bipy)3] 2+
	and [Fe(phen)3] 2+ complexes are for instance more accurately reproduced (within 0.2 eV
	only). For [Ni(phen)3] 2+ , using a DFTB3 does not reduce this error. The xTB method shows in
	general electronic gaps in better agreement with DFT than DFTB2 for closed-shell systems,
	but open-shell systems like [Fe(H2O)6] 2+ remain problematic.

Table 3 -

 3 dM-L mean, absolute mean, mean squared error (MSE) and mean absolute deviation around the mean value (MAD) for each method compared to DFT. All-systems (closed-shell and open-shell) or only closed-shell systems are considered in this analysis.

	Method	Descriptor	Mean (Å)	|Mean| (Å)	MSE (Å 2 )	MAD (Å)
	xTB	dM-L (closed-shell) dM-L (all-systems)	-0.045 -0.015	0.062 0.078	0.006 0.011	0.047 0.076
	DFTB2	dM-L(closed-shell) dM-L (all-systems)	0.011 0.043	0.054 0.073	0.007 0.013	0.070 0.076
	DFTB3	dM-L(closed-shell) dM-L (all-systems)	0.028 -0.024	0.030 0.060	0.002 0.012	0.053 0.063

Table 4 -

 4 HL gap mean, absolute mean, mean squared error (MSE) and mean absolute deviation around the mean value (MAD) for each method compared to DFT. All-systems (closed-shell and open-shell) or only close-shell systems are considered in this analysis. One tendency appears clearly from the study of Table3: having open-shell systems in the

	Method	Descriptor	Mean (eV)	|Mean| (eV)	MSE (eV 2 )	MAD (eV)
	xTB	HL (closed-shell) HL (all-systems)	0.332 -1.499	0.525 2.037	0.506 10.929	1.832 2.426
	DFTB2	HL (closed-shell) HL (all-systems)	0.279 -0.049	0.693 0.728	0.899 0.865	0.705 0.724
	DFTB3	HL (closed-shell) HL (all-systems)	0.545 -0.062	0.878 1.085	1.350 1.618	0.878 1.082

-6 1-T1-const 34 -34 1-S1 -26

Evariste, S.; Khalil, A. M.; Moussa, M. E.; Chan, A. K.-W.; Hong, E. Y.-H.; Wong, H.-L.; Le Guennic, B.; Calvez, G.; Costuas, K.; Yam, V. W.-W.; Lescop, C. Adaptive Coordination-Driven Supramolecular Syntheses toward New Polymetallic Cu(I) Luminescent Assemblies. J. Am. Chem. Soc. 2018, 140, 12521-12526. doi:10.1021/jacs.8b06901.

Czerwieniec, R.; Leitl, M. J.; Homeier, H. H. H.; Yersin, H. Cu(I) Complexes -Thermally Activated Delayed Fluorescence. Photophysical Approach and Material Design. Coord. Chem. Rev. 2016, 325, 2-28. doi:10.1016/j.ccr.2016.06.016.

Yersin, H. Highly Efficient OLEDS: Materials Based on Thermally Activated Delayed Fluorescence. Wiley-VCH Verlag GmbH & Co, 2019.

Nolas, G. S.; Sharp, J.; Goldsmid, J.; Thermoelectrics: Basic Principles and New Materials Developments.Springer, New York, 2001.

-1.0 0 -0.5 0 --0.5 0 -1.0 0 --1.0 𝜖 0 in eV; Γ in meV 𝜖 0 Γ

Quek, S. Y.; Khoo, K. H. Predictive DFT-Based Approaches to Charge and Spin Transport in Single-Molecule Junctions and Two-Dimensional Materials: Successes and Challenges. Acc. Chem. Res.

2014, 47, 3250-3257. doi:10.1021/ar4002526.

Furthermore, the fact that some of the I-V curves (Figure 8) are not symmetrical in regard to the voltage polarity is not expected for a symmetrical molecule (both in terms of anchoring and conjugated backbone). In the negative bias part, the shapes of the I-V curves of Type-1, Type-2 and Type-3 displayed in Figure 8 are similar to profiles found in presence of Kondo effect or Coulomb blockade regime which would implies at least one weak interfacial coupling. 14 We performed the fitting (SLM and TVS) procedures for the positive and negative bias for each type of I-V curve presented in Figure 8, and their results are displayed in Table 1. Table 1 -Summary of the parameters obtained by applying an SLM fit and TVS procedure on the four different types of I-V curves shown Figure 8. The missing values are due to failure of the SLM methods to fit the experimental data or when no minimum was found in the TVS graph. Note that when the SLM fit is performed in the 0 to 1.0 V range, only the higher bias part of the I-V curve (from ∼0.5 V to 1.0 V) is correctly fitted. A second fit was thus performed in a smaller bias range (0.0 to ± 0.5 V). The SLM fit was first applied in the negative bias up to Vmax = -0.5 V since it failed for larger ranges of voltage bias. For the Type-1, Type 3 and Type 4 curves, the SLM gives similar 𝜖 0 with 𝜖 0-𝑚𝑒𝑎𝑛 (V max = -0.5 V) = -0.28 ± 0.04 eV.

Part II

Method

For the positive bias, the SLM fits from 0 to 1.0 V lead to homogeneous results with 𝜖 0-𝑚𝑒𝑎𝑛 = -0.78 ± 0.03 eV for Type-1, Type-3 and Type-4 (no fit possible in Type-2). Interestingly, these fits poorly recover the I-V curves below 0.5 V bias but match very well above this threshold.

This can be indicative that the transmission is modified upon bias voltage increase (e.g. due to polarization effects).

It appears interesting to only fit the I-V curves up to 0.5 V. In that case, -0.59 eV and -0.34 eV are found for 𝜖 0 for Type-1 and Type-2 respectively. This is indeed in favor of a modification of the transmission feature upon application of the voltage.

II.3 -CP-AFM on Ru-dppe junctions (SAMs on Au substrate) -Group of D. Vuillaume

The SAMs were prepared by Vuillaume's group following a standard procedure. The I-V characteristics (back and forth) were measured at each point of a 10×10 grid with a pitch of 100 nm. About 200 I-V traces were recorded for each dataset. They repeated the measurements at several locations on the SAMs with different cantilever load forces (10-34 nN). The sweep voltage speeds ranged from 0.04 to 4.0 V/s. Importantly, the tip of the AFM cantilever is made of an alloy of Pt/Ir.

The SAM thickness determined by ellipsometry is 1.5 nm. A theoretical value of 2.1 nm is obtained considering that the EDMS groups are removed in the fabrication process. The thickness is moderately smaller compared to the theoretical value. The molecule main axis is probably tilted from the surface normal. Indeed, a tilting angle of 40° from the surface normal would induce a thickness of 1.5 nm. Another possibility could be a mixing between coordinated molecules (height 2.1 nm) and molecules lying on the substrate (height 0.8 nm, i.e., thickness of the dppe ligand around the ruthenium atom). Right: SLM fit of the positive bias part of the averaged I-V curve. Orange dotted curve: fit of the I-V from 0 to 0.5 V; Green dotted curve: fit of the I-V from 0 to 1.0 V.

The I-V curves obtained for this system are shown in Figure 9. The dataset presents a quite large dispersion. The maximum conductance was estimated from this data at 1.3.10 -3 G0. The molecular parameters 𝜖 0 and Γ were estimated from the averaged I-V curve (Figure 10) using the SLM method: 𝜖 0 = -0.32 eV and Γ = 0.89 meV in the bias range from 0 to 0.5 V. The fit performed in the 0 to 1.0 V bias range is poor in the small voltage region, as shown in Figure 10. Because the tip is not made of gold, the coupling values of the two electrodes are different (Γ 𝐿 ≠ Γ 𝑅 ). If this is considered in the fits, the obtained couplings are Γ 𝐿 = 0.81 meV and Γ 𝑅 = 0.98 meV.

Part

The extraction of 𝜖 0 from TVS done by the group of Vuillaume is complicated by the important noise (see Figure 11). The value 𝜖 0 (TVS) = -0.18 ± 0.07 eV emerges from a majority of the TVS showing a minimum. The dispersion of the TVS curves minima was also plotted, as shown in Figure 11. They also applied the SLM (up to 0.5 V) on a hundred of I-V curves to obtain a more statistical analysis leading to 𝜖 0 (SLM) = -0.32 ± 0.06 eV. The two methods are using the same dataset to normally extract the same molecular parameter 𝜖 0 . The discrepancy in the results suggests that the hypotheses and approximations are not equivalent and lead to a different descriptor. detected but published results of parent systems showed that they are extremely weak and could be hidden in the background. 1 The UPS measurements shown in Figure 12 allows to extract a "HOMO energy" (-0.34 eV) which is normally directly linked to 𝜖 0 . Even though the thickness of the SAM is smaller than expected, this 𝜖 0 (UPS) value is in adequacy with the value extracted from the statistical average of hundred I-V curves with the SLM fitting (𝜖 0 (SLM) = -0.32 ± 0.06 eV). As a reminder, the comparison between both methods should be pondered by an added image effect.

Group of E. Scheer

The same procedure as described for the Ru-dppe system was applied by the group of Scheer to study the conducting properties of Fe-cyclam-based MCBJ. In that case, different concentrations of solutions were used, from 0.05 to 0.5 mM, as summarized in Table 2. The measured conductance is spread between 10 -6 G/G0 to 10 -2 G/G0 with a main occurrence at of 10 -4 G/G0. Two series of I-V curves were measured, one from 0.0 V to 1.0 V and another from 0.0 V to 0.3 V (Figure 13). The IETS shown Figure 14 was realized on one of the representative conductance Fe-cyclam MCBJ (10 -4 G/G0). The IETS peaks are associated to specific vibrations in which the molecular signature can be extracted. 15,16,17 This technique is used to check that a single molecular junction is formed. The peaks below 20 mV are corresponding to Au bulk phonon modes. 18,19 The peaks in the 27-33 meV region are signature of Au-S vibrations. 20,21 By comparison with the vibrational properties of the isolated Fe-cyclam molecule, the peak at 100-110 mV can be attributed to molecular vibrations. 22 The CC vibrations are situated between 250-270 meV. 18,23 These results demonstrate that the Fe-cyclam is grafted to at least on electrode of the junction in this particular case.

The SLM method was applied to a sample of the high conductance I-V curves, i.e., with a conductance of 3.1.10 -4 G0. The resulting ϵ 0 and Γ are given in Figure 15. Since the curve is symmetrical only the positive bias was fitted. Depending on the integration range (0-0.5 V or 0-1.0 V), as it was the case for the Ru-dppe molecule, the energetic position ϵ 0 and the coupling strength Γ are modified (with ϵ 0 = -0.53 eV, Γ = 4.7 meV and ϵ 0 = -0.70 eV, Γ = 6.6 meV for 0 to 0.5 V and 0 to 1.0 V, respectively).

III.2 -CP-AFM on Fe-cyclam junctions (SAM on Au substrate) -Group of D. Vuillaume

The same procedure as for Ru-dppe was applied for the Fe-cyclam. The SAM thickness obtained by ellipsometry is d = 1.8 nm (theoretically ~2 nm without the EDMS protection groups). The thickness appears to be more in adequacy with a molecule grafted approximately perpendicularly to the substrate. The tilting angle to get d= 1.8 nm is 33° normal to the surface.

The I-V curves obtained for this system are displayed in Figure 16. The maximum conductance could be estimated at GHC = 6.4.10 -6 G0 (about 200 times lower than Ru-dppe). From the averaged I-V curve (Figure 17), using the SLM, the molecular parameters 𝜖 0 and Γ were estimated at: 𝜖 0 (0 -0.5 𝑉) = -0.49 eV and Γ mean (0 -0.5 𝑉) = 0.6 meV. For V = 1.0 V, the fitting is also pretty good and gives 𝜖 0 (0 -1.0 𝑉) = -0.72 eV and Γ mean (0 -1.0 𝑉) = 1.0 meV.

Using the TVS approach, |𝜖 0 | can be approximated to 0.65 eV. In Figure 18 reports the corresponding dispersions from the SLM (for 0 -0.5 V) and TVS models; a hundred of I-V curves was analyzed to provide a statistical average of 𝜖 0 , leading to 𝜖 0 (SLM) = -0.76 ± 0.18 eV and 𝜖 0 (TVS) = -0.70 ± 0.16 eV. For this molecule, both methods overlap although the variation is much more significant than for the Ru-dppe molecule.

VI -Conclusions

In this chapter, the objective was to compare experimental and computational results for junctions containing organometallic complexes. The first input of our study relates to the tests made for extracting the molecular junction characteristics from the experimental results. The SLM appears to be applicable in a restricted range of voltages for these organometallic systems in which at least two transmission regimes appear to be operative. The lack of reproducibility and the important dispersion in the experimental results are revealing that multiple configurations are involved, a feature not considered for the moment in our simulations.

Additionally, the comparison between the SAM and MCBJ measurements are revealing that the conducting features in SAM are not the simple addition of the property of the molecules incorporated in the junction.

In order to further perform predictions on the basis of computationally assisted molecular modeling to find systems presenting enhanced thermoelectric properties, a qualitative compatibility has to be at least attained between calculations and experiments on organometallic series. The computational study of the Ru-dppe led to quite satisfying results for two of the three series of measurements performed. The variation of the experimental results implies to be cautious before providing definitive conclusions. For the Fe-cyclam system, the question to be solved on both the experimental and computational point of views, is to know if the open-shell radical character of the molecule is kept in the junction configuration. We have implemented a set of protocols to access different electronic configurations. A first set of XPS measurements performed in the group of Vuillaume is showing that the signatures of Fe II and Fe III are detected. These results open interesting perspectives on the computational point of view and the enlargement of the configurational space of the junctions accessed via molecular dynamics simulations (MD) for instance would be of high interest. In the following part, the ability of TB-DFT to describe the geometry and the electronic features of metal containing systems has been investigated. This low-cost method is indeed well adapted to MD calculations.

The thermal conductance of the junctions started to be studied in the group of Vuillaume. They are currently analyzing these results in order to correlate the results to the intrinsic properties of individual molecular junctions.

Scheme 1 -Graph showing the type of methods that can be used depending on the size of the system and the time scale. Adapted from reference 1.

The semi-empirical quantum approaches (SE) are located between the ab initio and FF methods in this theoretical landscape (green in Scheme 1). They can be up to three orders of magnitude faster than DFT but remain more demanding than FF. Historically, they usually contain a smaller number of element-specific parameters than in FF frameworks. Generally, these parameters are also more transferable, as they are less dependent on the chemical nature of the systems. The major advantage of SE methods is that they can tackle almost all quantum effects described by DFT, HF or post-HF methods. The SE methods can be divided into two main families. The first is based on an approximation of the HF theory by neglecting and parameterizing some two-electron integrals (for instance via the Zero Differential Overlap approximation, ZDO 2,3 ) which leads to the neglect of diatomic differential overlap (NDDO 4,5 ) methods (MNDO, 6 AM1, 7,8 PM3 9 ). The second family of SE method is often, called "Tight-Binding (TB)" 10,11 methods which represent a DFT counterpart to HF-SE methods, and are based on the same integral neglecting/parameterization philosophy.

The two SE approaches that we will investigate in this chapter are DFT-based TB methods, namely the "Density-Functional Tight-Binding" (DFTB) and the "xTB" models. Both schemes are based on the use of a minimal valence atomic orbitals (AOs) basis set. 11,12 emission, which is highly similar to that observed for Cu4CN2 complex. By pouring a drop of solvent (dichloromethane) on the powder, a yellow emission is recovered. The computational study that was performed to date concerns the bonding dissociation energy of the pyridyl ligands to the copper to analyze the possible dissociation processes occurring in this system (concomitantly or not). The computation of the different excited states of the isolated units should help to rationalize the optical properties of such systems. The second part of this manuscript is devoted to molecular junctions. The different rationalizing schemes and methodologies in the field are given in the first chapter. The DFT/NEGF computational approach was applied to study the different molecular junctions. The main goal of this part of the manuscript is to identify the transmission features that can enhance thermoelectric properties, to compare and give insights to rationalize the experimental results, and to evaluate the potential of transition metal containing molecules for thermoelectric properties. Several factors were evaluated:

Part II -Chapter VI -Tight Binding as a prospecting method

Table S1 -dM-L values for the different complexes obtained with all the methods. The charge of the complex is given, along with the most stable spin state of the system.

Complex