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Molecular orbital (MO) approximation and basis functions

Once simplified the Hamiltonian expression, a further approximation is applied to define the functions Ψ . The orbitals are definable as the form of the wave-functions describing one of a pair of electrons in a atom. In particular, the spatial orbital is definable as a function of the position vector r that describes the spatial distribution of a given electron so that 2 i( i ) r dr is the probability to find the particle in a volume dr. Notably the spatial orbitals are orthonormals: where δij is the symbol of Kronecker (δij = 1 if i = j and δij = 0 if i ≠ j). Nevertheless, the specification of the spin state is also necessary in order to have a complete description of an electron. This condition is obtained through the definition of two orthonormal functions α(ω) et β(ω) so that two different states of spin are undertaken for each electron. Consequently, for each orbital, two different wave functions (called spin-orbitals) can be defined: Having the expression of the wavefunction describing one electron, the approximations used for the multi electron wavefunction can be considered. Notably, it can be described as the product of the mono electronic wavefunctions:        .

PT in imidazolium-imidazole complex

The stationary points (minima and transition states) of the complex ImH+-Im have been computed taking into account all the considered functionals. Figure III-3 shows the obtained structures: a perfectly orthogonal rearrangement of the two heterocycles is present in both minimum and TS. Such a conformation takes place at all the considered levels of theory, the planar rearrangement being higher in energy as for instance in the case of the TS computed at B3LYP level (1.5 kcal/mol less stable if the rings are in the same plane). A more detailed analysis of the obtained minimum shows a picture where the distance ND-H is overestimated by most of the considered functionals while an opposite tendency is observed for the distances NA-H and ND-NA . Exceptions are represented by B3LYP, BMK and LC-wPBE (Table III-4).
Concerning the structure of the TS, the differences among the considered functionals are less significant. Indeed, all the obtained N-H distances are in a small range going from 1.273 Å (LC-TCA0 and M06-HF) to 1.290 Å (TCA) while the ND-NA distance ranges from 2.545 Å to 2.580 Å but with most of the functionals giving values around 2.565 Å. Here, the best agreement with the MP2 data is given by the functionals having 22-25% of HF exchange (PBE0, mPW1PW91, and X3LYP).

PT in 1,2,3-triazolium-1,2,3-triazole complex

The stationary points of the TrH+-Tr complex are characterized at any level of theory by a planar rearrangement of the two rings (figure III-4). The different conformation respect to the ImH+-Im pair could be imputed to the reduced steric effects due to the substitution of the CH group in imidazole with a nitrogen atom. Such a replacement, moreover, is responsible of a diffirent H-bond bridge, the NN length being shorter of about 0.015 Å with respect to the ImH+-Im dimer (table III-6). This leads to shorter NA-H (about 0.015 Å) and larger ND-H (about 0.006 Å) distances, a tendency which is confirmed at each level of theory.
Concerning the performance of the considered functionals, the trend already discussed for the imidazolium-imidazole dimer is here almost respected. Again the ND-H distances of the minimum are overestimated while the NA-H and ND-NA lengths underestimated with exception represented in the first case by B3LYP and BMK and in the second one by B3LYP, BMK and LC-PBE. The largest deviations for the ND-H lengths are provided by M06HF and TCA functionals while BMK significantly overestimates both the NA-H and ND-NA distances. The best estimation is obtained using M06-2X and LC-PBE. The trends already discussed in the ImH+-Im system are confirmed also analyzing the energetic barriers. Indeed, as shown in Table III-5, the best agreement is again given by BMK (1.3 vs. 1.2 kcal/mol) and once again LC-wPBE determines the same results of MP2 (0.4 kcal/mol).
Finally, it is worth to underline that no significant variation has been found between the energetic barriers of the two analyzed complexes. The reference energy value in TrH+-Tr pair is only slightly lower (-0.2 kcal/mol) respect to ImH+-Im and this trend is confirmed at any level of theory. In short, the analyzed complexes of imidazole and 1,2,3-triazole are characterized by similar structural and energetic features.

Conduction mechanism in small models: DFT investigation

As above mentioned, in the Brédas mechanism the polymeric system is represented simply by a free chain of azole moieties interacting trough subsequent hydrogen bonds. Following this model, the excess proton is transferred between two nitrogen atoms which are equivalent at the beginning and at the end of the process (see figure IV-3, step 2). Nevertheless, as shown in figure IV-1, if the imidazoles are tethered in the polymeric backbone through a covalent binding involving the position 4 (as in P4VI), the positions 1 and 3 of the heterocycles are no longer equivalent so that different PT reactions mechanisms can be envisaged between adjacent imidazoles:
– a) Transfer between nitrogen 1 (N1) of the first cycle and nitrogen 1 (N1) of the second cycle.
– b) Transfer from N1 to N3.
– c) Transfer from N3 to N1.
– d) Transfer from N3 to N3.
Simple models of protonated dimers which explicitly include the polymeric backbone have been considered to analyze such PT reactions. Figure IV-4 shows the structures of minima and transition states (TS) obtained for all the investigated mechanisms: the proton is transferred from the donor nitrogen atom in position x (x=1,3) of the first ring to the acceptor nitrogen atom in position x (x=1,3) of the second ring, following the atom numbering listed in figure IV-1. Note that the starting and resulting minima for the PT reactions a (N1-N1) and d (N3-N3) are exactly alike (but with the two imidazoles interchanged) while the product minima by reaction b (N1-N3) is equivalent to the reagent minima of reaction c (N3-N1) and vice versa. In other words, mechanisms b and c simply correspond to two opposite directions of the same PT reaction.

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Conduction mechanism in large models: molecular dynamics simulations

According to the picture emerging from the discussed results, proton conductivity in P4VI should not be ruled by the energetic barrier of the PT reaction but by the rotation of the heterocycle bearing the excess proton. Following this hypothesis, the flexibility and in particular the reorientational dynamics of the whole polymeric chain should be crucial aspects in the conduction mechanism which, therefore, can be much better described by including the chemical environment in the investigated model. In order to explore this point, classical MD simulations have been carried out on a larger model system having 15 imidazole moieties. These simulations have been performed using a modified version of the GAFF force field, developed to reproduce the available DFT data.

Table of contents :

1. The Hydrogen Economy
1.1. Hydrogen production
1.2. Hydrogen storage
1.3. Hydrogen use
2. Fuel cells: working principle
2.1. High temperature fuel cells
2.2. An example of low temperature fuel cell: the proton exchange membrane fuel cell (PEMFC) Working principle and applications
Reducing catalyst poisoning
Increasing the catalyst activity
Increasing operative temperature
A PEMFC subcategory: DMFC
2.3. Other low temperature fuel cells
3. The key component of PEMFCs: the Proton Exchange Membrane
3.1. Mechanisms of proton transport in PEMs
3.2. Other desired properties of the membrane
3.3. The PEM state of the art: Nafion
3.4. Alternatives PEMs for high temperature PEMFCs
3.5. Azoles and azole-based polymers as proton conductors
Azoles as liquid solvents
Azole-based polymers as anhydrous proton conductors
4. Objectives
5. References
1. Quantum Mechanics (QM)
1.3. Molecular orbital (MO) approximation and basis functions
1.4. The Hartree-Fock Theory
The Hartree-Fock Equations
1.5. The correlation energy
1.7. The Coupled-Cluster (CC) theory
1.8. The Density Functional Theory
The Hohenberg-Kohn theorems
The Kohn-Sham method
Exchange-correlation Functionals
1.9. Statistical thermodynamics and partition function
Electronic partition function
Translational partition function
Rotational partition function
Vibrational partition function
2. Molecular Mechanics (MM)
2.1. Force Field
Basic principle
The Verlet integration algorithm
Choosing the time step
The thermodynamic ensembles
Molecular Dynamics at constant temperature
2.3. Free energy calculations: the umbrella sampling technique
3. References
1. Modeling proton transfer in imidazole-like dimers
1.1. Introduction
1.2. Methodological details
1.3. Identification of the PT reaction mechanisms
1.4. Basis Set selection
1.5. The reference energy values
1.6. PT in imidazolium-imidazole complex
1.7. PT in 1,2,3-triazolium-1,2,3-triazole complex
1.8. PT in tetrazolium-tetrazole complex
1.10. The BMK/B3LYP model
2. PT reactions: a benchmark study
2.1. Introduction
2.2. Methodological details
2.3. Results and discussion
Proton transfer barriers at given structure
Standard energy barrier evaluations on the DBH24/08 database
Optimized structures: PT barriers & H-bond structural parameters
3. Conclusions
4. References
1. Introduction
1.1. Poly-(4-vinyl-imidazole)
1.2. The Brédas mechanism
2. Methodological details
3. Results and discussion
3.1. Conduction mechanism in small models: DFT investigation
Protonated dimers
Protonated trimer
3.2. Conduction mechanism in large models: molecular dynamics simulations
Force field parametrization
Analysis of the trajectory
3.3. A new charge-transport mechanism
3.4. Support from experimental evidences
3.5. Charges and electrostatic potential: support to the MD simulation quality
4. Conclusions
5. References
1. Introduction
2. Results and discussion
2.1. Identification of a Gotthuss chain in the starting complex
2.2. Proton transfer reactions in the protonated model
2.3. Investigation of the rate-limiting step
3. Conclusions
4. References
1. Introduction
2. Methodological details
3. Results and discussion
3.1. Conduction mechanism in small models: DFT investigation
PT in protonated dimers
Cooperative reorientation in a trimeric model
3.2. Conduction mechanism in large models: Molecular Dynamics simulations
Force Field calibration
Analysis of trajectory
4. Comments and experimental evidences
5. Conclusions
6. References
ANNEX I: Supplementary information for chapter III
ANNEX II: Supplementary information for chapter IV
ANNEX III: Supplementary information for chapter VI
Synthèse générale
1. Contexte et objectifs
1.1. Les piles à combustible
La membrane d’échange de protons (PEM)
1.2. Les azoles et les polymères à base d’azoles comme conducteurs de protons
1.3. Objectifs de la thèse
2. Étude DFT des réactions de transfert de proton dans les azoles
2.1. Le modèle BMK/B3LYP
3. Une étude de référence de réactions PT
4. Étude du transport de protons dans P4VI
4.1. Étude DFT du mécanisme de conduction dans les petits modèles
4.2. Étude MD du mécanisme de conduction dans les grands modèles
4.3. Un nouveau mécanisme de transport de charge
5. Conduction protonique des P4VI dopés avec du H3PO4
6. Transport des protons dans les systèmes avec attache en position 2
6.1. Étude DFT du mécanisme de conduction dans les petits modèles
6.2. Étude MD du mécanisme de conduction dans les grands modèles
7. Conclusion générale
8. Références


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