Marie-Morgane Paumard

David Picard

Hedi Tabia

JIGSAW PUZZLE SOLVING USING LOCAL FEATURE CO-OCCURRENCES IN DEEP NEURAL NETWORKS

Keywords: Cultural heritage, fragment reassembly, jigsaw puzzle, image classification, deep learning

Archaeologists are in dire need of automated object reconstruction methods. Fragments reassembly is close to puzzle problems, which may be solved by computer vision algorithms. As they are often beaten on most image related tasks by deep learning algorithms, we study a classification method that can solve jigsaw puzzles. In this paper, we focus on classifying the relative position: given a couple of fragments, we compute their local relation (e.g. on top). We propose several enhancements over the state of the art in this domain, which is outperformed by our method by 25%. We propose an original dataset composed of pictures from the Metropolitan Museum of Art. We propose a greedy reconstruction method based on the predicted relative positions.

INTRODUCTION

The reconstruction of pieces of art from shards is a timeconsuming task for archaeologists. Close to puzzle solving problems, it may be automated. On the one hand, the computer vision algorithms struggle with those tasks. The dataset has to be scrupulously annotated by experts to reach a plausible solution, especially when the object fragments are lost, degraded or mixed among non-relevant fragments. Even so, the false-positive rate is still significant. On the other hand, deep learning algorithms are seen as an promising alternative, as they surpass other methods in most image-related tasks.

In this paper, we consider the image reassembly, which can be seen as solving a jigsaw puzzle: given two image fragments, we want to predict the relative position of the second fragment with respect to the first one. To solve this task, we investigate the setup proposed by Doersch et al. [START_REF] Doersch | Unsupervised visual representation learning by context prediction[END_REF]. Given an image, we extract same-size squared 2D-tiles in a randomized grid pattern (see Figure 1). Visual features are then extracted from both fragments using a Convolutional Neural Network (CNN). These features are the combined and fed to a classifier in order to predict the correct relative position.

However, the authors of [START_REF] Doersch | Unsupervised visual representation learning by context prediction[END_REF] are not interested in solving the jigsaw puzzle problem per se, but merely in using it as a pretraining of generic visual features.

In order to solve the jigsaw puzzle, we extend the work of Doersch et al. [START_REF] Doersch | Unsupervised visual representation learning by context prediction[END_REF]. Our contributions are the following: First, we propose a simpler, yet more effective, CNN architecture to extract visual features. Then, we propose a new combination scheme based on the Kronecker product that is able to better take into account correlations between localized parts of the fragments. We also propose a new dataset more closely related to cultural heritage puzzle solving tasks, which consists of 14,000 images from the Metropolitan Museum of Art (MET) archives. With our contributions, we obtain state of the art results on both the original ImageNet dataset proposed in [START_REF] Doersch | Unsupervised visual representation learning by context prediction[END_REF] and our new MET dataset.

The remaining of this paper is organized as follows: in the next section, we present related work and contextual information on puzzle solving and fragment reassembly. Then, we detail our method in Section 3. Finally, we describe our new dataset and we examine our experimental results in Section 4.

RELATED WORK

Whether it comes to precisely align fragments or approximate a relative position, archaeological object reconstruction attracts numerous researchers, as shown by Rasheed and Nordin in [START_REF] Rasheed | A survey of computer methods in reconstruction of 3d archaeological pottery objects[END_REF][START_REF] Rasheed | A survey of classification and reconstruction methods for the 2d archaeological objects[END_REF]. When most reassembly work is based on semi-automated methods, some stands out by the use of automated reconstruction, such as [START_REF] Mcbride | Archaeological fragment reconstruction using curve-matching[END_REF][START_REF] Jampy | 3d puzzle reconstruction for archeological fragments[END_REF][START_REF] Zhu | A partial curve matching method for automatic reassembly of 2d fragments[END_REF][START_REF] Stanco | Virtual restoration of fragmented glass plate photographs of archaeological repertoires[END_REF]. These are mainly inspired by the research on puzzle solving, especially on jigsaw puzzles [START_REF] Hammoudeh | Clustering-based, fully automated mixed-bag jigsaw puzzle solving[END_REF][START_REF] Andaló | Psqp: Puzzle solving by quadratic programming[END_REF][START_REF] Lifang | A new intelligent jigsaw puzzle algorithm base on mixed similarity and symbol matrix[END_REF][START_REF] Gur | From square pieces to brick walls: The next challenge in solving jigsaw puzzles[END_REF]. Those methods study missing fragments or various-sized tiles, belonging to annotated datasets. They perform well on small datasets but poorly when the fragments are mixed from similar sources. Moreover, these methods are often very slow.

Since 2015, the deep learning community uses puzzle solving as a pretext task, proposing a reasonable alternative to data labeling. In [START_REF] Doersch | Unsupervised visual representation learning by context prediction[END_REF], Doersch et al. introduce puzzle solving as a pretraining task for objects classification and detection. Their algorithm outperforms other unsupervised methods, which illustrates that CNN are able to learn object representations from the relative positions of the image part.

Based on [START_REF] Doersch | Unsupervised visual representation learning by context prediction[END_REF], Noroozi and Favaro [START_REF] Noroozi | Unsupervised learning of visual representations by solving jigsaw puzzles[END_REF] propose a network that observes all the nine tiles arranged in a grid pattern to obtain a precise object representation. They claim that the ambiguities may be wiped out more effectively when all fragments are examined. However, this leads to a much more complex classification due to the huge number of fragment orderings.

In our case, as we are not interested in building generic image features, but in solving approximatively the jigsaw puzzle itself. As such, the setup of [START_REF] Noroozi | Unsupervised learning of visual representations by solving jigsaw puzzles[END_REF] is impractical as it requires all fragments in order to make a prediction. This is unrealistic in cultural heritage where missing fragments are very common. However, the correlations between localized parts of the fragments (e.g., the correlation between the right part of the central fragment and the left part of the right fragment) are not taken into account in [START_REF] Doersch | Unsupervised visual representation learning by context prediction[END_REF][START_REF] Noroozi | Unsupervised learning of visual representations by solving jigsaw puzzles[END_REF], whereas we argue this information is essential to successful classification. This is what we investigate in our method.

PROPOSED METHOD

In this section, we detail our proposed method. We start by presenting the Feature Extraction Network that is shared between the two fragments. Then, we describe our propositions to combine the features of both fragments leading to the classification stage. Finally, we present a greedy algorithm to solve the jigsaw puzzle given any number of fragments.

Feature Extraction Network architecture

We use a CNN to compute the features associated with the fragments. Each fragment of size 96 × 96 pixel is processed such that pixels values vary between -1 and 1. Our architecture takes inspiration from VGG [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF] and is composed of a sequence of 3 × 3 convolutions followed by batch-normalizations [START_REF] Ioffe | Batch normalization: Accelerating deep network training by reducing internal covariate shift[END_REF], ReLU activations [START_REF] Nair | Rectified linear units improve restricted boltzmann machines[END_REF] and maxpoolings. The full architecture is shown on Table 1.

Our Feature Extraction Network ends with a fully connected layer that allows preserving the spatial layout of the input fragment. Although it is more costly than the worse popular global pooling layers [START_REF] Lin | Network in network[END_REF], we conjecture that keeping the layout is essential to successfully predict the relative position of fragments.

Combination Layer

To predict the relative position of a fragment with respect to another one, we extract features for both fragments using the same feature extraction network. These two features are then combined using a combination layer and processed by a neural network as shown in Figure 3.

In [START_REF] Doersch | Unsupervised visual representation learning by context prediction[END_REF], the authors propose to concatenate both features in the combination layer. This leads the subsequent fully con- nected layer to perform a linear combination of the fragment features:

∀i,

y i = m α i,m φ m (f 1 ) + m β i,m φ m (f 2 )
where y i is the output of the neuron i in the fully connected layer, φ(f 1 ) is the feature of the first fragment (respectively, φ(f 2 ) for the second fragment) and α i,m , β i,m are the weights of neuron i. Such a linear combination does not highlight co-occurrences of features, although it can be argued that subsequent layers with non-linearities may eventually be able to achieve similar results.

To circumvent these problems, we propose to use the Kronecker product in the combination layer. The output of the fully connected layer is then:

∀i, y i = m,n α i,m,n φ m (f 1 )φ n (f 2 ),
with α i,m,n being the weights of neuron i. The Kronecker product enables to explicitly model the co-occurrences between features of both fragments. This comes however at the cost of an increased number of parameters.

The output of the full network consists of a fully connected layer with k neurons followed by a Softmax activation, corresponding to the probabilities of the k possible relative locations. The full network (the Feature Extraction part and the Classification part) is trained at once using stochastic gradient descent on batches of fragments pairs.

Puzzle solving

In order to solve the puzzle problem, we consider the case where given a central fragment, we want to assign each of the remaining 8 fragments to its correct location. For each fragment, we compute the probabilities of assignment using the full network. This results in an 8 × 8 matrix where each row is a fragment and each column is a possible location. Solving the puzzle problem corresponds then to an assignment problem where we have to pick 8 values from the matrix (only one per row/column) such that their sum is maximized. We propose a greedy algorithm where we iteratively pick the maximum value and remove its corresponding row and column. Remark that this problem corresponds to a graph-cut problem for which much more involved algorithms exist. However, we found out that our greedy algorithm provides correct results in our case.

EXPERIMENTS

In this section, we first describe our new dataset and experimental setup. Then, we comment on the results comparing our approach to [START_REF] Doersch | Unsupervised visual representation learning by context prediction[END_REF]. Finally, we give some qualitative results on puzzle solving using the greedy algorithm.

MET dataset and experimental setup

In order to be closer to our aimed application regarding cultural heritage puzzle solving, we propose a new dataset consisting of images from the Metropolitan Museum of Art. We collect 14,000 open-source images of paintings and pieces of art. Contrarily to ImageNet, the quality of the sensors taking the pictures allows avoiding the unbalanced colors distribution. Images from the MET dataset are shown in Figure 2.

We use 10k images for training and evaluate the performances on the remaining 4k. During training, each image from the training set is resized and square-cropped so that its size is 398 × 398 pixels. We divide it into 9 parts separated by a 48-pixels gap, mimicking an erosion of the fragments. This value was the one used by Doersch et al. in [START_REF] Doersch | Unsupervised visual representation learning by context prediction[END_REF]. Then, we extract the center fragment and one of its 8 neighbors. Each fragment is of size 96 × 96 pixels, and we randomly move the location of the fragment by ±7pixels in each direction. The learning rate is 0.1 For the validation, we only consider a single pair of random fragments per image.

In the evaluation, we consider the following three setups: 1) we train the neural network on ImageNet and evaluate it on ImageNet ; 2) we train the network on ImageNet, fine-tune it on MET and evaluate it on MET (transfer setup) ; 3) we train the network from scratch on MET (MET setup). The results are compared using the accuracy of correct location prediction.

Results

On Figure 5, we show the evolution of the validation accuracy on ImageNet for a network similar to that of [START_REF] Doersch | Unsupervised visual representation learning by context prediction[END_REF], compared to our proposed architecture using either the concatenation or the Kronecker combination layer. Our implementation of the network proposed in [START_REF] Doersch | Unsupervised visual representation learning by context prediction[END_REF] has fewer parameters (we reduced the fully connected layers from 4k to 512 neurons) but nonetheless outperforms what is reported in [START_REF] Doersch | Unsupervised visual representation learning by context prediction[END_REF], achieving 57% accuracy on ImageNet compared to the 40% reported in the paper. Our proposed architecture significantly outperforms that of [START_REF] Doersch | Unsupervised visual representation learning by context prediction[END_REF] by a 25% margin. In accordance with our intuition, the Kronecker combination consistently outperforms the concatenation combination, reaching a validation accuracy of 65% after 100 epochs. All the results were obtained through a single run. Finally, we show on Figure 4 examples of puzzle solving using the greedy algorithm on several images taken from the MET dataset. As we can see, most of the predictions are correct. In the case where the algorithm wrongly predicts the position of the fragments, we can see that the corresponding fragments are visually close to what is expected to be in that location (e.g., sky and could pattern in the first example).

Using this greedy algorithm, we are able to solve the jigsaw puzzle perfectly 28.8% of the time. The proportion of correctly placed fragments is 68.8%, which means that on average only 2 fragments are swapped per image which is consistent with the accuracy at predicting individual positions of our neural network.

CONCLUSION

We proposed a robust deep learning method to classify the position of two neighboring fragments, which outperforms the state of the art by 25%. We successfully apply it to solve 9-tiles puzzles, and we show promising results on a new proposed dataset composed of images taken from the Metropolitan Museum of Art.

In the future, we plan to add a ninth class describing the not-neighbor relationship. Such class will allow us to solve more challenging puzzles. We are extending our method to non-squared fragments with irregularities.

Fig. 1 .

 1 Fig. 1. Example of fragments extracted using the randomized grid pattern on the MET dataset. Labels are the classes of the relative position with respect to the central fragment.

Fig. 2 .

 2 Fig. 2. Examples of images from the MET dataset.

Fig. 3 .

 3 Fig. 3. Full network architecture. FEN: Feature Extractor Network. CL Combination Layer. FC: Fully Connected. BN: Batch-Normalization. R: ReLU activation. S: Softmax activation.

Fig. 4 .

 4 Fig. 4. Examples of reconstructions using the greedy algorithm on images taken from the MET dataset. The red outlined fragments are misplaced.

Fig. 5 .

 5 Fig. 5. Evolution of the validation accuracy on ImageNet.

Table 1 .

 1 Architecture of the Feature Extraction Network.

	Layer	Shape	# parameters
	Input	96 × 96 × 3	0
	Conv+BN+ReLU	96 × 96 × 32	1k
	Maxpooling	48 × 48 × 32	-
	Conv+BN+ReLU	48 × 48 × 64	19k
	Maxpooling	24 × 24 × 64	-
	Conv+BN+ReLU	24 × 24 × 128	74k
	Maxpooling	12 × 12 × 128	-
	Conv+BN+ReLU	12 × 12 × 256	296k
	Maxpooling	6 × 6 × 256	-
	Conv+BN+ReLU	6 × 6 × 512	1.2M
	Maxpooling	3 × 3 × 512	-
	Fully Connected+BN	512	2.4M

Conv: 3×3 convolution, BN: Batch-Normalization, ReLU: ReLU activation.

s

Table 2

 2 the validation accuracy of the MET dataset for both Combination Layers and comparing the MET setup to the transfer setup (training on ImageNet, validation on MET). As we can see, training on ImageNet followed by a fine tuning on MET provides better results than training on MET alone. In this setup, we also remark that the Kronecker combination performs significantly better than the concatenation layer, which confirms the soundness of the approach.

	ImageNet → MET	MET setup
	concat	kron	concat	kron
	Accuracy 59.7%	64.9%	48.9% 47.9%

Table 2 .

 2 Comparison between the transfer setup (ImageNet → MET) and the MET setup for various combination layers.

This work is supported by the Fondation des sciences du patrimoine (*EUR-17-EURE-0021*)


Haiet Douib

Louis Cornet

Jessica Flores Gonzalez

Elzbieta Trzop

Vincent Dorcet

Abdelkrim Gouasmia

Lahcène Ouahab

Olivier Cador

Fabrice Pointillart

Spin Crossover and Field-Induced Single-Molecule Magnet Behaviour in Metal(II)-Dipyrazolylpyridine Complexes

The reaction between the 2,6-di(pyrazol-1-yl)-4-(bromomethyl)pyridine (L) ligand and divalent transition metal salts led to the formation of three mononuclear complexes of formula [M(L)2](BF4)2xMeNO2 (M = Fe II (1xMeNO2), x = 0 and 4, M = Co II , x = 2 (22MeNO2) and). Static magnetic measurements highlighted thermal spin crossover at high temperature (320-340 K) for the two Fe II solvatomorphs nevertheless both the solvation degree and crystal packing had drastic influence on the shape and width of the hysteresis loop. Thus the 1 displayed abrupt and straight (6 K) hysteresis loop while for 14MeNO2 the thermal spin transition could be described as a six-step gradual transition with the widest magnetic bistability range (35 K) for this kind of complexes. Irradiation at 530 nm of 14MeNO2 provoked an efficient LIESST effect at low temperature (TLIESST = 55 K). Finally dynamic magnetic measurements of 22MeNO2 shown a field-induced mononuclear Single-Molecule Magnet behavior with  = 14.6(4) cm -1 and 0 = 6.28(8)  10 -7 s.

10.

Introduction

Molecular magnetic bistability is intensively studied by both chemist and physicist communities of the molecular magnetism due to the wide potential applications such as high-density memory capacities. [1] Such magnetic behaviour could be observed under different stimuli. Thus under magnetic field, Single-Molecule Magnets (SMMs) [2] is targeted while Spin CrossOver (SCO) [3] is the intended behaviour under temperature variation or light irradiation among other stimuli (pressure…).

The SCO behaviour is a well-known phenomenon since the 1930s. [START_REF] Cambi | [END_REF] By restricting the external stimuli to the temperature and light irradiation, the magnetic bistability can be thermodynamically (thermal SCO) and kinetically (photo-induced SCO called lightinduced excited spin state trapping, LIESST) [5] reached. The magnetic bistability is driven mainly by the cooperativity induced by intermolecular interactions in the crystal lattice. All the metallic ion with the d 4 -d 7 electronic configuration are potential candidates to SCO behaviour nevertheless the most popular metal ion is unambiguously the Fe II because it allows the largest variation of the spin value involving paramagnetic and diamagnetic states (S = 2  S = 0). Obviously, the choice of the nature of the ligand is crucial to impose an adequate crystal ligand field to the Fe II . Thus the bpp-R ligand (where the bpp-R is a 2,6-di(pyrazol-1yl)pyridine derivative is one of the most used ligand to be associated with the Fe II ion leading to [Fe(bpp-R)2] 2+ series of complexes. [START_REF]Spin Crossover in Transition Metal Compounds I-III[END_REF] The reason of the high popularity of such series of complexes is due to its ability to show thermal (near room temperature) and photo-induced spin crossover as well as the chemical-dependence of the spin transition. Therefore the observation or not of the spin transition and the temperature at which the spin transition could occur strongly depend of the substituent of the bpp moiety, [7] the nature of the solvent of the crystal lattice, [8] the nature of the anions [9] and the Jahn-Teller distortion [10] .

Since the discovery of magnetic bistability at low temperature for a mononuclear lanthanide complex, [11] the interest for such simple systems were growing rapidly. Thus chemist community started to study mononuclear complexes of 3d ions to design SMMs. These systems mainly involved highspin Mn III , [12] Fe III , [13] Fe II , [14] Fe I , [15] Ni I , [16] Ni II , [17] Co I [18] and Co II[19] ions. The last Co II -based SMMs highlighted a large number of coordination for the metal ion ranging from two to eight [20] nevertheless the slow magnetic relaxation is usually observed under applied DC field to cancel the quantum tunnelling relaxation except for few examples of organometallic complexes [21] . Slow magnetic relaxation of the magnetization was rarely observed in system combining one 2,6-di(pyrazol-1-yl)pyridine derivative (bpp-R) and Co II [22] and the probe of magnetic properties for [Co(bpp-R)2] 2+ complexes is very scarce in literature [23] .

The aim of our final project is to combine both SMM and SCO behaviour in a unique system by functionalizing the 2- (4,5-(4,5-bis(propylthio)-tetrathiafulvalenyl)-1H-benzimidazol-2-yl)pyridine molecular skeleton by alkylation of the amine with a bromo-methyl-derivative such as the 2,6-di(pyrazol-1-yl)-4-(bromomethyl)pyridine (L). [24] A new approach is now to prepare the metal complexes involving the L ligand which could be therefore used as precursors to perform metallo-alkylation of our molecular skeleton. In the following lines, we reported the X-ray structures and the (photo)magnetic properties of two of such potential metallo-precursors of formula [M(L)2](BF4)2xMeNO2 (M = Co II , x = 2 and M = Fe(II), x = 0 and 4).

FULL PAPER

Results and Discussion

Crystal structure

Slow diffusion of Et2O vapor into MeNO2 solution of [Fe(L)2](BF4)2 yields mixtures of brown plate [Fe(L)2](BF4)2 and prism [Fe(L)2](BF4)24MeNO2 single crystals with variable ratio.

[Fe(L)2](BF4)2 (1). Compound 1 crystallizes in the orthorhombic space group Pbcn (N°60) (Table 1). The ORTEP drawing of 1 is shown in Figure 1a. The asymmetric unit is composed of one half molecule of [Fe(L)2] 2+ and one BF4 -anion. The Fe II centre is coordinated to two trischelating ligands conferring a N6 surrounding. The average Fe-N bond lengths is equal to 1.947(3) Å with a slight difference between the Fe-N(py) distances (1.888(3) Å) which are shorter than the Fe-N(pz) distances (1.969(3) Å) due to the difference of Lewis basis strength between a pyridine and a pyrazole unit. Such bond lengths strongly suggest that the Fe II ion is in its low-spin state (S = 0) at 200 K in the Fe compound. The characteristic N(pz)-Fe-N(pz) and N(py)-Fe-N(py) angles have been measured to 160.3(1)° and 174.6(1)° respectively. The two coordinated L ligands are perpendicular. The CH2Br substituent is localized in the plan formed by the bpp fragment (Figure 2b).

The crystal packing of 1 highlights the formation of stack of molecules into a pseudo "terpyridine embrace" lattice i.e. a packing which is dominated by face-to-face - interactions and edge-to-face C-H contacts between the pyrazole moieties of neighbouring complexes. [25] The regular "terpyridine embrace" lattice form a stacked layers of molecules [26] while for the 1 compound the - stacking takes only place along one direction (Figure 2a). Interestingly, in a perpendicular direction (along the a axis) to the pseudo "terpyridine embrace" lattice, BrBr short contacts (3.617 Å) are identified and could have consequences in the magnetic behavior. [7] Figure 2. Crystal packing of 1 highlighting the pseudo "terpyridine embrace" (a) and the BrBr short contacts (b).

Table 1. X-ray crystallographic data for the complexes 1, 14MeNO2 and 22MeNO2.

Compounds

[Fe(L)2](BF4)2

(1) [Fe(L)2](BF4)24MeNO2 (14MeNO2). 14MeNO2 crystallizes in the orthorhombic space group Iba2 (N°45) (Table 1). The ORTEP drawing of 14MeNO2 is shown in Figure 1b. The asymmetric unit The presence of nitromethane molecules in the crystal induced drastic change for the stack of the complexes since the  and BrBr contacts are lost (Figure 3). The crystal packing highlighted perpendicular arrangement of the complexes. The section on the magnetic properties will show the consequence of the structural changes between the two complexes.

[Fe(L)2](BF4)24MeNO2 (14MeNO2) [Co(L)2](BF4)22MeNO2 (22MeNO2) Formula C24H20F8FeN10Br2B2 C28H32F8FeN14O8Br2B2 C26H26F8CoN12O4Br2B2 M / g.
[Co(L)2](BF4)22MeNO2 (22MeNO2). Co2MeNO2 crystallizes in the triclinic space group P-1 (N°2) (Table 1). The ORTEP view of Co2MeNO2 is shown in The Figure 5 highlighted a regular "terpyridine embrace" crystal packing for 22MeNO2 with CH2Br groups localized at 68.8° and 81.4° of the plane formed by the bpp fragments.

Magnetic measurements

[Fe(L)2](BF4)2 (1). Magnetic susceptibility measurements of 1 shows an abrupt HS (S = 2, MT = 3.95 cm 3 K mol -1 ) to LS (S = 0) transition centred around 324 K with a straight thermal hysteresis (4 K). On cooling, the transition occurs at TC = 326 K while increasing the temperature the transition occurs at 322 K. Such abrupt transition with a straight magnetic bistability are associated with the "terpyridine embrace" crystal packing form and an octahedral coordination sphere characterized by the two angles ϕ = 180° and θ = 90° [ 9] in the HS state of 1 complex. The spin transition temperature is higher than the room temperature which is surprising since the analogue complex with two hydrogen atoms instead of the bromide atoms highlighted a spin transition centred at 207 K. [9] Such substitution and the resulting BrBr short contacts clearly stabilize the LS state of 1.

[Fe(L)2](BF4)24MeNO2 (14MeNO2). The magnetic properties of the solvated derivative showed a more progressive spin transition around 340 K with a wider thermal hysteresis (35 K) compared to the unsolvated derivative (Figure 6). Based on the previous 10.1002/ejic.201800819

Accepted Manuscript

FULL PAPER

Halcrow's works, the difference of hysteresis width and temperature of transition could be attributed to both solvation [8] and a moderate Jahn-Teller distortion of the HS state [10] for 14MeNO2. Unfortunately the complete thermal spin crossover is localized at to high temperature (T > 400 K) to perform data collection and refinement of the HS state structure for 14MeNO2. It is worth to notice that the 14MeNO2 compound presents the widest hysteresis loop measured for a [Fe(bpp-R)2] 2+ derivatives since the previous record was measured at 18 K for the methylsubstituted analogue. [9] The form of the crystal packing imposed by the orientation of the bromide substitutions and the presence of nitromethane molecules of crystallization might be the origin of a such large hysteresis loop. In addition, one can observe that the MT product decreased in cooling mode in a six-step manner centred at 265, 280, 295, 315, 335 and 355 K (see the derivative of the data plot in Figure S1). The hysteresis loop remains open after several warming cycle and it cannot be attributed to desolvation or irreversible crystallographic phase transition (Figure S2). Such multiple spin-state conversions was already observed in warming mode for the [Fe(bpp)2][Ni(mnt)2]2MeNO2 compound (where mnt = maleonitriledithiolate) [27] and in both cooling and warming modes for the [Fe(bppMe)2](BF4)2xH2O (bppMe = 2,6-bis(3-methylpyrazol-1-yl)pyridine) [28] . The red data plots of the Figure 6 shows the results of the irradiation at 660 nm in the tail of the absorption band which corresponds to the d-d and/or Metal-to-Ligand Charge Transfer (MLCT) transitions of the LS state of 14MeNO2 (Figure S3). After 5 min of light irradiation at 10 K, almost all the LS fraction is converted in HS by population of the metastable 5 T2g (Oh) state. Switching off the light leads to a population of the metastable 5 T2g which persists up to TLIESST = 55 K at a sweep rate of 0.3 K min -1 . This value is lower than the TLIESST measured under a light irradiation of 510 nm for the two solvatomorphs of the methylsubstituted-bpp analogues (87 and 112 K). [9] This experimental observation is perfectly in agreement with the linear relation (TLIESST = T0-0.3T1/2) [28] which connects photomagnetism and thermal spin crossover and saying that higher the T1/2 is found lower the TLIESST will be determined. T1/2 is the thermal spin transition temperature, TLIESST is the light-induced excited spin state trapping temperature and T0 is an empirical parameter determined equal to 150 K for meridional tridentate ligands [29] . Using such relation, a T1/2 = 340 K should lead to a TLIESST = 48 K which is a value close to the experimental TLIESST = 55 K.

[Co(L)2](BF4)22MeNO2 (22MeNO2). The magnetic properties of 22MeNO2 were probed measuring the magnetic susceptibility for an immobilized crystalline powder of the sample. The temperature dependence of the MT product is depicted in the Figure S3. The room temperature value of 3.19 cm 3 K mol -1 is larger than the expected spin-only value (1.875 cm 3 K mol -1 ) for high-spin Co II ion. Nevertheless this experimental value is in agreement with the value range of 2.1-3.4 cm 3 K mol -1 for an highly anisotropic Co II ions including a strong orbital angular momentum contribution. [29] Upon cooling, the MT(T) curve decreases monotonously to reach a value of 2.30 cm 3 K mol -1 at 2 K. The decreasing is mainly due to the magnetic anisotropy of the Co II with possible additional antiferromagnetic intermolecular interactions at low temperature.

Accepted Manuscript

FULL PAPER

The magnetic field dependence of the magnetization at 2 K between 0 and 50 kOe is depicted in inset of the Figure S4. At the maximum field value the magnetization takes the value of 2.50 B but without reaching a saturation sign of significant magnetic anisotropy. Consequently the magnetic relaxation dynamics were probed by measuring the frequency-dependence ac susceptibility below 15 K. No out-of-phase component of the magnetic susceptibility was detected under zero dc field. Nevertheless application of a moderated dc field provokes the appearance of an out-of-phase which is centred at 160 Hz at 2 K for the optimal dc field of 1000 Oe (Figure S5). Such cancelling of the tunnelling magnetic relaxation is very common for Co II -based SMM. The compound 22MeNO2 displayed typical field-induced mononuclear SMM under an applied dc field of 1000 Oe (Figures S6 and7a). From the frequency dependence of the magnetic susceptibility is generated the Cole-Cole plots below 14 K (Figure S7). The values (α) and the distribution of relaxation time () were extracted using a generalized Debye model (Table S1) [START_REF] Cole | [END_REF] while the energy barrier  was evaluated from the linear fit of the high temperature region of the Arrhenius plot (Orbach regime) (Figure 8b). Using the following equation  = 0exp(Ueff/kBT), the best fit gives Ueff = 14.6(4) cm -1 and 0 = 6.28(8)  10 -7 s. The α values range from 0.02 to 0.12 suggesting a narrow distribution of the relaxation time.

Conclusions

The association of the ligand 2,6-di(pyrazol-1-yl)-4-(bromomethyl)pyridine (L) with the divalent metallic ion Fe(II) and Co(II) led to the formation of mononuclear complexes of formula [M(L)2](BF4)2xMeNO2. The Fe II based compound crystallized in two solvatomorphs [Fe(L)2](BF4)2 (1) and [Fe(L)2](BF4)24MeNO2 (14MeNO2). The LS Fe II crystal structure of the unsolvated derivative highlighted a partial "terpyridine embrace" type crystal packing while the solvated derivative did not. As a consequence, both solvatomorphs displayed thermal Spin Crossover at high temperature (T1/2 = 320-340 K) nevertheless the one for 1 presented an abrupt spin transition with a straight hysteresis loop while 14MeNO2 presented a more gradual spin transition with the widest thermal hysteresis (35 K) for this type of complexes. In addition the spin transition in cooling mode operated in a six-step manner. Finally the 14MeNO2 solvatomorph is photo-active with an efficient LIESST effect under light irradiation at 530 nm and TLIESST = 55 K. The [Co(L)2](BF4)22MeNO2 (12MeNO2) complex did not show any spin transition but the dynamic magnetic properties highlighted a mononuclear SMM behaviour under an applied dc field of 1000 Oe with an effective energy barrier of 14.6(4) cm -1 and a relaxation time of 6.28(8)  10 -7 s. Both Fe II and Co II derivatives shown promising magnetic properties and they could be used as metallo-precursors for further functionalization thank the bromo-methyl groups.

Experimental Section

Synthesis. General Procedures and Materials. The ligand 2.6di(pyrazol-1-yl)-4-bromomethyl)pyridine (L) [14] was synthesized following previously reported methods. All other reagents were purchased from Acros, Alfa Aesar, Aldrich Co., Ltd. and used without further purification.

Synthesis of [Fe(L)2](BF4)2xMeNO2 (x = 0 (1) and x = 4 (14MeNO2). 50.6 mg of Fe(BF4)26H2O (0.15 mmol) and 91.2 mg of L (0.30 mmol) were dissolved in 15 mL of CH3NO2 and heated to reflux for 3 h under argon atmosphere. After cooling to room temperature, the solution was stirred overnight and then concentrated to a volume of c.a. 7.5 mL. The resulting brown solution was divided in 5 glass hemolysis tubes and slow diffusion of Et2O vapour led to the formation of brown plate (x=0) and prism (x=4) single crystals suitable for X-ray analysis. Yield: 73 mg (about 54 % based on single crystals (m), 918 (w), 860 (w), 789 (m), 764 (s), 672 (w), 643 (w), 606 (w) and 566 (w) cm -1 .

Synthesis of [Co(L)2](BF4)22MeNO2 (22MeNO2).

Similar experimental procedure than for the Fe II derivatives was used except that 51.3 mg of Co(BF4)6H2O was employed instead of Fe(BF4)26H2O. The resulting yellow-brownish solution was divided in 5 glass hemolysis tubes and slow diffusion of Et2O vapour led to the formation of yellow board single crystals suitable for X-ray analysis. Yield: 85 mg (about 59 % based on single crystals). Anal. Calcd (%) for C26H26F8CoN12O4Br2B2: C 34.42, H 1.68, N 4.15; found: C 34.69, H 1.74 N, 4.14. I.R. (KBr): 3130 (w), 3089 (w), 3063 (w), 3039 (w), 3009 (w), 2915 (w), 1638 (s), 1580 (s), 1525 (m), 1471 (s), 1404 (s), 1339 (m), 1310 (m), 1285 (m), 1174 (m), 1114 (w), 1084 (m), 1042 (s), 969 (m), 915 (w), 860 (w), 770 (m), 678 (w), 641 (w), 598 (m), 524 (w) and 448 (w) cm -1 . X-ray diffraction. Single crystal of 1, 14MeNO2 and 22MeNO2 were mounted on a APEXIII D8 VENTURE Bruker-AXS diffractometer for data collection (MoK radiation source,  = 0.71073 Å), from the Centre de Diffractométrie (CDIFX), Université de Rennes 1, France (Table 1). Structure was solved with a direct method using the SHELXT program [32] and refined with a full matrix least-squares method on F2 using the SHELXL-14/7 program [33] . Complete crystal structure results as a CIF file including bond lengths, angles, and atomic coordinates are deposited as Supporting Information.

Physical Measurements. The elementary analyses of the compounds were performed at the Centre Régional de Mesures Physiques de l'Ouest, Rennes. Single crystals of 1, 14MeNO2 and 22MeNO2 are manually selected and their cell parameters are checked before magnetic measurements. The dc magnetic susceptibility measurements were performed on grinded previously selected single crystal in Teflon tape with a Quantum Design MPMS-XL SQUID magnetometer between 2 and 300 K in applied magnetic field of 0.02 T in the temperature range 2-20 K, 0.2 T in the temperature range 20-80 K and 1 T above 80 K. These measurements were all corrected for the diamagnetic contribution of the Teflon tape and the intrinsic diamagnetism calculated with Pascal's constants.

FULL PAPER

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Figure 1 .

 1 Figure 1. Ortep View of the structures of Fe (a) and Fe4MeNO2 (b). Atomic displacement ellipsoids are drawn at the 30% probability level. Hydrogen atoms are omitted for clarity.

  one half molecule of [Fe(L)2] 2+ , one BF4 -anion and two nitromethane molecules of crystallization. Both solvated 14MeNO2 and unsolvated 1 complexes are very similar. No significant difference of Fe-N distances are identified between the two complexes (Fe-N(py) = 1.895(3) Å and Fe-N(pz) = 1.942(3) Å) and thus the Fe II centre is in its LS state at the temperature of the X-ray structure resolution (150 K). The characteristic N(pz)-Fe-N(pz) and N(py)-Fe-N(py) angles have been measured to 161.3(1)° and 179.6(1)° respectively. The two coordinated L ligands formed an angle of 79.2(4)°. Significant local structural changes are identified compared to 1 which should induce significance variation in the crystal filed around the Fe II ion. The CH2Br substituent is localised at 58.7° of the plan formed by the bpp fragment which is one more structural difference compared to 1.

Figure 3 .

 3 Figure 3. Crystal packing for 14MeNO2.

Figure 4 .

 4 Figure 4. Ortep View of the structure of Co2MeNO2. Atomic displacement ellipsoids are drawn at the 30% probability level. Hydrogen atoms are omitted for clarity.

  Figure 4. The asymmetric unit is composed of one molecule of [Co(L)2]2+, two BF4 -anion and two nitromethane molecules of crystallization. The characteristic compressed octahedral polyhedron is observed with Co-N(py) = 2.066 Å and Co-N(pz) = 2.143(3) Å. Such Co-N bond lengths designated a HS state Co II centre. The longer Co-N distances compared to the Fe-N distances lead to smaller N(pz)-Co-N(pz) (149.5(1)° and 150.8(1)°) and N(py)-Co-N(py) (170.2(1)°) angles compared to its Fe II analogue. The two L ligands are close to be perpendicular with an angle of 83.0(1)°. Nevertheless while one L is planar (as observed for the two Fe II solvatomorphs) the second one is not. In fact the two plans formed by the pyridine ring and one of the pyrazole unit display an angle of 16.0(1)°.

Figure 5 .

 5 Figure 5. Crystal packing for 22MeNO2.

Figure 6 .

 6 Figure 6. Thermal dependence of MT for 1 (full circles), 14MeNO2 (full square). The thermal dependence of MT 14MeNO2 after light irradiation (660 nm, 5 mW) is represented with red squares.

Figure 7 .

 7 Figure 7. (a) Frequency dependence of the out-of-phase component of the magnetic susceptibility for 22MeNO2 between 2 and 14 K under an applied DC field of 1000 Oe. (b) Arrhenius plot of the thermal dependence of the relaxation time of the magnetization. The red line represents a fit for the high-temperature region.

  complexes of Fe II and Co II involving the 2,6di(pyrazol-1-yl)-4-(bromomethyl)pyridine (L) displayed thermal and photo-induced spin transition, and field-induced Single-

Acknowledgements

We acknowledge financial support from CNRS, Université de Rennes 1 and ERC Consolidator Grant MULTIPROSMM (Proj. no. 725184).

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