Structural investigation of the High-Spin→Low-Spin relaxation dynamics in spin crossover compounds

Thesis

Reference

Structural investigation of the High-Spin→Low-Spin relaxation dynamics in spin crossover compounds

DELGADO PEREZ, Maria Teresa

Abstract

In the course of the work of this thesis we have tried to describe the main mechanisms of the HS⟶LS relaxations after photo-excitation at low temperatures in spin-crossover compounds with the fundamental objective of understanding the processes that govern the apparition and growing of domains of centres either in the HS or in the LS state and their relationship with the cooperativity that takes place in these compounds during both, the light-induced and thermal-induced spin transitions. With this purpose in mind we have focused our work on the analysis of the structural evolution of two different systems during the HS⟶LS relaxation: The Hofmann clathrate [Fe(pz)Pt(CN)4], which is very well known for its excellent spin-crossover properties, since it shows a very cooperative transition around room temperature, and the [Fe(n-Bu-im)3(tren)](PF6)2 compound, for which two different thermal spin crossover behaviors have been observed depending on the scan rate of the temperature.

DELGADO PEREZ, Maria Teresa. Structural investigation of the High-Spin→Low-Spin relaxation dynamics in spin crossover compounds. Thèse de doctorat : Univ. Genève, 2017, no. Sc. 5094

DOI : 10.13097/archive-ouverte/unige:95674 URN : urn:nbn:ch:unige-956744

Available at:

http://archive-ouverte.unige.ch/unige:95674

Disclaimer: layout of this document may differ from the published version.

UNIVERSITÉ DE GENÈVE FACULTÉ DES SCIENCES

Section de Chimie et Biochimie Professeur Andreas Hauser Département de Chimie Physique Docteure Céline Besnard

Structural Investigation of the High-Spin→Low-Spin Relaxation Dynamics in Spin Crossover Compounds

THÈSE

présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès Sciences, mention Chimie

par

Maria Teresa Delgado Pérez d’Espagne

Thèse N° 5094 GENÈVE

Atelier d’impression ReproMail 2017

Acknowledgements

Thanks to my supervisor Professor Andreas Hauser for giving me the opportunity to enter in his group, for explaining me complex things in an understandable way, for inspiring me with his example to be perfectionist and patient and overall for giving all of us the opportunity to work in an enjoyable environment. It is not surprising the value and importance that everyone here gives to your words. Thanks to my supervisor Doctor Céline Besnard for being always so accessible to any question, for dedicating part of your time to explain me all about the crystallography concepts we used in this thesis and encouraging me to treat the data by myself.

I have a lot of fun while learning and I felt very lucky for working with such a clever woman like you.

Thanks to Dr Marie-Laure Boillot and Professor Cristian Enachescu for accepting to evaluate my thesis and being the jury members of my thesis defence.

Thanks to all the researchers who collaborated with us during this thesis. Thanks to Cristian for his results about the mechanoelastic model, to Jose Antonio for sending some of the spin crossover samples that we worked with and to Phil and Dimitry for his great help during the synchrotron measurements at SNBL.

I would like to thank as well to Professor Hans Hagemann for his continues help and his fruitful ideas during seminars and journal clubs and to all the crystallography team of Professor Radovan Cerny, especially to Dr Laure Guénée who helped us with all the synchrotron measurements and subsequent data treatment. I also would like to thank all the professors in the Physical Chemistry department for their inspiring lectures during the forming period and for the departmental seminars. Thanks as well to two wonderful persons, Catherine and Nahid, who always help us more than they should. Thanks to Laurant, Patrick and Dominique who improve our measurements with their knowledge.

Thanks to all the former and current members of the group. Thanks to Antoine who dedicated part of his postdoc time to kindly show me several techniques to synthesize spin crossover compounds and how to use the main instruments in our lab. It was a pleasure to work with someone so talented. Likewise thanks to Pradip who at the beginning and also in this last period of my PhD shared with me all his knowledge and passion about spin crossover in a very transparent way. Thanks to Enza who always made me feel in family here, integrated and happy, as she always does with everyone. Thanks to my confidants and friends Jihane and

Yolanda for sharing with me the best and worst moments during this PhD. Thanks to my very good friends and colleagues Manish, Daniel Qinchao, Rania… who always supported me and take care of me. I would like to thank specially to Qinchao for helping me always with absolutely any stuff in the office and in the lab. Thanks to Jan for sharing with me this knowledge and gave me an example of goodness. Thanks to Max for having always an interesting and enriching conversation to share with me. Thanks to Jacob, Angelina, Romain, Andrea, Elia, Anne-Laure, Jakob B., Yan, Bei, Noelia, Annelis, Martin, Hugo, Jiri, Roberto, Ulf, Matteo…for all the good moments we spent together.

Thanks to my parents for their continue support during all my studies and their unconditional love and protection during all my life, to my mother for teaching me to be stubborn on my ideas and to my father who always looks for my happiness overall. Thanks to my brother for making our lives much funnier, to Mireia for becoming a lovely sister, to my nieces Andrea and Maria for bringing back to our lives the innocence and tender that so soon is loss when we grow up. Thanks to my grandparents for taking care about me, for the kindness and love.

Thanks to Melquiades, Lola, Lucia and Vicky for welcoming like a daughter and make me feel always so calm and comfortable. Thanks to my childhood friends Asun, Almu y Jess for being always there when I come back like if anything ever changed, for being my family, and to my university friends, especially to Diego for making me laugh always non-stop.

Thanks to Pablo for convincing me to leave our comfortable life and experience this adventure, for being my honest friend, my psychologist, my cooker, my GPS, my companion of adventures, the best player of my team and my best supporter.

Thanks to all of you from the bottom of my heart, I am not good at writing but I am good at faces, moments and feelings and I know I will never forget any of you.

Résumé

Les composés à transition de spin font partie des matériaux bistables les plus connus et les plus étudiés en science des matériaux. Il s’agit généralement de composés de complexes hexacoordinés d’ions de métaux de transition 3dⁿ (n = 4 - 7). L’état électronique fondamental de ces complexes est soit l’état dit bas-spin (BS), caractérisé par un appariement maximum des électrons des niveaux métalliques 3d, soit l’état dit haut-spin (HS), caractérisé par un non- appariement maximum des électrons de ces niveaux 3d. La transition de spin s’observe pour les complexes à état électronique fondamental BS. Il s’agit d’un équilibre thermique, qui admet l’entropie comme force motrice, et qui amène le système de l’état BS, exclusivement peuplé aux très basses températures, à l’état HS exclusivement peuplé aux températures élevées. Cet équilibre peut être influencé par des perturbations externes telles que la température, la pression ou la lumière. Une très grande variété de phénomènes peut être observée. L’un des plus connus est l’apparition de cycles d’hystérésis autour de la température de transition. L’hystérésis est une manifestation de la coopérativité qui résulte des interactions intermoléculaires de courtes et longues portées entre les centres métalliques. Signalons que la conversion BS⟶HS peut aussi être induite par photo-excitation et donner lieu aux températures cryogéniques à un phénomène dont l’étude est essentiel à la compréhension de la bistabilité dans les composés à transition de spin, à savoir l’effet LIESST (“light-induced exited spin state trapping”).

Les transitions de spin sont accompagnées de grands changements structuraux.¹ Ainsi, pour les complexes de fer(II), la longueur de la liaison métal-ligand varie de ~0.2 Å : des interactions élastiques apparaissent entre les centres métalliques et influencent la transition de spin thermique et photo-induite.² Par conséquent, l'analyse structurale est une approche essentielle pour la compréhension des phénomènes de transition de spin. En outre, l’utilisation de l’analyse structurelle dans l’étude de la relaxation HS⟶BS suite à la photo-excitation BS⟶HS acquiert une importance particulière pour la compréhension de la physique associée aux effets coopératifs.

C’est la raison pour laquelle, dans nos études, la diffraction sur poudres de rayons X obtenus par rayonnement synchrotron a été utilisée pour suivre aux températures cryogéniques les changements structuraux qui accompagnent la relaxation HS⟶BS dans différents composés à transition de spin.

La première partie de cette thèse porte sur le clathrate de Hofmann [Fe(pz)Pt(CN)4], pz = pyrazine. Plus précisément, une étude complète de la relaxation a été réalisée pour six différentes tailles de particules de [Fe(pz)Pt(CN)4], et une forte dépendance du mécanisme de relaxation vis-à-vis de la taille des particules a été trouvée. La diffraction de rayons X sur poudres nanocrystallines et microcrystallines de ce composé révèle une conversion photo- induite quantitative BS⟶HS à 10 K (LIESST). Les mesures résolues dans le temps confirment que la relaxation HS⟶BS dépend de la taille des particules. En effet, on observe pour les

particules microcrystallines une relaxation en deux étapes consistent en une conversion aléatoire HS⟶BS suivie d’un processus de nucleation-croissance³ ; tandis que pour les particules nanocrystallines la nucléation n’est plus observée (Figure 1).

Cette dépendance de la transition vis-à-vis de la taille des particules peut être également observée lors de la transition thermique. Ainsi, pour des particules micrométriques, la transition thermique est basée uniquement sur un mécanisme de nucléation et croissance. En revanche, pour des échantillons nanométriques, ce mécanisme de nucléation et croissance reste le principal mécanisme mais il est accompagné de façon minoritaire par une transition aléatoire de certains centres de spin.

Cette étude a été étendue à des particules nanométriques du composé [Fe(pz){Pt(CN)4(I)p}]

obtenu par addition oxydante d’iode à [Fe(pz)Pt(CN)4]. Pour les particules nanométriques de [Fe(pz){Pt(CN)4(I)p}], le mécanisme de la transition de spin thermique est similaire à celui observé pour les nanoparticules de [Fe(pz)Pt(CN)4]. Étonnamment, le passage du macroscopique aux nanoparticules n’amène pas une diminution des effets coopératifs pendant la transition thermique qui reste abrupte et a lieu aux mêmes températures. Cela est très intéressant du point de vue fonctionnel car l’utilisation des composés à transition de spin dans des dispositifs fonctionnels requiert la synthèse de nanoparticules dont les propriétés restent celles observées à l’échelle macroscopique.

La dispersion sur support solide de nanoparticules de [Fe(pz)Pt(CN)4] et [Fe(pz){Pt(CN)4(I)p}]

et l’impact significatif que cela a sur les propriétés de spin ont également été étudiés en profondeur.

La seconde partie de la thèse porte sur l’étude de la relaxation dans le composé [Fe(n-Bu- im)3(tren)](PF6)2, n-Bu-im = n-butyle imidazole et tren = tris(2-éthylamino)amine. Pour ce composé, deux types de transition thermiques sont observés en fonction de la vitesse de balayage en température.⁴ Les courbes de relaxations obtenues par spectroscopie d’absorption UV-visible présentent un plateau lorsqu’environ la moitié des centres ont relaxés vers l'état BS : cela indique une caractéristique structurelle spécifique à cette composition.

Les études par diffraction de rayons X sur monocristaux révèlent l'apparition d'une transition ordre / désordre qui dépend significativement de la température (Figure 2).

Figure 1. Comparaison des évolutions temporelles des pics de diffraction des particules de 1.3 µm et de 50 nm de diamètres du [Fe(pz)Pt(CN)4] pendant la relaxation HS⟶LS qui suit la photoexcitation dans l'état HS à 15 K.

Figure 2.Évolution temporelle à différentes températures de la fraction HS normalisée dans le composé [Fe(n-Bu-im)3(tren)](PF6)2 pendant la relaxation HS⟶LS qui suit la photoexcitation dans l'état HS à 10 K ; et évolution de la structure du complexe [Fe(n-Bu-im)3(tren)]²⁺ au cours de la transition de phase de type ordre/désordre.

Table of Contents

Chapter 1. Introduction ____________________________________________________ 13 1.1. Electronic structure of coordination compounds _____________________________ 13 1.2. Spin-crossover phenomena _____________________________________________ 16 1.2.1. Thermal transitions ________________________________________________ 18 1.2.1.1. Collective behaviour of the spin transition centres. ____________________ 20 1.2.1.2. Cooperative effects ____________________________________________ 21 1.2.2. Light-induced spin transition: LIESST and reverse-LIESST ________________ 22 1.2.3. The HSLS relaxation after LIESST __________________________________ 25 1.2.3.1. Non-adiabatic multiphonon relaxation _____________________________ 25 1.2.3.2. Cooperative effects in the HSLS relaxation ________________________ 27 1.3. The mechanoelastic model ______________________________________________ 29 1.4. Hofmann Clathrates ___________________________________________________ 32 1.4.1. Historical background ______________________________________________ 32 1.4.2. SCO Hofmann clathrates as chemosensors ______________________________ 38 1.4.3. SCO Hofmann clathrates at the nanoscale ______________________________ 43 1.5. X-ray powder diffraction as a key tool to study spin crossover phenomena ________ 49 Chapter 2. Structural investigation of the HSLS relaxation dynamics on the porous coordination network [Fe(pz)Pt(CN)4]·2.6H2O _________________________________ 51

Abstract ________________________________________________________________ 51 2.1. Introduction _________________________________________________________ 51 2.2. Experimental part _____________________________________________________ 53 2.2.1. Sample preparation ________________________________________________ 53 2.2.2. Powder X-ray diffraction ____________________________________________ 54 2.3. Results _____________________________________________________________ 56 2.3.1. The thermal spin transition __________________________________________ 56 2.3.2. The photo-induced spin transition _____________________________________ 60 2.4. Conclusions _________________________________________________________ 69

Chapter 3. Structural investigation of the thermal spin transition and HSLS relaxation dynamics on [Fe(pz)Pt(CN)4] nanoparticles and comparison with the bulk material. _________________________________________________________________ 71

Abstract ________________________________________________________________ 71 3.1. Introduction _________________________________________________________ 71 3.2. Experimental part _____________________________________________________ 73 3.2.1. Sample preparation ________________________________________________ 73 3.2.2. Optical spectroscopy _______________________________________________ 77 3.2.3. Powder XRD studies _______________________________________________ 78 3.2.4. Magnetic susceptibility measurements _________________________________ 79 3.3. Results _____________________________________________________________ 82 3.3.1. Thermal spin transition _____________________________________________ 82 3.3.1.1. Magnetic susceptibility measurements _____________________________ 82 3.3.1.2. Absorption spectroscopy ________________________________________ 84 3.3.1.3. X-ray diffraction study based on synchrotron radiation ________________ 86 3.3.1.4. Comparison of results obtained by absorption spectroscopy, powder X-ray diffraction and magnetic susceptibility measurements. _______________________ 92 3.3.1.5. Simulations of the hysteresis of spin crossover nanoparticles on a surface _ 95 3.3.2. Photo-induced LSHS spin conversion and the HS LS relaxation ________ 103 3.3.2.1. X-ray diffraction based on synchrotron radiation ____________________ 103 3.3.2.2. LS to HS irradiation followed by XRPD based on synchrotron radiation __ 122 3.3.2.3. HSLS relaxation followed by absorption spectroscopy ______________ 124 3.3.2.4. Evolution of the diffraction patterns in the framework of the Mechanoelastic Model ____________________________________________________________ 127 3.4. Conclusions ________________________________________________________ 131

Chapter 4. Hofmann-type MOF nanoparticles for iodide adsorption and sensing. ___ 133 Abstract _______________________________________________________________ 133 4.1. Introduction ________________________________________________________ 133 4.2. Experimental part ____________________________________________________ 137 4.2.1. Sample preparation _______________________________________________ 137 4.2.2. Variable temperature measurements and Light Induced Excited Spin State

Trapping by optical spectroscopy _________________________________________ 139 4.2.3. X-ray powder diffraction ___________________________________________ 139 4.2.4. Fourier transform infrared spectroscopy _______________________________ 140 4.2.5. Differential scanning calorimetry ____________________________________ 140 4.2.6. Magnetic susceptibility measurements ________________________________ 140 4.3. Results ____________________________________________________________ 141 4.3.1. FTIR changes associated with the oxidative addition of iodide _____________ 141 4.3.2. Quantity of iodide incorporated ______________________________________ 143 4.3.3. Entropy and enthalpy differences associated with the spin transition _________ 143 4.3.4. [Fe(pz){Pt(CN)4(I)0.67}] structural characterization ______________________ 146 4.3.5. Absorption spectra of [Fe(pz){Pt(CN)4(I)0.67}] __________________________ 147 4.3.6. Thermal spin transition ____________________________________________ 148 4.3.6.1. Thermal spin transition by absorption spectroscopy __________________ 149 4.3.6.2. Thermal spin transition by magnetism _____________________________ 150 4.3.6.3. Thermal spin transition by XPRD ________________________________ 152 4.3.6.4. Thermal spin transition by FTIR _________________________________ 158 4.3.7. Photo-induced LS  HS spin conversion and the HS LS relaxation _______ 160 4.4. Conclusions ________________________________________________________ 163 Acknowledgements ______________________________________________________ 164

Chapter 5. Spectroscopic and structural investigation of the thermal and photoinduced spin transition on single crystals and powder of the multi-stable molecular material [Fe(n-Bu-im)3(tren)](PF6)2 ________________________________________________ 165 Abstract _______________________________________________________________ 165 5.1. Introduction ________________________________________________________ 165 5.2. Experimental part ____________________________________________________ 168 5.2.1. Single crystal absorption spectroscopy ________________________________ 168 5.2.2. Synchrotron-based X-ray diffraction __________________________________ 169 5.3. Results ____________________________________________________________ 170 5.3.1. Thermal spin transition ____________________________________________ 170 5.3.1.1. Phase I (LS1) ________________________________________________ 171 5.3.1.2. Phase II (LS2) _______________________________________________ 173 5.3.1.3. HS phaseLS2 phase thermal relaxation __________________________ 175 5.3.1.4. Thermal transition at different scan rates___________________________ 177 5.3.2. Kinetics of the photo-induced HSLS relaxation _______________________ 180 5.3.2.1. Phase I _____________________________________________________ 180 Irradiation at 10 K ___________________________________________________ 180 Irradiation at 80 K ___________________________________________________ 184 5.3.2.2. Phase II_____________________________________________________ 198 Irradiation at 10 K ___________________________________________________ 198 5.3.3. Powder X-ray diffraction ___________________________________________ 203 5.4. Conclusions ________________________________________________________ 211 Acknowledgements ______________________________________________________ 212 General conclusions and outlook ____________________________________________ 213

Bibliography ____________________________________________________________ 217

Chapter 1. Introduction

1.1. Electronic structure of coordination compounds

The crystal field theory is one of the theoretical models to describe the nature of the metal- ligand bonding of transition metal coordination compounds. It is based on an electrostatic model of the bonding where the ligand lone pair (L:) is modelled as a negative charge, which repels the electrons of the d orbitals of the central metal ion. This theory is mainly focused on the resulting splitting of the d orbitals into groups with different energies in order to rationalize the optical spectra, the thermodynamic stability and the magnetic properties of the transition metal complexes.

For instance, in the presence of an octahedral crystal field the d orbitals of the metal are split into a lower-energy triply degenerate set (t2g) and a higher-energy doubly degenerate set (eg) separated by an energy Δ0 (known as a the ligand-field splitting parameters or crystal field splitting parameter)⁵ as illustrated in Figure 1.1b due to the different interactions of the electrons in the d orbitals of the metal with the ligands. As observed in Figure 1.1a in the case of the dz2 and dx2- y2 orbitals the electrons of these orbitals are concentrated close to the ligands along the axis and consequently they are repelled strongly, whereas the electrons of the dzx, dyz

and dxy, orbitals are concentrated in regions that lie between the ligands and in consequence the repulsion is lower, giving rise to the different energy destabilization and stabilization of these orbitals according to Figure 1.1b.

According with the crystal field theory the Δ0 value changes with the identity of the metal atom and increases along the so-called spectrochemical series of ligands as follows:

I^- < Br^- < S^2- < SCN^- < Cl^- < NO2- < N3- < F^- < OH- < C2O42- < O^2- < H2O < NCS^- < CH3C≡N <

py < NH3 < en < bpy < phen < NO2- < PPh3 < CN^- < CO

Some other generalizations can be made as for instance the fact that the Δ0 value increases with increasing oxidation state of the metal ion due to the smaller size of the ion and consequently shorter metal-ligand distance, which induces stronger interaction energies, that is to say, greater repulsion. For the same reason the Δ0 value increases when going down in a group in the periodic table due to the larger size of the 4d and 5d orbitals that creates stronger interactions compared to 3d metal ions.

However, despite of these generalizations the crystal-field theory is only useful to interpret magnetic, spectroscopic and thermochemical data by using empirical values of Δ0 and it does not explain the above mentioned ligand spectrochemical series since it just considers the ligands as point charges.

Figure 1.1. a) Orientation of the five d orbitals with respect to the ligands of an octahedral complex and b) energies of the d orbitals in an octahedral crystal field. Note that the mean energy remains unchanged relative to the energy of the d orbitals in a spherically symmetrical environment (such as in a free atom).⁶

Contrary to the crystal-field theory, the ligand-field theory takes into account the overlap of the ligand and metal orbitals of the coordination compounds to explain the nature of the metal- ligand bonding and sheds light on the real meaning of Δ0.

In the ligand-field theory the valence orbitals on the metal and ligand are used to form symmetry-adapted linear combinations (SALCs) giving rise to molecular orbitals whose relative energies are estimated by using empirical energy and overlap considerations and then verified by UV/visible absorption and photoelectron spectroscopy data.

For instance, in the case of octahedral complexes, the molecular orbitals are formed by combining ligand orbitals (SALCs) and metal atomic orbitals of the same symmetry and each combination gives rise to two different molecular orbitals, bonding and antibonding. The calculations of the resulting energies result in the molecular orbital energy level diagram represented in Figure 1.2a. Qualitatively the concept is similar to the crystal-field theory. In the ligand-field theory, Δ is the separation between the molecular orbitals largely (but not

completed) confined to the metal atom, and HS or LS complexes are obtained depending on the relative values of Δ0 and the pairing energy . As mentioned above the principal difference with respect to the crystal-field discussion is that with the ligand-field theory it is possible to understand why some ligands provide stronger and others weaker values of the ligand-field splitting. Indeed good -donor ligands provide larger Δ0 values due to the strong metal-ligand overlap that gives rise to a more strongly antibonding eg orbital.

In addition, the ligand-field theory explains the role of the π bonding and its relation with the Δ0 values. In the case of octahedral complexes as observed in Figure 1.2b the π donor ligands such as Cl^-, Br^-, OH^-, O^2- or H2O decrease the Δ0 value since when the π orbitals of the ligand are filled the non-bonding metal t2g orbitals become more anti-bonding and move closer in energy to the anti-bonding eg orbitals. In contrast as observed in Figure 1.2c π acceptor ligands such as CO, N2 or phosphines (PR3) increase the Δ0 value since the π acceptor orbitals of the ligand form bonding molecular orbitals (t2g), which are largely of metal d-orbital character and in consequence lie at lower energy than the d orbitals themselves. Indeed the order in the spectrochemical series that could not be explained by the crystal-field theory is a function of the strengths with which the π orbitals of the ligand participate in the metal-ligand  bonding.⁶

Figure 1.2. a) Molecular orbital energy levels of a typical octahedral complex. The frontier orbitals re inside the tinted box, b) the effect of π bonding on the ligand-field splitting parameter. Ligands that act as π donors decrease Δ0. Only the π orbitals of the ligand are shown, and c) ligands that act as π acceptors increase Δ0. Only the π orbitals of the ligand are shown.

Adapted from reference [6].

1.2. Spin-crossover phenomena

Spin-crossover complexes present different magnetic, optical, electrical and structural properties depending on the electronic configuration that can switch from low spin (LS) to high spin (HS) and vice versa by changes of the temperature,⁷ pressure,⁸ magnetic fields⁹ and also by light irradiation.^{2, 10} This spin transition can be followed by different techniques according to the changes occurring: Magnetic susceptibility measurements, Mössbauer spectroscopy, UV-Vis-NIR absorption spectroscopy, vibrational spectroscopy, X-ray diffraction studies, synchrotron radiation studies, heat capacity measurements, magnetic resonance studies, etc. As can be observed in Figure 1.3 the electronic configuration change between the LS and the HS state is generally associated with a colour change, which usually is even visually detectable.

Figure 1.3. Electronic configurations of the LS and HS states of Fe²⁺ and corresponding colour changes of the complex [Fe(pz){Ni(CN)4} ] from red (LS) to yellow (HS).

The spin-crossover phenomena are present in d⁴-d⁷ transition metals such as Fe²⁺, Fe³⁺, Co²⁺, etc. In Figure 1.3 the d⁶ electronic configuration of Fe²⁺ is represented. The ligand field strength, as mentioned above, depends on the nature of the ligand and metal ions. However for a given combination of ligands and metal, the ligand-field splitting Δ0, referred to in the drawing as 10Dq, depends on the metal-ligand distance as 1/rⁿ, with n = 5 - 6.¹¹ In addition, in order to analyse the electron-electron repulsion for systems with more than one d electron, the concept of spin-pairing energy, 𝛱, is used. Thus if 𝛱 is large compared to 10Dq, or in other words, 10Dq is lower than 1/rij, the six d electrons of the Fe²⁺ will be distributed in the molecular orbitals with maximum value of the multiplicity according to Hund’s rule giving rise to a paramagnetic complex with S = 2. In contrast, when the value of Π is low compared to 10Dq

(10Dq is higher than 1/rij), the electrons will accommodate only in the t2g orbitals giving rise to a diamagnetic complex with S = 0.

An easy way to visualize the possible electronic transitions occurring on a transition metal complex depending on the relative values of the ligand field strength and the pairing energy is by using the Tanabe-Sugano diagrams.¹² In these diagrams the splitting of the different energetic states of a free metal ion under the influence of an octahedral ligand field are represented for a given dⁿ configuration. In the case of Fe²⁺ complexes, the corresponding Tanabe-Sugano diagram for a d⁶ configuration is displayed in Figure 1.4. As observed, on the y-axis the electronic energies of the excited states relative to the ground state in units of the Racah parameter B (which account for the electron-electron repulsion) are represented. These terms take into account the electron-electron repulsion present in the ion and are written in the form of ^2S+1L according to the Russel-Saunders-coupling scheme.¹³ On the x-axis the value of the ligand field strength with respect to the same Racah parameter B is represented. Thus when the value of 10Dq/B is zero on the left side of the x-axis the terms indicated on the y-axis are the ones of the free metal ions whereas when the value of 10Dq/B increases the terms split. In the case of the Fe²⁺ complexes as mentioned before for low values of 10Dq/B, this splitting gives rise to a new ground state ⁵T2g(t2g4eg2) with a corresponding excited state ⁵Eg(t2g3eg3) in the HS state, whereas above a critical 10Dq/B value where 10Dq = 𝛱, the new ground state is

1A1g(t2g6eg0) in the LS state. Thereby it is possible to assign the corresponding spectroscopic d- d transitions of the compound with the help of the Tanabe-Sugano diagrams. The energy value of the d-d band in absorption can be directly related to the 10Dq/B value of the diagram and analysed for the corresponding vertical transition in the diagram because according to the Frank-Condon principle the geometry of the molecule, and therefore the ligand field strength, do not change during the absorption process.¹⁴ Likewise the value of 10Dq can be extracted from the spectra by calculating the value of B in the complex with the help of the value of B in the free ion and the value of the nephelauxetic parameter, . This later parameter represents the ratio B(complex)/B(free ion) and it gives an idea of how much the electron-electron repulsion decreases when going from the free ion to the complex due to the delocalization of the d electrons on the ligands. Thus it depends on the identity of the ligand and of the metal in such a way that a lower value is obtained when the complex presents a higher covalent character.

In this regard, in 1958 Schäffer et al.¹⁵ established that according to the nephelauxetic series the value of for a series of ligands increases as follows: Br^-< CN^-<, Cl^-< NH3< H2O< F^-

Figure 1.4. Tanabe-Sugano diagram for the octahedral d⁶ electron configuration. Adapted from reference [14].

1.2.1. Thermal transitions

The case of Fe²⁺ is of particular interest since, when changing the electronic configuration from the diamagnetic LS ^𝟏𝑨_𝟏𝒈(𝒕_𝒆𝒈^𝟔 𝒆_𝒈^𝟎) state to the paramagnetic HS ^𝟓𝑻_𝟐𝒈(𝒕_𝒆𝒈^𝟒 𝒆_𝒈^𝟐) state, two electrons move into the eg orbitals, which have antibonding character. Thus the metal-ligand (usually nitrogen atom) bond length increases by 𝜟𝒓_𝑯𝑳 = rHS – rLS= 0.2 Å.^1-2 This means that important structural chances are occurring during the transition.

Knowing that 10Dq, depends on the metal-ligand distance as 1/rⁿ, with n = 5 - 6 and that 𝛥𝑟_𝐻𝐿 is approximately 0.2 Å, if the two potential wells of the HS and LS are depicted along the

totally symmetric stretching vibration of the metal-ligand bond they must be displaced relatively to each other both horizontally and vertically as shown in Figure 1.5. Indeed thermally the complexes remain in the LS state at low temperatures and at high temperature there is an entropy-driven population of the HS state due to the higher electronic degeneracy and the higher density of vibrational states in the HS state. Thus when the thermal energy kBT is of the order of the zero point energy difference between the two states, 𝜟𝑬_𝑯𝑳^𝟎 = 𝑬_𝑯𝑺^𝟎 − 𝑬_𝑳𝑺^𝟎 , it is possible to induce the thermal spin transition. During the spin transition the metal-ligand bond length increases therefore abruptly and the same occurs with the ligand field value. In this regard the relation between the 10Dq value in the HS and LS state is given by equation 1.1. Thus by considering 𝑟_𝐻𝑆 ≈ 2.2 Å and 𝑟_𝑙𝑆≈ 2.0 Å , the ratio 10Dq^LS/10Dq^HS gives a value of around 1.75. This value has been in fact verified experimentally.¹⁴

10𝐷𝑞^𝐿𝑆

10𝐷𝑞^𝐻𝑆

= (

^𝑟𝐻𝑆

𝑟_𝐿𝑆

)

^𝑛

;

n = 5 - 6 (1.1)

Figure 1.5. Adiabatic potentials for the HS and the LS state along the most important reaction coordinate for spin crossover, namely the totally symmetric metal-ligand strech vibration r(Fe- L). As an example of the electronic configuration of the LS and HS states of Fe²⁺ are schematized.

It is important to underline that even if at the crossing point in the Tanabe-Sugano diagram 10Dq = 𝛱, the value of the spin pairing energy is actually not changing significantly from the LS to the HS state, which makes sense considering the strong reduction of the electron-electron repulsion from the free ion to the complex of around 70-80%. 𝛱 is just slightly larger in the HS state due to the larger bond-length, but the parameter that is really changing is 10Dq in such a way that in the LS state the 10Dq value is substantially larger than 𝛱 and in the HS state the 10Dq value is substantially smaller than 𝛱. Thus it is possible to evaluate whether the transition will take place for a giving system with a given value of 𝛥𝐸_𝐻𝐿⁰ just by analysing the values of the ligand-field strength. Indeed the spin crossover phenomena takes place only for a narrow range of 𝛥𝐸_𝐻𝐿⁰ between 0 and 2000 cm^-1, corresponding to a 10Dq values between 19000 - 20000 cm^-1 for the LS state and 11000 - 12500 for the HS state.¹⁴

1.2.1.1. Collective behaviour of the spin transition centres.

The structural changes in spin-crossover (SCO) systems can be spread cooperatively through the whole solid through elastic interactions. When the cooperativity between the SCO centres is weak like in diluted compounds, the thermal spin transition is gradual following just the temperature dependence of a Boltzmann distribution between the molecular states, whereas when the cooperativity is strong as in concentrated solids, first-order phase transitions accompanied by hysteretic effects can occur. In addition there are some especial cases with two or more steps in the transition curve that can be due to the presence of different lattice sites, or preferential formation of HS/LS pairs, etc. Finally it can also occur that the transitions are incomplete due to the presence of different crystallographic sites. This phenomenon is typical of nanoparticles where the HS centres on the surface have different coordination and normally lower values of the ligand field strength so they remain in the HS state even at low temperatures. All the different types of thermal spin transitions described above are represented in Figure 1.6, where 𝛾_𝐻𝑆 represents the high spin fraction, and 𝑇_1/2 is defined as the temperature at which 𝛾_𝐻𝑆= 0.5, sometimes also called the average of the critical temperatures between 𝑇_{𝑑𝑜𝑤𝑛} (critical temperature during the cooling) and 𝑇_𝑢𝑝 (critical temperature during the heating) for systems with a hysteresis.

Figure 1.6. Representation of the collective behaviour of the spin crossover centres in different types of systems.¹⁶

1.2.1.2. Cooperative effects

In the case of gradual transitions the thermal LS↔HS equilibrium is simply described by equation 1.2.

𝛥𝐺_𝐻𝐿^𝑥→0 = 𝛥𝐻_𝐻𝐿^𝑥→0− 𝑇𝛥𝑆_𝐻𝐿^𝑥→0 = −𝑘_𝐵𝑇 𝐿𝑛 ( ^𝛾𝐻𝑆

1−𝛾_𝐻𝑆) , (1.2)

where 𝛥𝐺_𝐻𝐿^𝑥→0, 𝛥𝐻_𝐻𝐿^𝑥→0 and 𝛥𝑆_𝐻𝐿^𝑥→0 are the standard differences in Gibbs free energy, enthalpy and entropy respectively between the HS and LS state in an infinite solution, for instance for the iron(II) complex doped into an inert host lattice of the corresponding zinc(II) complex.

Consequently in the thermal equilibrium between the two states when 𝛥𝐺_𝐻𝐿^𝑥→0 = 0, 𝑇_1/2 can be calculated as indicated in equation 1.3.

𝑇_1/2 = ^{𝛥𝐻𝐻𝐿𝑥→0}

𝛥𝑆_𝐻𝐿^𝑥→0

^(1.3)

However in the case of concentrated systems a new parameter that accounts for the cooperativity should be introduced. This cooperativity is the result of the difference in size between the molecule in the HS state and the molecule in the LS state. Thus it has an elastic origin which is transmitted by long-range interactions though the whole material. Thereby in

the mean-field approximation the thermal equilibrium for neat systems is described as in equation 1.4.¹⁷

𝛥𝐺_𝐻𝐿 = 𝛥𝐻_𝐻𝐿^𝑥→0− 𝑇𝛥𝑆_𝐻𝐿^𝑥→0+ 𝛥_𝐿− 2Γ (γ_𝐻𝑆− 1/2) , (1.4) where Γ is the above mentioned parameter of cooperativity that represents the tendency of a molecule in a given spin state of being surrounded by other molecules in the same spin state.

Thus Γ represents the efficiency with which the structural changes associated with the spin transition are transmitted through the system. On the other hand 𝛥_𝐿 represents the difference in interaction of the two states with the reference or host lattice chosen, with which 𝛥𝐻_𝐻𝐿^𝑥→0, 𝛥𝑆_𝐻𝐿^𝑥→0 and 𝛥𝐺_𝐻𝐿^𝑥→0 have been calculated.

1.2.2. Light-induced spin transition: LIESST and reverse-LIESST

The irradiation into the characteristic absorption bands of the LS species at low temperatures results in a light-induced population of the HS state at the expense of the population of the LS state. At cryogenic temperatures the HSLS relaxation slows down to such an extent that it is possible to trap some spin crossover compounds in the HS state. This phenomena is known as

“light-induced exited spin state trapping” (LIESST).² In the case of Fe²⁺ complexes it is possible to achieve a quantitative conversion by irradiating the sample in the UV-Vis or near- IR regions at low temperature, typically at 10 K, that means, in the LS state, to populate different excited states of the molecule (the ¹MLCT and ¹T1,2 states from the ¹A1 ground state via a spin-allowed and the ³T1,2 states via spin-forbidden transitions). Then the excited molecule relaxes non-radiatively to the ⁵T2 HS state by intersystem crossing through the triplet states as it is schematized in Figure 1.7. At cryogenic temperatures due to the large difference in geometry between the two states (large difference in the metal-ligand bond length, ∆𝑟_𝐻𝐿 ) and the small value of the zero-point energy difference between the HS and the LS state, 𝛥𝐸_𝐻𝐿⁰ , the energetic barrier is too high for the system to come back to the ground state ¹A1. Thus the relaxation is slow enough to trap the HS state quantitatively. Likewise, it is possible to pump the system back to the LS state by selective irradiation into the spin-allowed ⁵T2  ⁵E absorption band of the HS species in the near-IR.

For some systems, particularly for those with intense charge transfer bands in the visible range, certain precautions have to be taken: First, when the extinction coefficient, , is of the order of 10⁴ M^-1 cm^-1 the penetration depth of the light is lower than 1 µm, then the LIESST effect is

more effective at the surface and it creates concentration gradients, which may lead to the destruction of the crystals. Thus it is often advisable to irradiate the sample into the tail of an absorption band rather than at the maximum. Secondly, if the value of  of the HS species is different from zero at the maximum of the MLCT band of the LS species and if the HSLS relaxation is not extremely slow, it may be rather difficult to achieve full HS state trapping.

Figure 1.7. LIESST and reverse-LIESST mechanisms in Fe²⁺ spin crossover complexes.

The quantum efficiency in the case of the LIESST effect is defined as the number of Fe(II) centres converted to the HS state per absorbed photon. Several experimental parameters have to be controlled with precision during LIESST:

a) The absorption cross section of the LS state at the irradiation wavelength should be known precisely and in turn the absorption of the HS state at this wavelength should be minimum.

b) In order to avoid concentration gradients during the irradiation the crystal should be thin enough in such a way that the optical density remains below 0.1 at the irradiation wavelength.

c) The possible baseline shifts due to deterioration in crystal quality appearing during the optical density measurements should be corrected by recording the full absorption spectra of the sample.

Taking into account all these parameters the bleaching of the LS state is given by the differential equation 1.5.

𝑑𝛾_𝐿𝑆

𝑑𝑡 = − 𝑘_𝑒𝑥 𝜂𝛾_𝐿𝑆 = − 𝑘_𝑜𝑏𝑠 𝛾_𝐿𝑆(1.5) where 𝑘_𝑒𝑥 is the excitation rate constant, which in turn can be written as 𝜎_𝐿𝑆𝜙, with 𝜎_𝐿𝑆 the absorption cross section of the LS species (𝜎_𝐿𝑆 = 3.82^.10^-21 𝜀_𝐿𝑆) and 𝜙 the photon flux (𝜙 = 𝐼/ℎ𝜈_𝑒𝑥), and is the quantum efficiency of the double intersystem crossing step.

The solution of equation 5 is given by equation 1.6.

𝛾_𝐿𝑆 = 𝛾_𝐿𝑆⁰ 𝑒^{−𝑘𝑜𝑏𝑠 𝑡} = 𝛾_𝐿𝑆⁰ 𝑒^{−𝜂 𝑘𝑒𝑥 𝑡} = 𝛾_𝐿𝑆⁰ 𝑒−𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝜂 𝜀 𝐼 𝑡 , (1.6) where 𝛾_𝐿𝑆⁰ is the initial LS fraction before irradiation.

In addition, when the lifetime of the excited HS state is short, the HSLS relaxation competes with the LSHS excitation and therefore it should be included when analysing the bleaching of the LS state upon irradiation as indicated in equation 1.7.

𝑑𝛾_𝐿𝑆

𝑑𝑡

= − 𝑘

_𝑒𝑥

𝜂𝛾

_𝐿𝑆

+ 𝑘

_𝐻𝐿

𝛾

_𝐻𝑆



(1.7)

It is important to mention that in the case of thick crystals, the gradient of the concentration created in the sample upon irradiation should be considered in the calculation. Thus the intensity of the irradiation and the concentration of the sample will depend on x and time.

In order to calculate the reverse-LIESST efficiencies the same procedure is applied.

Regarding the concrete values of quantum efficiencies the typical value for the LIESST effect is around unity although this value can slightly change depending on the compound, whereas the quantum efficiency of the reverse-LIESST effect is generally smaller. For systems with only high-energy MLCT states it is around 0.1, for those with lower-energy MLCT states it usually lower. Thus the light-induced return to the LS state (reverse-LIESST) is not fully quantitative and even after prolonged irradiation a LS fraction of only 0.85 is obtained for the former due to the spectral overlap between the ⁵T2  ⁵E transition band of the HS state and the

1A1 to ³T transition bands of the LS state, which leads to a steady state-type situation. In any case LIESST and reverse-LIESST are properties of the individual complex.

1.2.3. The HSLS relaxation after LIESST

1.2.3.1. Non-adiabatic multiphonon relaxation

In Figure 1.8 the potential wells of the HS and LS states are schematized. In this figure the reaction coordinate Q is described by the totally symmetric breathing mode of the spin crossover compounds and the difference value between both states is typically around 0.5 Å.

Nevertheless in many occasions the reaction coordinate can be directly found in the drawings in terms of metal-ligand bond lengths difference between the HS and LS states, which is typically around 0.2 Å, keeping a relation with Q given by 𝛥𝑄_𝐻𝐿 = √6𝛥𝑟_𝐻𝐿.

Figure 1.8. Schematic representation of HS and LS potential wells along the reaction coordinate Q. The zero-point energy difference 𝛥𝐸_𝐻𝐿⁰ depends primarily on ligand characteristics. In concentrated spin-crossover compounds it is strongly influenced by cooperative effects. In the mean-field approximation, 𝛥𝐸_𝐻𝐿⁰ is regarded as a linear function of the LS fraction.¹⁸

Classically, the complexes trapped in the HS state would have to acquire enough thermal energy to pass the top of the energy barrier between the two potential wells of the HS and LS state. Quantum mechanically the tunnelling probability for a spontaneous, non-radiative process from a given vibrational level m of the HS state to a vibrational level m’ of the LS state is given by Fermi’s Golden Rule (Equation 1.8)¹⁸:

𝑊_𝑚𝑚′ = ^2𝜋

ℏ²𝜔 𝛽_𝐻𝐿 ² |⟨𝑋_𝑚|𝑋_𝑚′⟩|² 𝛿 (𝐸_𝑚^′𝐸_𝑚)

,

(1.8) where ℏ𝜔 represents the vibrational energy of the active vibration, 𝛽_𝐻𝐿 is the electronic coupling matrix element ( 𝛽_𝐻𝐿 = ⟨𝜙_𝐿𝑆|𝐻_𝑠𝑜|𝜙_𝐻𝑆⟩ ), |⟨𝑋_𝑚|𝑋_𝑚^′⟩|² is the Frank-Condon factor of the overlap between the vibrational wave functions of the HS and the LS state at the corresponding energy, and 𝛿(𝐸_𝑚^′, 𝐸_𝑚) ensures the energy conservation.

For simplicity the force constant and the vibrational frequencies of the potentials of both states are assumed to be equal according to harmonic approximation. Thus the energy conservation principle requires that:

𝑚^′= 𝑚 + ^{𝛥𝐸𝐻𝐿0}

ℏ 𝜔 = 𝑚 + 𝑛

,

(1.8b)

where n is called the reduced energy gap and represents the measure of the vertical displacement of the potential wells relative to each other.

The HSLS relaxation rate constant, 𝑘_𝐻𝐿, can be then be expressed as a thermal average over all vibrational levels of the HS state according to equation 1.9.

𝑘_𝐻𝐿(𝑇) = ^2𝜋

ℏ²𝜔 𝛽_𝐻𝐿 ² 𝐹_𝑛 (𝑇)

,

(1.9) where 𝐹_𝑛 (𝑇) is the average of the Frank-Condon factor over all the vibrational levels of the HS state m.

By using equation 1.9 it is possible to calculate the HSLS relaxation rate constant. In Figure 1.9 the 𝑘_𝐻𝐿 values on a logarithmic scale vs the inverse of the temperature are plotted.¹⁹ At very low temperatures (T → 0) the HSLS relaxation is temperature independent and is based on a pure tunnelling process in which the electronic energy of the HS state is transformed into vibrational energy in the LS state followed by rapid and irreversible dispersion of this energy into the surrounding medium. On the other hand for higher temperatures (generally higher than 50 K) a thermally activated process is observed which should be understood as a tunnelling process from thermally populated vibrational levels of the HS state.¹⁹

Figure 1.9. The calculated relaxation rate constant kHL plotted on a logarithmic scale as a function of 1/T and the reduced energy gap n according.¹⁹

1.2.3.2. Cooperative effects in the HSLS relaxation

In the strong vibronic coupling limit when T → 0, the Frank-Condon factor, 𝐹_𝑛 (𝑇), can be reformulated as shown in equation 1.10 where as observed, |⟨𝑋_𝑛|𝑋₀⟩|² depends exponentially on n. In other words, 𝑘_𝐻𝐿(𝑇) depends exponentially on 𝛥𝐸_𝐻𝐿⁰ . Thus as observed in Figure 1.8, 𝛥𝐸_𝐻𝐿⁰ increases as a function of the LS fraction due to cooperative effects according to equation 1.11, which results in the dependence of the HSLS relaxation rate constant with the LS fraction described by equation 1.12 within the framework of the Mean-Field approximation.

𝐹_𝑛 (T → 0) = |⟨𝑋_𝑛|𝑋₀⟩|² = ^{𝑒−𝑆𝑆𝑛}

𝑛! , (1.10) where the Huang-Rhys factor S (S = QHL2/) is the reorganisation energy expressed in units of the vibrational frequency and is a measure for the horizontal displacement of the two potential wells relative to each other.

𝛥𝐸_𝐻𝐿⁰ (γ_𝐿𝑆) = 𝛥𝐸_𝐻𝐿⁰ ( γ_𝐿𝑆 = 0) + 2Γ γ_𝐿𝑆, (1.11) where Γ is the interaction constant.

𝑘_𝐻𝐿 ( 𝑇, γ_𝐿𝑆) = 𝑘_𝐻𝐿⁰ (𝑇) 𝑒^{𝛼(𝑇)γ𝐿𝑆} , (1.12) where 𝛼 is the acceleration factor.

This dependence of 𝑘_𝐻𝐿⁰ on γ_𝐿𝑆 gives rise to sigmoidal relaxation curves in concentrated systems as illustrated in Figure 1.10, where different relaxation curves for the neat compound [Fe(ptz)6](BF4)2 (ptz = 1-propyltetrozle) are displayed. From the least squares numerical fit of the experimental data it is possible to extract the corresponding values of 𝑘_𝐻𝐿⁰ (𝑇) and 𝛼(𝑇).

The physical behaviour of the lattice during the relaxation in neat compounds responsible of the described sigmoidal behaviour is schematized in Figure 1.11. As observed in Figure 1.11, at the beginning of the relaxation with all the HS centres sitting in a pure HS lattice the internal pressure experienced by the complexes is very low and therefore the value of 𝛥𝐸_𝐻𝐿⁰ is low too and the activation energy is high. Once that the relaxation starts to proceed and the lattice gets more and more dense, the internal pressure increases and so does the value of 𝛥𝐸_𝐻𝐿⁰ . As a consequence the activation energy decreases and the relaxation is accelerated according to equation 1.12.

Figure 1.10. HS→LS relaxation curves for [Fe(ptz)6](BF4)2 in the supercooled high- temperature phase between 50 and 60 K following a quantitative light-induced population of the HS state. (—) Experimental, (---) least-squares fit according to Equation 12.¹⁸

Figure 1.11. Effect of different lattice pressures on the HS→LS relaxation.

1.3. The mechanoelastic model

The simplest model to analyse the macroscopic behaviour of cooperative spin-crossover systems is the mean-field model, which implies random distribution of HS and LS species during the transition. However the mean-field model does not distinguish between short and long-range interactions, and therefore does not allow an advanced analysis of the propagation of spin flipping and first order crystallographic phase transitions inside the sample. In order to realize such an analysis more complex models should be used. From a historical point of view, the first one of these models was the Ising model.^20-21 The Ising model to study SC complexes implies the introduction of short and long-range intermolecular interactions between spin crossover units through an Ising-like Hamiltonian.

Due to the difference of molecular sizes between LS and HS states, the lattice is distorted during the transition. The larger volume of HS molecules as compared to LS molecules induces elastic stresses in the crystalline network, thus changing the probability of other molecules to switch to the LS state with the decreasing of during HSLS relaxation. Therefore, we have to assume that the intermolecular interactions themselves arise from stresses due to changes of the volume, shape and elasticity of the lattice. This behaviour described as an

“elastic interaction” between the molecules appears also as a total interaction, which is the sum between an infinite range contribution, originating in lattice strains and a direct contribution from pair interactions.

Therefore, the Ising-like models have some limits as they keep constant the intermolecular interactions and do not consider distortions explicitly. In order to correct for this problem a new class of models, the elastic models, have been elaborated.^22-24 In these models, the short and long-range interactions are replaced by purely elastic interactions. The molecules, considered as rigid spheres, are linked by springs, which can be compressed or elongated during the transition, resulting in practically an infinite number of possible interaction terms between molecules (Figure 1.12).

Figure 1.12. Molecules connected by springs in the frame of the mechanoelastic model.²⁵ g_HS

The Hamiltonian for such an elastic model can be simply written as indicated in equation 1.13:²⁶

, (1.13)

where the first term corresponds to the classical Hamiltonian used to treat the spin-crossover Ising-type systems ( is the HS-LS energy difference and can be regarded as the difference in enthalpy between the two states, is the vibronic degeneracy ratio such that kBlng corresponds to the difference in entropy between the two states of an non-interacting system) and the second term stands for the elastic energy, calculated as the sum of energies for all the springs in the system, with the elastic spring constant and the individual spring elongations or compressions between all neighbouring molecules in the system. Here, we shall use the mechanoelastic model, which implies that after every switching the new positions of all molecules are calculated, by solving systems of differential equations, in order to reach the mechanical equilibrium. For the sake of simplicity here we shall study the evolution of the system during the HSLS relaxation, which implies that the clusters will be produced at the same temperature for all the simulations (here considered 100 K). The evolution of the system will be discussed through Monte Carlo Arrhenius dynamics, in which the transition probabilities are modulated by an activation energy barrier . In this approach the transition probabilities can be written:²⁷

(1.14a)

, (1.14b) where is a scaling constant, chosen so that the above probabilities are well below unity at any temperature,  is a scaling factor between the local pressure and the activation energy of the individual molecule, and is the local pressure acting on molecule , defined as , with taken positive for compressed springs and negative for elongated ones.

 

²

,

1 ln

2  2 





 ^{B i}



^ij

i i j

H D k T g k x

D

g

k x_ij

Ea

ln

1exp exp

2



 

     

    

   

i B a i

HS LS

B B

E p

D k T g

P k T k T

1 ln

exp exp

2



 

     

    

   

i B a i

LS HS

B B

E p

D k T g

P k T k T



pi i









i ij

neighbours springs

p k x x_ij

If the springs near a HS molecule are compressed in the mechanoelastic case, that is, most of neighbours are already in the LS state, the probability that this molecule passes to the smaller volume LS state is higher.

The simulation procedure is the following:

(i) At the beginning all the molecules are considered to be in the HS state.

(ii) The transition probabilities (1.14) for a random molecule in the system are calculated.

(iii) A random number is generated. If this number is smaller than the transition probabilities then the switch is accepted and the molecules flips to the new state; otherwise the molecule keeps its previous state.

(iv) The steps (ii)-(iii) are repeated for all molecules in the system; when this step is finished a Monte Carlo step (MCstep) has been realized.

(v) After every MCstep, the position of all molecules in the system is updated by solving a system of differential equations which takes into account the pressure forces determined by the springs for every molecules in the system.²⁴ As a state change of a molecule results in a volume change of the molecular sphere, an instantaneous forces will appear inside neighbouring springs and will determine at first the shift in position of neighbour molecules and then progressively of all other molecules in the system. The molecules stop moving when all of them are in mechanical equilibrium, i.e. the resulting force on every molecule is zero.

(vi) The steps (ii)-(v) are repeated until all molecules have switched to the LS state.

 

^0,1

 r