Andrei Mitriakov

Panagiotis Papadakis

Jérôme Kerdreux

Serge Garlatti

Reinforcement Learning based, Staircase Negotiation Learning in Simulation and Transfer to Reality for Articulated Tracked Robots

Autonomous control of reconfigurable robots is crucial for their deployment in diverse environments. The development of such skills is however hampered by the diversity in hardware and task constraints. We advocate the use of artificial intelligence-based approaches to improve scalability to different conditions and portability to platforms of comparable traversability skills. In particular, we succeed in tackling the problem of staircase traversal via a reinforcement learning-based control framework applicable to different articulated tracked robots, powerful enough to generalize to varying conditions learnt in simulation and to transfer to reality in a zero-shot setting. Our extensive experiments demonstrate the robustness of the framework in learning tasks with increased risk and difficulty induced by platform diversification and increased control dimensionality.

I. INTRODUCTION

Autonomous 3D navigation has a tremendous potential for robotic applications that range from operation in hazardous environments to providing assistance to humans. In the latter case, a robot may encounter steps and staircases that may have to be avoided or traversed by passive adaptation that palliates minor collisions. Most often, however, ensuring robot and environment safety requires active robot control.

Staircase negotiation has been predominantly addressed using conventional control, assuming proper knowledge of robot kinematics and dynamics. As a result, rudimentary changes of the platform or the task may render the adaptation of such solutions cumbersome if not impossible. Domain expertise is essential yet scarce and hard to obtain in realistic conditions due to the elevated accident risk.

On the other hand, machine learning-based approaches are better suited as they lower the need for ad-hoc solutions customized to specific platforms and endow a robot with the capability of safe and autonomous obstacle negotiation [START_REF] Chatzilygeroudis | A survey on policy search algorithms for learning robot controllers in a handful of trials[END_REF]. Using machine learning then raises the question of the type of learning to be sought, such as learning from demonstration (LfD), behaviour cloning [START_REF] Torabi | Behavioral cloning from observation[END_REF], reinforcement learning (RL) or their combination (as for example in [START_REF] Ho | Generative adversarial imitation learning[END_REF]). For the staircase negotiation task that we consider, it is impractical, if not impossible, to obtain sufficiently rich demonstrations. In this respect, RL-based approaches are particularly well suited while allowing the emergence of novel behaviors.

The work is performed in the context of the project REACT, project VITAAL and is financed by Brest Metropole, the region of Brittany and the European Regional Development Fund. In our current work we pursue a RL-based paradigm for the acquisition of robot controllers with desired behaviour properties such as safety and bumpiness through reward function design, in line with our earlier works, [START_REF] Mitriakov | Staircase traversal via reinforcement learning for active reconfiguration of assistive robots[END_REF], [START_REF] Mitriakov | Staircase negotiation learning for articulated tracked robots with varying degrees of freedom[END_REF]. In particular, we consolidate the training of effective controllers in simulation, by developing and comparing these skills on two distinct articulated, tracked robot vehicles in simulation and further successfully transferring the skill onto one of the real robots. On top of attaining our main goal that consists in successful staircase negotiation by the real robot, we also present new insights that are the product of quantitative as well as qualitative cross-comparison of behaviors among task variants and between robots. Concretely, the current work advances the state-of-the-art in the following points:

• We successfully acquire RL-based staircase negotiation controllers for two robotic platforms in simulation, both exhibiting the desired behavior properties. • We quantitatively compare controllers obtained for different robots and task variants via Kullback-Leibler (KL) policy divergence. • The efficacy of the obtained controllers is demonstrated in reality by transfer to a commercial robot (see Fig. 1).

The rest of the paper is organized as follows. Section II presents an overview of earlier approaches for robot control in staircase negotiation by articulated tracked robots. In section III we detail our framework for RL-based staircase negotiation learning in simulation, followed by the procedure for transferring and evaluating the obtained behaviors onto a real robot in section IV. Finally, in section V we present experiments that demonstrate the effectiveness of the framework in simulation as well as in reality through multiple variants of the main task.

II. CONTROL METHODS IN OBSTACLE NEGOTIATION

Robot navigation is dictated by the environment where the robot operates. As obstacle negotiation in outdoor as well as indoor environments requires advanced traversability skills, tracked robots equipped with additional sub-tracks (flippers) have been developed. More recently, the obstacle negotiation potential of such platforms is being studied for improving the autonomy of frail people, either for transporting humans or objects in 3D environments.

Therefore, a strong interest has been shown by the scientific community in modelling the kinematics and dynamics of such robots, in order to exploit their full potential. Although articulated front and/or rear flippers drastically increase the negotiation capability of complex obstacles, they inevitably increase complexity of control, kinematics calculation and estimation of robot dynamics. The presence of a robotic arm further elevates control difficulty due to additional degrees of freedom (DOF) and safety constraints.

Among multiple control methods reviewed by [START_REF] Tzafestas | Mobile robot control and navigation: A global overview[END_REF], neuralnetwork based control that relies on unsupervised learning or reinforcement learning allows to learn effective controllers for high-dimensional observation and action spaces or decision making from raw input data such as pixels [START_REF] Chatzilygeroudis | A survey on policy search algorithms for learning robot controllers in a handful of trials[END_REF] in contrast to learning-free control methods. In the sequel, we review representative works from both categories that treated the problem of indoor navigation of tracked robots, while highlighting limitations and open challenges. For reference, Table I provides a coarse overview of those works. Standard control Preliminary works on autonomous staircase traversal were devoted to highly customized, standard robot control. One of the first such works was presented in [START_REF] Mouriakis | Autonomous stair climbing for tracked vehicles[END_REF] relying on estimation of step edges and gyroscopic data. A highly customized platform was presented in [START_REF] Ben-Tzvi | A mobile robot with autonomous climbing and descending of stairs[END_REF], where thanks to a sophisticated track platform the problem of staircase traversal is tackled, further aided by an actively controlled pendulum. Unfortunately, approaches that are highly customized to specific hardware bear limited potential in general.

To palliate the increased cognitive load experienced by users that tele-operate such platforms, the authors of [START_REF] Nagatani | Semi-autonomous traversal on uneven terrain for a tracked vehicle using autonomous control of active flippers[END_REF] propose a motion planning framework where the robot exploits side laser sensors in order to perceive the geometry of ahead obstacles and adapt flippers so that they are tangential to the surface. Upon the definition of the robot negotiation limits, the authors consider that the system is able to negotiate any uneven terrain, yet being based on exhaustive sensory data. The framework of [START_REF] Suzuki | Remote control system of disaster response robot with passive sub-crawlers considering falling down avoidance[END_REF] is comparatively superior as the system is endowed with passive flippers that push against a traversed obstacle to improve traction, combined with an accident warning system that is based on estimation of the normalized energy stability margin (NESM). Still, the robot is not autonomous and the overall approach is weakly generalizable.

Another approach relying on precise estimation of the staircase configuration was proposed in [START_REF] Endo | Stair climbing control for 4dof tracked vehicle based on internal sensors[END_REF] where the robot could autonomously negotiate a staircase using proprioceptive state estimation. That approach showed prominent performance, however, the system could only negotiate a staircase known beforehand. One of the most elaborate analysis was presented in [START_REF] Zhang | System centroid position based tipover stability enhancement method for a tracked search and rescue robot[END_REF] using a common tracked platform with 2 additional DOF due to a controllable arm. The derived analytical solution is indicative of the high complexity of modelling the interaction between a multi-DOF system and the traversed surface. That system lacked congruent control of tracks, flippers and the arm and needed full knowledge of geometry, kinematics and physical characteristics which can be difficult to acquire.

Learning-based control Learning-based approaches appear more suitable for developing controllers when robot kinematics and dynamics are harder to model. The authors of [START_REF] Guastella | Learning-based methods of perception and navigation for ground vehicles in unstructured environments: A review[END_REF] identify two main categories of learning-based methods: terrain traversability analysis and end-to-end while [START_REF] Chatzilygeroudis | A survey on policy search algorithms for learning robot controllers in a handful of trials[END_REF] presents the diversity of employed learning techniques along with challenges. One of the main requirements is scalability to highdimensional state and action spaces. Priors could bootstrap learning, still, this demands more attention in the design of the search policy algorithm in order to ignore irrelevant policy properties. Generalization and robustness are relevant for robots that are required to learn various tasks under varying conditions.

To the best of our knowledge, earlier work on learning-based control for actively articulated tracked robots is relatively scarce. At the same time, in staircase ascent and descent where it is hard to model the dynamics of physical interaction, learning-based techniques are particularly promising.

Policy endowment with properties such as bumpiness or stability can be performed in different ways. Constraints may be introduced in a RL optimization problem as in [START_REF] Pecka | Autonomous flipper control with safety constraints[END_REF] while safety and reward can be associated in a single function [START_REF] Zimmermann | Adaptive traversability of unknown complex terrain with obstacles for mobile robots[END_REF]. Another more sophisticated approach was demonstrated in [START_REF] Pecka | Autonomous flipper control with safety constraints[END_REF] where authors incorporated constraints into the optimization problem of relative entropy policy search using RL. Still, results were limited to the traversal of a palette.

Authors of [START_REF] Ejaz | Vision-based autonomous navigation approach for a tracked robot using deep reinforcement learning[END_REF] use images for the navigation of a tracked robot in indoor environments, improving dueling double deep Q-network model with layer normalization and noise injection and propose an original algorithm for its training. Still, this approach is only employed for navigation on 2D surfaces.

Deep Q-Network (DQN) was one of the first deep RL algorithms and had shown fascinating results at that time [START_REF] Sutton | Reinforcement Learning: An Introduction[END_REF]. Over the years, RL algorithms have witnessed a significant evolution with the current state-of-the-art being composed of actor-critic algorithms which benefit from state-action value estimation as in pure value-based DQN and directly optimize policy parameters through gradient ascent.

Indicatively, the off-policy Deep Deterministic Policy Gradient (DDPG) algorithm [START_REF] Azar | Drone deep reinforcement learning: A review[END_REF] moved discrete action space to a continuous one as an approximate DQN. Still the major improvement of DDPG, which has introduced Twin-Delayed DDPG (TD3), corresponds to enhanced DDPG with delayed policy improvement, clipped double Q-Learning and target policy smoothing. In parallel to the latter algorithm, Soft Actor-Critic (SAC) was proposed which contains the same features combined with a novel policy entropy based maximization. An earlier on-policy algorithm is Proximal Policy Optimization (PPO) which yields more conservative updates.

III. ROBOT CONTROL FRAMEWORK

Following a typical RL-based problem formalization, given a state s t ∈ S the agent selects actions a t ∈ A according to a policy π and transits to the next state s t+1 ∈ S receiving the reward r t ∈ R. The array of consecutive state-action pairs τ = {s 0 , a 0 , .., s T -1 , a T -1 } creates the trajectory known as roll-out. The set of s t , a t , r t , s t+1 observed along a roll-out is used by RL algorithms to perform policy optimization. In the sequel, we detail the chosen action and state spaces, present reward function design and employed controller for the problem of the staircase traversal, the way KL divergence is used to compare policies and finally their transfer to reality.

A. Problem description

We consider a robot that can minimally control 3 DOF corresponding to front and rear flipper angles and linear velocity, occasionally complemented with 2 extra DOF of a robotic arm whose use can prove particularly useful in staircase ascent and descent. Therefore, the entire continuous 5 DOF action space forms the following action vector:

a = (ψ f ront a , ψ rear a , v a , φ 1 a , φ 2 a ) (1)
a ∈ [ψ min a , ψ max a ] 2 × [v min a , v max a ] × [φ min a , φ max a ] 2
where ψ f ront a and ψ rear a are front and rear flipper angles, v a is the applied main tracks velocity and φ 1 a and φ 2 a are the angles of the two arm joints. Superscripts min and max denote the kinematic limits of the corresponding DOF. We deliberately exclude yaw control since the traversal of a staircase prescribes that a robot maintains a fixed, orthogonal to the steps orientation along the staircase.

The observation state vector s ∈ S is defined as follows:

s = (p f ront x , p f ront y , p rear x , p rear y , v s , ψ f ront s , ψ rear s , φ 1 s , φ 2 s ) (2)
where p f ront s , φ 2 s are the arm joint angles (see Fig. 2). State parameters were determined as a group rather than distinctively, in order to avoid redundancy and maximize complementarity between state dimensions. As a result, the pitch was excluded from the state because eq. ( 2) already allows to capture that parameter indirectly. Specifically, as the state contains the flipper angles and the coordinates of the front and rear step edges, this corresponds in reality to a specific pose of the platform onto the staircase at a certain pitch.

B. Reward function design

In this section we present reward functions used to obtain policies allowing ascent and descent staircase traversals. To guide learning we assign the following positive reward proportional to the traversed path from start to finish:

R tr (τ ) = Nτ t=1 x t D max (3)
where R tr (τ ) represents the episode return along a trajectory τ whose maximal value can attain 1, x t is the traversed distance during time step t, N τ is the number of time steps in the current episode and D max is the total distance between start and finish. While only positive rewards can produce policies that accomplish a given task, those may be sub-optimal in terms of features such as safety. Furthermore, the situations that may compromise the safety of the robot differ between ascending and descending a staircase. To account for robot safety in different contexts we introduce negative penalties alongside with the positive reward.

In detail, to reduce the risk of tip-over during ascent we assign a negative reward proportional to the deviation of the robot centroid. On the other hand, when the robot descends a staircase accidents are mostly due to platform shaking and bumps, therefore we tackle this via a negative reward proportional to platform acceleration. In either case, the scale of a negative reward is normalized to avoid biasing. Independently of the chosen negative reward r t , the overall time step return R t then becomes:

R t = x t D max + r t . (4)
The next two subsections present the design of the aforementioned negative rewards. 1) Center of gravity stabilization: Whenever the center of gravity (COG) destabilizes it may lead to robot tip-over. In detail, during ascent the COG position leads to tip-over only when its projection point C x crosses the lowermost edge of the robot support polygon, namely, the point of contact of the rear flipper with the staircase (cf. Fig. 2 (b)). In accordance with [START_REF] Mitriakov | Staircase negotiation learning for articulated tracked robots with varying degrees of freedom[END_REF], the COG-based penalty r D t is set as:

r D t = -1, if tip over -K D * D t , otherwise (5)
where K D is the normalizer and D t the COG deviation. We assign this penalty only in ascent tasks and when the robot COG projection is located between staircase step edges.

In [START_REF] Mitriakov | Staircase traversal via reinforcement learning for active reconfiguration of assistive robots[END_REF] a robot learnt a stable behavior by favoring poses with low COG with respect to the underlying surface A that represents the Normalized Energy Stability Margin (NESM) [START_REF] Garcia | A classification of stability margins for walking robots[END_REF]. The presence of an arm nonetheless may place the COG low but also close to the "safety zone" border [START_REF] Zhang | System centroid position based tipover stability enhancement method for a tracked search and rescue robot[END_REF], without violating the NESM. To overcome this, we use the Stability Margin (SM) that estimates the distance between the lowermost robot edge and the COG projection on the ground.

Since it is difficult to estimate where exactly the lowermost robot edge touches the ground, we instead minimize the deviation of the COG projection C x on the staircase from the projection point O of the robot geometric center. We can thus guarantee that the robot respects the SM criterion. We consider that the arm has to place the COG as close as possible to the point O -the most stable point along both X and Y axes, optimizing both the SM and the NESM. Thus, in eq. ( 5) the deviation is set as D = √ d 2 + h 2 where d and h are the distances between O and the projections of the COG on the X and Y axis respectively.

Besides safety, we further observed that the contact of the robot tracks with the surface cannot alone guarantee proper traction. In reality, it is essential that sufficient force is exerted from the robot to the ground which is possible when the projection of the COG C x is located near the perpendicular projection of the geometrical center O on the staircase. This leads to more uniformly distributed traction along the entire platform. Finally, we further penalize the robot when it loses stability and tips over that happens when its pitch reaches π/2.

2) Platform shaking reduction: The effect of gravity to the movement of the robot switches between ascent and descent, by hindering ascent and accelerating descent that may endanger the robot and lead to tip-overs. Even if no accident happens, every staircase edge traversal causes platform shaking and collisions which influence the robot and potentially an object that is being transported. These events are repetitive when the robot traverses every step, but the greatest impact can be observed in the beginning of the traversal when the robot COG traverses the uppermost stair edge. At that moment the robot velocity can be instantaneously increased leading to a shock, slippage and, at worst, tip-over. The end of traversal is also crucial because if the rear flippers are pushed down the rear robot part just falls down from the step.

Such behaviours lead to distinctive pitch velocity peaks, therefore the controller should mitigate such events. Letting W t denote pitch velocity and K W the normalization coefficient, the corresponding negative reward is:

r W t = -1, if tip over -K W * W t , otherwise (6)
This penalty is assigned only in descent tasks and when the robot traverses the staircase, namely, when the COG projection on the staircase is located between the staircase nosing limits.

C. Controller learning details

In designing a controller learnt via RL there are several alternatives that can be considered [START_REF] Sutton | Reinforcement Learning: An Introduction[END_REF]. Approximators such as tile or coarse coding tend to suffer from the curse of dimensionality and do not generalize well due to sensitivity to grain size. The main advantage of artificial neural networks over other function approximators is their generalization capability and straightforward increase of depth which was well studied.

Another common choice is radial basis function neural networks (RBF-NN) which were widely applied before the emergence of deep neural networks. RBF-NN converges quickly to global optima with less trials and errors, still, they have proved of limited use due to absence of tractable and stable integration into more complex deep neural network architectures.

Shallow networks with one hidden layer tend to have more parameters to approximate the same function in contrast to deep networks with less parameters and the additional ability to learn different representations at intermediate levels. Thus, we have opted for a multi-layer perceptron (see Fig. 1). The first layer is composed of 9 neurons on which the state vector s is received every time step. In the sequel, the input is forwarded through 2 dense layers to the output layer which forms the action vector a.

Following up on the discussion at section II, SAC and TD3 are considered to be more sample efficient than PPO, since they are off-policy and converge faster to higher returns. Still, we use PPO in our work to optimize the parameters of the policy because it is more straightforward to implement, less sensitive to hyper-parameter changes and shown the most favorable results in deep RL.

D. KL divergence-based policy comparison

We can quantitatively compare a pair of trained policies via a commonly used probability distribution divergence measure, namely, the KL divergence. Often, RL algorithms exploit the divergence as metric to be optimized or a constraint of policy updates as for example in PPO where updates of the policy parameters are constrained through a penalty on KL divergence.

We denote the reference policy for a given task as π q and we wish to find how the policy π p diverges from π q (assuming same observation and action spaces). The process of learning a policy amounts to learning the mean and variance statistics of the underlying probability distribution, rather than the exact probability distribution for each possible state-action pair. Making the hypothesis of a normally distributed policy with mean and variance that depend on the state, a policy can then be sampled randomly over the total space of observations with actions chosen according to that policy. Thus, once the total set of observations has been sampled, the corresponding actions are prescribed by one of the two policies being compared, namely, the reference policy π q . After having determined the state-actions pairs over which the policies will be compared, we then calculate the divergence as: Training a policy is always a stochastic procedure and as such a policy is a random variable itself, together with the number of state-action pairs that we choose to calculate divergence. We currently determine empirically the number of times needed to train a policy as well as the number of samples taken in the calculation of the divergence, though a more consistent estimation is possible if required.

D KL (π p , π q ) = s,a π p (s, a)log π p (s, a) π q (s, a) (7)

IV. CONTROLLER DEPLOYMENT ON REAL ROBOT

System description We begin by presenting the experimentation set-up used to obtain the ground-truth robot and environment state. This allows to establish an upper bound on the effectiveness of a controller in reality with moderate state errors while allowing the development and evaluation of alternative robot localization approaches later on. On the other hand, we opted for injecting noise in the state during policy learning in simulation (cf. [START_REF] Mitriakov | Staircase traversal via reinforcement learning for active reconfiguration of assistive robots[END_REF]) to make it more robust to errors in general as well as to reduce the risk of accidents.

To obtain the robot pose we employed visual markers whose pose can be reliably obtained via calibrated cameras and standard toolkits. We distinguish markers into static and dynamic with associated coordinate frames S m and D m (see Fig. 3). We use the notation j T i for the transformation between coordinate frames, denoting translation and rotation of the coordinate frame i within the coordinate frame j. We can then obtain transformations C T Sm and S T Sm , the latter being measured manually. The dynamic marker is placed onto the robot and used to locate its base coordinate frame with respect to camera C, via a constant transformation R T Dm set manually.

The camera was placed in front of the operational space of the robot, ensuring proper coverage of the robot as well as the staircase. Its purpose is to associate the robot and the staircase into a common hierarchy of coordinate frames, allowing at any moment to retrieve the coordinates of the front and rear step edges with respect to the robot (recall eq (2)). The staircase is then represented as a series of static transformations Si T S where S i is the coordinate frame associated to a step edge.

The staircase perception task could also be performed by a robot using its on-board sensors but we refrain to do so to keep our set-up generic and independent of the robot. Perception of a staircase before descent is more challenging but this problem resides out of our current scope.

Policy deployment from simulation to reality In the beginning of an experiment the robot faces the staircase and the controller is activated upon the approaching of front and rear step edges. For the task of staircase negotiation, it turns out that only a subspace of the entire action space spanned by the flipper and arm joints is useful. This is something that we take into account by constraining the action space from which actions can be sampled. As a direct consequence of the fact that joints are allowed to move in a significantly smaller space, this provides sufficient time to perform a necessary action and adaptation of low-level controllers from the simulated platform to the real one and makes this transfer zero-shot. Crucially, constraining the action space further serves in ensuring safety of the platform, since learning is stochastic and non-previously encountered conditions may lead to accidents. For example, overly raising rear flippers while ascending or front flippers while descending, combined with acceleration can provoke a tip-over. To prevent such behaviours, the limits of rear flippers while ascending and front flippers while descending were set as ψ rear s , ψ f ront s ∈ [-π/4, 0]2 respectively. The deployed system architecture is shown in Fig. 3 (b). The Marker monitor detects markers in the camera image. The Environment monitor builds and maintains the geometric relationships between the staircase and the robot, within a single transformation hierarchy and provides the front-most and rearmost step coordinates (recall eq. ( 2)). The Robot state monitor provides information about the robot flippers and arm configuration, its velocity and IMU data. The Safety estimator receives the output of Environment monitor and Robot state monitor, evaluates safety metrics such as the COG deviation and angular velocity and its output is concatenated with data provided by Environment monitor. That output is fed to Execution controller which decides whether to block motors in the case of upcoming accidents or halt the system when the experiment terminates. Otherwise, state output data pass to Policy which samples the action vector a which is sent to the robot actuators. The latter executes those actions and Marker monitor observes changes in the environment. Finally, the output of Safety estimator is logged via Logger.

V. EXPERIMENTS

In this section we present the experiments that we performed with two different platforms, namely, Absolem1 [START_REF] Pecka | Autonomous flipper control with safety constraints[END_REF] and Jaguar V4 Manipulator 2 . Extensive simulated and real-world experiments were performed with varying degree of difficulty for the task of staircase negotiation, allowing us to successfully deal with the associated challenges (cf. video3 ).

A. Description of platforms and associated tasks

In Table II we juxtapose the two articulated tracked robots for which we trained the various controllers. Their simulated models matched the geometric and mass characteristics of the real ones, making use of vendor provided geometric models. The two robots possess similar obstacle negotiation properties thanks to the presence of flippers, with the main difference being that Absolem can independently move each flipper in Jag-Asc-3-Def Abs-Asc-3-Def Ascent, 3DOF, COG Jag-Asc-3-COG Abs-Asc-3-COG Descent, 3DOF, Ang.

Jag-Des-3-Ang Abs-Des-3-Ang Ascent, 5DOF, COG Jag-Asc-5-COG None Descent, 5DOF, Ang.

Jag-Asc-5-Ang None contrast to Jaguar where front and rear flippers are coupled.

To make consistent comparisons, we therefore chose to couple Absolem flippers as well.

It follows that each robot has at most 4 DOFs corresponding to linear velocity, angular velocity, front and rear flipper angles, with Jaguar having 2 additional DOFs due to its arm. For the staircase traversal task though we have deemed that yaw angular velocity control was unnecessary during traversal after the following observations; (i) a robot can easily align itself with the staircase before traversal and (ii) the robot remains aligned during traversal thanks to the coupled front and rear flippers grip on the steps. Thus, the maximum number of DOF is 3 for Absolem and 5 for the Jaguar.

Each task of Table II is thus characterized by the direction of traversal (Ascent or Descent), the number of DOF of the action space and the type of reward used to guide learning (Default refers to the use of eq. ( 3), COG refers to the use of eq. ( 4) and [START_REF] Mitriakov | Staircase negotiation learning for articulated tracked robots with varying degrees of freedom[END_REF] and Ang refers to the use of eq. ( 4) and ( 6)).

B. Simulation environment

The simulation environment used for training is shown in Fig. 4. The Gazebo simulator (gazebosim.org) is employed using the Contact Surface Motion (CSM) model of tracks [START_REF] Pecka | Fast simulation of vehicles with non-deformable tracks[END_REF] for both robots. Adding a new robot has no impact on the overall training time since each robot controller can be trained independently and all of them in batch, server-side simulations. The simulation environment further allows to vary staircase size in ranges of real-world staircases (cf. [START_REF] Mitriakov | Staircase traversal via reinforcement learning for active reconfiguration of assistive robots[END_REF] for details). We note that this is an essential step in succeeding to transfer the policies trained in simulation into the real-world and can be thought of as a data augmentation step, that favors generalization over multiple obstacle shapes and further deals with the influence of noise in the state estimation process.

The robot starts off at the usual location at either the ground floor or the first floor depending on traversal direction, aligned with the staircase and has to achieve the goal position by respecting the preset criteria (both positive and negative rewards are assigned during traversal). The episode ends if the robot tips over, exits the training area, reaches the goal or exceeds the maximal episode time step length.

We evaluate 8 task variants as listed in Table II differing by traversal direction, DOF variation, applied criterion and Fig. 4: Simulation environment robotic platform. The Jaguar robot model starts always with a vertically stretched arm that is fixed in 3 DOF tasks, flippers of both platforms are parallel to the floor in the beginning and

φ 1 a , φ 2 a ∈ [-90 • , 90 • ] 2 .
Finally, penalties are evaluated and assigned only when robots reside on the staircase.

C. Results

1) Performance for ascent tasks: Fig. 5 (a) shows the mean smoothed episode return over independent learning trials with min-max bands, for (Jag-Asc-3-COG) and (Abs-Asc-3-COG).

Mean episode return starts from -0.23 and -0.44 for the two tasks, because of the negative penalty and the failure of the robots to advance to the goal, and converges to 0 by the end of learning for both policies. The max bands can attain the values of 0.26 and 0.28 episode return respectively. This means that the robots learn to control flipper angles and the linear velocity. The episode return for the policy (Abs-Asc-3-COG) is lower in the beginning because the Absolem COG is higher and car tip-over more easily. None of the two learning curves reaches the maximum return 1 as it is impossible to put the COG closer to the point O (see Fig. 2) than a certain distance. This is normal since the robots cannot be entirely parallel to the ground while they traverse the staircase. Fig. 5 (b) shows the evolution of the COG during learning. Both policies reduce the COG deviation at the same pace and reach minimal values after 8000 time steps, or 120 episodes. The Absolem policy exhibits higher COG deviation equal to 0.19 m due to the higher COG placement in the initial configuration where flippers are extended, still, it decreases the initial COG by 17%. The Jaguar policy decreases the COG deviation by 36% down to 0.075 m that approximately represents the COG deviation along the Y axis.

The learning process for two robots is fairly similar. Furthermore, the same value of episode return is achieved by the end of training. The COG curves exhibit similar behaviour and converge at different values, due to differences in mass distributions. Those differences also change robot control features, still both robots achieve the goal by minimizing the COG deviation. Note, however, that the absolute COG deviation difference between Absolem and Jaguar noticed at Fig. 5: Ascent task learning analysis the end of learning in Fig. 5 (b), is not transposed to an equivalent absolute difference in the total reward in Fig. 5 (a), thanks to the normalization of each negative penalty. These results strongly indicate that the framework is applicable to different robot models for the ascent task.

2) Performance for descent tasks: Figure 6 (a) presents learning curves for tasks (Jag-Des-3-Ang) and (Abs-Des-3-Ang). The firsts starts at -0.1 episode return and converges to the value 0.1. The second policy convergences similarly but starts at -0.3 episode return and ends at -0.1. Both platform policies exhibit similar learning behaviour separated by around 0.2 that can be explained by robot dynamics, because the Absolem is exposed to higher pitch angular velocity. This is seen in Fig. 6 (b) where the mean angular velocity of a platform during an episode is higher for the Absolem, due to a higher centroid position over the surface in the resting state. The Absolem policy decreases its angualr velocity by the end of learning by 13%, while the Jaguar policy drops it by 10%. Minimal angular velocity values are attained by 5000 steps, but from this moment onward the learning curves slight increase. This may happen due to optimization of traversal time as the robot spends less time in staircase traversal. As before, these results suggest that the framework is effective for both robots.

3) KL divergence between policies: Figure 7 presents KL divergences for learned controllers in the 3 DOF tasks. We omitted 5 DOF tasks since the simulated model of Absolem robot does not include an arm. An (i, j) th cell of the heat map represents the divergence between a controller with row index i and a controller with column index j, namely, to D KL (π i , π j ). If the KL divergence is low for two policies, then this implies that very similar actions are chosen for the same observations.

The heat map allows the extraction of some very useful insights. For example, for a given staircase traversal direction and total reward design, the two robots develop different policies. This can be particularly seen in divergences (1,4) and (2,5) and demonstrates the importance of separate controller training for the different robots. Despite the fact that, in simulation, the robots share the same motors, the same observations and the same action space, the differences in geometry and mass distribution result in different necessary flipper actions throughout the traversal. It further suggests an operator that is sufficiently skilled to operate a given platform, will not be necessarily able to operate a slightly different platform. We can further observe that for a given robot and traversal direction, alternating the total reward function also induces a quantitative change in the employed actions, for example in divergences (1,2) and [START_REF] Mitriakov | Staircase traversal via reinforcement learning for active reconfiguration of assistive robots[END_REF][START_REF] Mitriakov | Staircase negotiation learning for articulated tracked robots with varying degrees of freedom[END_REF]. This justifies our interest for optimizing staircase negotiation with more elaborate criteria than merely arriving to the goal.

Finally, the highest divergences are noticed between policies where the staircase traversal is different, namely for (1,6) and [START_REF] Torabi | Behavioral cloning from observation[END_REF][START_REF] Tzafestas | Mobile robot control and navigation: A global overview[END_REF]. This clearly suggests that staircase ascent and descent require independent treatment, as dynamics and risks signifi-Fig. 7: Mean Kullback-Leibler divergence between policies We finally performed a quantitative analysis of policy adequacy on different platforms provided in Fig. 8. This experiment is destined to assess how well a policy trained on one robot would perform when deployed to another robot. We performed 10 such trials (rollouts) for a given task. Policies (Jag-Asc-3-Def) and (Abs-Asc-3-Def) are evaluated on the basis of time steps spent in a rollout. As we can see, if we deploy a policy on a platform for which it was trained, then the robot spends less episodes until task completion. Characteristically, rollous tested on Absolem last around 47 time steps if we use the (Abs-Asc-3-Def) policy against 80 time steps if (Jag-Asc-3-Def) policy is used where performance is significantly degraded. Similarly, deployment of the (Jag-Asc-3-Def) on Jaguar requires on average 39 time steps as opposed to 51 when the (Abs-Asc-3-Def) policy is applied.

Deployment of (Jag-Asc-3-Cog) on Absolem further reveals a slight performance degradation in terms of COG deviation measured at 0.206 against 0.188 when (Abs-Asc-3-Cog) is deployed. In contrast, the deployment of these two policies on Jaguar seems equally performant.

With respect to descent, the deployment of the (Jag-Des-3-Ang) and (Abs-Des-3-Ang) policies on the Jaguar robot clearly favors the robot specific policy, but the same policies perform equally well on Absolem. We can conclude that a policy trained and deployed on the same robot provides the best performance, otherwise performance can only be degraded.

4) Real-world performance: Policies (Jag-Asc-5-COG) and (Jag-Des-5-Ang) were tested in two staircases presented in Table III (we recall that the arm is actively involved in the optimized controllers). The respective experiments are included in the supplementary video. Overall, we have performed 10 trials for each task id (5 trials per staircase) discussed in this section, all of which were successful.

Policy (Jag-Asc-5-COG) was tested on the big staircase, the robot achieving the task by keeping the rear flippers pushed down, the front flippers up, the first arm joint was inclined clockwise and the second arm joint counter-clockwise. The flipper configuration ensures that the robot has the contact points with rear and front stair steps. The entire arm is moved forward to decrease COG deviation, which minimizes tipover risks or getting stuck. A point of ambiguity emerges for the arm as certain configurations decrease C y and increase C x at the same time, or vice versa, hence minimizing COG deviation D is not straightforward. Remarkably, the obtained results of the learnt controller show that the robot learnt to incline forward the first arm link and backward the second link, which is indeed the best configuration. For reference, Table IV provides average observed values for some key parameters. The accomplishment of the descent task (Jag-Des-5-Ang) on the big staircase merits more attention. One of the most important aspects is the linear velocity control (see provided video), where the robot moves smoothly and adapts its velocity very attentively that leads to increased time in the staircase traversal. Naturally, this greatly reduces the mean perceived angular velocity by 0.1 rad/s, as opposed to the ascent task.

The ascent and descent tasks were further successfully accomplished on the small staircase composed of 3 steps, each of height 0.17 m and length 0.305 m. The values reported in Table IV allow us to draw the same conclusions between ascent and descent, as for the big staircase.

VI. CONCLUSION

We have presented an effective RL framework for control learning of staircase negotiation in multiple task variants. In particular, we trained controllers for two robotic platforms in simulation with reward function designs suited for ascent and descent and different staircases. Results show learning convergence and optimization of targeted behaviour features for both platforms within 150 episodes. We employed KL divergence to quantitatively compare the obtained policies, allowing us to consolidate earlier empirical findings. Most importantly, we succeeded in deploying the controllers learned in simulation to a commercial robotic platform in firstly encountered realworld staircases. The robot was able to successfully ascend and descend while respecting the underlying criteria.

Our framework can accommodate further improvements that would further increase the autonomy of the deployed system. Although we managed to train effective controllers for normal situations, accidents might still happen as the robot may encounter new situations not explored in training, hence incremental learning is an interesting perspective. Also, full 3D navigation autonomy implies the development of multiple individual controllers that need to be coordinated with each Bottom row; descent by minimizing pitch angular velocity other. We seek to address these challenges in the future as well as explore the possibility to use raw sensory data as input to the learnt controllers.

Fig. 1 :

 1 Fig. 1: Pipeline overview: the desired behavior is developed via RL in simulation and transferred in reality

  coordinates of the next and previous step edge respectively, v s is the robot linear velocity, ψ f ront s and ψ rear s are the flipper angles and φ 1

Fig. 2 :

 2 Fig. 2: (a) Front and rear obstacle (step edge) coordinate systems; (b) side schematic view of the robot on a staircase

Fig. 3 :

 3 Fig. 3: (a) Experimentation setup along with coordinate frame hierarchy; (b) architecture of deployed robot system

Fig. 6 :

 6 Fig. 6: Descent task learning analysis

Fig. 8 :

 8 Fig. 8: Performance evaluation of policies when deployed on Absolem (left) and Jaguar (right) robots.

Fig. 9 :

 9 Fig. 9: Snapshots of staircase negotiation with congruent control of 5 DOF. Top row; ascent by minimizing COG deviation.Bottom row; descent by minimizing pitch angular velocity

Table I :

 I Overview of related works

	Work	Method	DOF	Input data sensors
	Mourikis et al. [7]	State-feedback linearization	3	RGB camera, gyroscope
	Nagatani et al. [8]	Fuzzy, vision-based	4	Laser sensors
	Ben-Tzvi et al. [9]	Fuzzy	3	Motor encoders, compass
	Zhang et al. [10]	Fuzzy, resolved motion rate controller	3	Gyroscope
	Pecka et al. [11]	Neural, RL	2	IMU, laser sensors
	Endo et al. [12]	Proportional-integrative	4	IMU
	Ejaz et al. [13]	Neural, RL, end-to-end	2	Depth camera

TABLE II :

 II Robot characteristics and task variants

	Description	Jaguar	Absolem
	Photo
	Mass, kg	41	29
	Length, m	0.98	1.196
	Width, m	0.7	0.61
	Height, m	0.4	0.46
	Task	Task id	Task id
	Ascent, 3DOF, Default

TABLE III :

 III Staircases configurations

	Name	Number of steps	Height	Length
	Big	5	0.195	0.275
	Small	3	0.17	0.305

TABLE IV :

 IV Observed mean target values on real staircases

	Task id	Stair	Cy, m	D, m	Ang. vel., rad/s
	Jag-Asc-5-COG	Big	0.11 ± 0.01	0.12 ± 0.01	0.33 ± 0.06
	Jag-Asc-5-COG	Small	0.1 ± 0.01	0.11 ± 0.01	0.36 ± 0.09
	Jag-Des-5-Ang	Big	0.16 ± 0.02	0.09 ± 0.02	0.23 ± 0.06
	Jag-Des-5-Ang	Small	0.12 ± 0.02	0.05 ± 0.02	0.23 ± 0.09

bluebotics.com, github.com/mariogianni/trav nav indigo ws

jaguar.drrobot.com/specification V4Arm.asp

partage.imt.fr/index.php/s/LDyp6QRp4nGGKd2/download


Mohamed Darari

Edoardo Domenichini

Antonio Francés-Monerris

Cristina Cebrian

Kevin Magra

Marc Beley

Mariachiara Pastore

Antonio Monari

Xavier Assfeld

Stefan Haacke

Philippe C Gros

Iron(II) complexes with diazinyl-NHC ligands: impact of π-deficiency of the azine core on photophysical properties

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Photoactive metal complexes play a pivotal role in several excitonic devices. 1,2,3 Such applications require the smart design of ligands to finely control the optical and electrochemical properties of the metal complexes. Due to their optimal photophysical features, noble metals, such as ruthenium or platinum allows the design and fabrication of devices having impressive properties in terms of efficiency or tenability. [START_REF] Longhi | Iridium(III) in Optoelectronic and Photonics Applications[END_REF][START_REF] Hagfeldt | [END_REF] However, such success comes with a price to pay; notably the low abundance, relatively high cost and toxicity of such metals, that may strongly limit their employment in larger scale industrial productions. Thus, finding alternative routes, by using less toxic and more abundant metals such as iron, would highly reduce the cost and the environmental impact of manmade technology. However, the fine control of the photophysical behaviour of first-row transition metal complexes is particularly challenging. Indeed, and as a most significant example, while the excited state evolution of ruthenium (II) polypyridyl complexes is dominated by longlived metal-to-ligand (MLCT) triplet states, 6 a sub picosecond relaxation to metal centred (MC) states followed by an ultrafast ground-state recovery is observed for the iron analogues. 7,8,9,10 Most notably, this ultrafast non-radiative deactivation, is the main reason precluding the successful utilization of iron complexes in photochemical applications requiring stabilized MLCT states, such as dye sensitizes solar cells (DSSCs), [START_REF] Hagfeldt | [END_REF] or in emissive devices. Recently, several groups have raised the challenge and proposed the design of photoactive iron complexes by increasing the ligand field splitting with the aim to destabilize the metal centred states. 11 In this context, a considerable breakthrough has been achieved using pyridyl N-heterocyclic carbenes (NHC) as ligands combining the -deficiency of pyridine and the -donating property of NHC. 12,13 Indeed, excited-state lifetimes of tens of picoseconds, have been obtained for iron (II) complexes with tridentate 14,15 and bidentate 16,17 pyridyl-NHC ligands. The influence of the conjugation pattern and peripheral substituents on the electronic properties of NHC has also been investigated by switching from imidazolylidene to benzimidazolylidene (BIm) or modifying the alkyl chain showing a notable improvement of the MLCT lifetime. 14 In contrast, the effect of the nature of the central azine core has not been investigated yet, while the increase of its -deficiency might be a promising route to promote an additional stabilization of the MLCT states when combined with the NHC-induced destabilization of the MC states. In this contribution we report the synthesis of new diazinyl-NHC ligand precursors L1-L5 containing -deficient pyrazine and pyrimidine and their corresponding iron (II) complexes C1-C4. The excited-states decay of the organometallic complexes was characterized by ultrafast time-resolved spectroscopy and by modelling the potential energy surfaces (PESs) via timedependent density functional theory (TD-DFT). Fig. 1 Complexes studied in this work. In all cases, counter ion is PF6

-in any case

Results and discussion

Synthesis

The ligand precursors L1-L5 required for the preparation of the target complexes have been obtained in good yields from both dichloro pyrazine and pyrimidine according to Scheme 1. The ligands have been designed bearing short and long alkyl chains and different counter-anions to anticipate solubility concerns as well as anion effects along the coordination process with iron (II). The coordination of the new ligands with iron (II) to obtain the complexes C1-C4 is performed as reported in Scheme 2. The ligands L1, L2 and L3 were reacted with a mixture of FeBr 2 and hexamethyldisilazide (KHMDS) leading to the target complexes C1, C2 and C3 respectively. The process used above was not effective for the synthesis of C4. In this case, the carbene was first generated by initial deprotonation of L5.Br by LiHMDS followed by reaction with FeBr 2 . In this case, due to solubility issues, only the ligands bearing the hexyl chain led to the expected complex C4.

Electronic and electrochemical properties

The electronic properties of the complexes C1-C4 were preliminary investigated by means of steady-state UV-vis spectroscopy (Figure 2) and cyclic voltammetry (Table 1). The data related to complex C0 have been also included for the sake of comparison. For all the complexes, the absorption spectra are composed of three bands. Intense band in the UV region and below 350 nm (not shown here) corresponds to * transitions centred on the ligands, and as expected, they are much more intense for C4 bearing the four BIm moieties. At longer wavelengths, all the complexes exhibit two distinct bands associated to MLCT transitions and presenting significant shifts and changes in the band-shape as a function of the ligands. The bands in the 360-420 nm range are related to carbene-Fe MLCT transitions 14 , 20 while the bands in the 430-600 nm area are due to the azine-Fe MLCTs. As a general trend, a notable red-shift of 30-40 nm of the azine-Fe band as compared to C0 was observed for all the new complexes. This observation is in full agreement with the increased -deficiency of the central azine when switching from pyridine to pyrazine and pyrimidine. Furthermore, BIm promotes a blue shift of both azine-Fe and carbene-Fe transitions, as can be noticed by comparing the absorption spectra of C1 and C4, as a result of an increased -back donation from the metal, thus stabilizing the occupied iron t 2g orbitals and as a consequence the ground state. 14 While the carbene-Fe transition was not affected when going from pyridine to pyrazine (compare C0, C1 and C2), the use of pyrimidine as the central azine induced a non-negligible red-shift of 30 nm, and a general broadening of the 430-600 nm band, with the appearance of a broader shoulder around 450 nm. This may be due to the reduced point group symmetry of C3 as compared to the other complexes. Figure S17 shows the TD-DFT spectra of C1, C2, C3 and C4.

Even though the lowest-energy bands are blue shifted by around 0.3 eV with respect to the experimental results, the trends described above are correctly reproduced by the computational protocol. The analysis of the natural transition orbitals (NTOs) 21 further confirms the MLCT nature of the excitations involving the lowest-energy bands (see Figures S18-20) and their assignment, whereas some contributions of interligand (IL) states are predicted for the intense band of C4 appearing at higher energies (see Figure S20).

The electrochemical profiles (Table 1) globally confirm the trends underlined by the absorption spectroscopy studies. Due to their higher -deficiencies, pyrazine and pyrimidine promoted an increase of the Fe II /Fe III oxidation potential as compared to C0, an effect that is even higher for C4 due to the electron-accepting effect of BIm . The reduction potentials were also strongly affected and generally increased going from C0 to C4. Since reduction reflects the injection of one electron into the antibonding π* orbitals these results can, in a first approximation, be rationalized as a more efficient stabilization of the LUMO orbital by pyrazine than pyridine. However, C3 represents an exception to the trend, indeed the reduction potential was less significantly modified with respect to the other complexes.

Ultrafast transient absorption spectroscopy

In order to get further insights into the impact of the new ligands on the photophysical properties, the four complexes were investigated by femtosecond transient absorption spectroscopy (TAS) at room temperature. All samples are dissolved in CH 3 CN and excited at 480 nm. For C4, Figure 3A, three signals are observed, two positive bands (excited state absorption ESA) and one negative (ground state bleach GSB). Between such signals, two isosbestic points are identified at 428 and 500 nm. The dot-dashed line corresponds to the inverse of the normalised steady-state absorption (SSA) scaled on the TA spectra at 484 nm, the maximum of GSB. This spectrum has a perfect overlap with the TA-spectra in the range 440-491 nm, while outside, the ESA signals are significantly adding up to the GSB, which implies a decrease of the GSB intensity. The positive bands have a different trend, indeed the blue one (λ<428 nm) does not present any time-evolution, in contrast to the redder one (λ<500 nm), which can be split into two ranges: 500-590 nm where the intensity purely decays (shape conserved in time) while in the range 590-690 nm the spectra evolve within ≈1 ps before reaching a stable shape. This trend is highlighted in the spectra normalised at 473 nm (Fig SI 23). All the spectra at t>1 ps overlay over the full wavelength range regions, while, at earlier times, a slight spectral evolution is visible.

In Fig 3B, selected spectra of C2 are shown. In variance with C4, two negative bands (GSB) are observed, peaking at 388 and 487 nm, and three positives (ESA) at λ<370 nm, 410 and 578 nm. Four isosbestic points are identified at 370, 400, 435 and 516 nm. The SSA, normalized at 487 nm, overlaps perfectly in the red side of the main GSB (λ>487 nm) while, in the blue part (λ<487 nm), the intensity is ≈30% lower than the SSA. This small discrepancy, also in this case, points to an overlapping ESA at these wavelengths. The shape of the long-wavelength ESA presents a small evolution in time, which is better seen in Fig SI 25, where the normalized spectra show for the long-wavelength ESA, at λ>525 nm narrowing, and a small dynamic shift. This causes the isosbestic point at 516 nm to shift by ≈5 nm in the first 20 ps. The kinetics of C4, at relevant wavelengths, are plotted in a semi-log scale and reported in Fig. 4. They are normalised and multiplied by an arbitrary factor. The dashed lines highlight the mono-exponential decay associated with a lifetime of ~30 ps. The kinetic traces are well approximated, on this time scale, by the mono-exponential decay, but some present small deviations i.e. additional short decay components at t < 5 ps with small amplitudes.

For C2, the kinetics are processed as before and plotted in Figure SI27. Most of the kinetics appear to be monoexponential, but for a precise analysis three lifetimes are necessary (see below). For C4 and C2, the minor dynamic shifts of the TA spectra are compatible with the use of a global fit approach. 22 The obtained In analogy to previous work 14,23 , the decay times are interpreted along the reaction scheme (1).

(

) 1
Intersystem crossing (IC) occurs on the ≈0.1 ps time scale, 7,8,9,10 leading to a non-emissive triplet species (no stimulated emission). It can be associated with a non-equilibrated state of MLCT, MC or mixed character (see computational part below, 16 which decays within  2 into the vibrationally equilibrated ( 3 MLCT/ 3 MC) eq . This relaxation step is manifested by the small spectral shifts (Figure 3), which were not analysed in previous work 23,24 . After this relaxation step, related to structural or solvent relaxation, the population lives 32 ps for C4 and 22 ps for C2 (Table 2). Pronounced spectral shifts in compounds C1 and C3. Unlike the previous cases, the data for complexes C1 and C3 show pronounced dynamic shifts and narrowing of the ESA. This is highlighted in Figure 3C for C1, which has the same chemical structure as C2, except that the Hex chains are replaced by methyl groups (cf. scheme 2). C1 and C2 have a very similar excited state decay times ( 3 = 21-22 ps, Table ), but obvious differences in their ESA spectra (Figures 3B and3C). Not only are the ESA transition energies slightly higher for C1, the spectra display pronounced dynamic shifts (no isosbestic point) and narrowing of the long-wavelength ESA band, indicative of a more significant excited state relaxation on time scales > 3ps. This is best highlighted by the normalised data (Figure SI28). The data for C3 display even stronger relaxation effects, and a shorter  3 (11-12 ps), as summarized in the SI. For C1, the modifications of the ESA spectrum in the initial 0.1-0.2 ps are due to ultrafast IC as above. For delays until ≈ 10 ps, the ESA decays faster on the long wavelength side (≥ 530 nm), flattens and shifts to shorter wavelengths. This gives rise to a decay at > 530 nm, and a rise component in the 490-520 nm range (cf. Figure 6).

The result is the progressively shifting ESA/GSB crossover point around 500-510 nm. A similar decay/blue-shift behaviour is also observed in the minor ESA band at 410-430 nm. The ESA difference spectra steepen progressively in the 500-525 nm range (Figure SI28). Even though the effect may appear small, global fitting cannot be used any longer since the ESA extinction coefficient is time dependent , i.e. the oftenmade assumption of separability of wavelength and time dependencies does not hold. Note that in the present case, spectral evolution and excited state decay occur on similar time scales, particularly for C3 (Table ).

The results of 20 three-component fits performed for individual wavelengths are given in Figure 7. The wavelength-dependent lifetimes are highlighted in the top panel, especially for  2 and  3 . The latter represents the excited state lifetime of C1, characterised by the average ground state bleach recovery of 21±1 ps in the 430-490 nm range. In the ESA region,  3 is below 20 ps for > 600 nm and rises slightly but significantly for shorter wavelengths, up to 24±1 ps for 505-520 nm. This is consistent with the different longtime slopes of the 519 and 630 nm traces in Figure 6. The above spectral shifts occur on the time scale of  2 , which follows a similar trend, i.e. increasing up to 6-8 ps in the 520nm region. The present situation of excited state kinetics with spectral shifts is similar to our recent report on bidentate Fe(II) complexes, and poses the question whether  2 and  3 represent the decays, i.e internal conversion into S 0 , of two separate excited state populations. For C1, we rather opt for a sequential scenario, as for C2 and C4, and assign the small amplitude  2 to a slow excited state relaxation for two main reasons. Firstly,  2 is an ESA decay time for > 530nm and an ESA rise time for 500-530 nm (Figures 6 and7). There, its amplitude is too large compared with what it would be for a GSB recovery (Figure 7, bottom panel). In the minor ESA band (400-430 nm),  2 appears to be a rise time (cf. 420 nm trace in Erreur ! Source du renvoi introuvable.), even though the fit gives a small positive amplitude due to the high noise level (see Figure SI29 for a better illustration of the rising ESA in the 490-510 nm and around 410-410 nm range). Second, comparing C1 and C2, one can envisage that the lateral hexyl/methyl chains have an influence on details of the excited state structure suggesting an effect in the structural dynamics, thus modulating the energies of ESA transitions and the value and amplitude of  2 in a sequential reaction scenario. On the contrary, we do not expect the hexyl/methyl substitution to open a second faster excited state decay channel ( 2 in addition to  3 ), which would lead to the "parallel" excited state relaxation scenario observed for bidentate complexes. 17 Therefore, C1 obeys the same excited state relaxation scheme (1) as C2 and C4. For C3, it is likely that  2 is also associated with IC into S 0 since this component appears as a recovery time throughout the GSB band (cf. Fig. SI33). We will address the origin of the different lifetimes and the visibility of the  2 -related structural relaxation process in the discussion section.

Computational analysis of the excited-state decay

The photophysics of the new dyes have been analysed by means of density functional theory (DFT) and its timedependent DFT (TD-DFT) extension, given its proven good description of the electronic structure and the excited-state decay of related Fe(II) complexes. 13 Figure 10 shows the calculated PESs for the lowest-lying triplet 3 MLCT and 3 MC states and the quintet [START_REF] Hagfeldt | [END_REF] MC state for compounds C1, C3 and C4. Note that the electronic structures of C1 and C2 are very similar since they only differ in the length of the alkyl chain, therefore only the C1 complex has been modelled. Previous works on bidentate Fe-NHC systems reported that Fe-N or Fe-C bond elongations actually drive the excited-state decay also allowing singlet-triplet crossings with the ground state. 13,17,25 Thus, the curves have been built by means of relaxed scan calculations along one of the Fe-N bonds in order to explore the role of this coordinate in the excited-state decay in tridentate Fe(II) complexes. Note that such an approach while it has the advantage of a reduced computational cost compared to the full characterization of the PES by the calculation of minimum energy paths (MEP) only provides upper bounds to the energetic barriers encountered during the path. It can be readily seen that the 3 MLCT minima are close to the Franck-Condon region, whereas the lowest point in the 3 MC curves is placed at larger Fe-N distances due to its inherent Fe-N antibonding character. It shall be noted that the adiabatic energy difference between the 3 MLCT and 3 MC lowest energy points in the PESs, for instance of C4 (0.19 eV, see Figure 10c), is in excellent agreement with the 0.12 eV value reported for a related compound with similar excited-state lifetime. 14 Results displayed in Figure 8 clearly indicate that the Fe-N stretching mediates the photoresponse also in tridentate Fe-NHC complexes, giving access to the 3 MC/S 0 intersystem crossings. Striking differences are however found between tridentate and bidentate complexes. 16 The ps range excited-state lifetime in the former can be explained by the exploration of planar or quasi planar triplet PES regions where S 0 lies above the triplet state manifold, acting as energy traps (spin crossover). 17 In tridentate compounds, the Fe-N bond stretch clearly destabilizes both S 0 and the triplet states, even though the destabilization is not uniform between the two states, hence allowing their crossing. The reason for the extended excitedstate lifetime beyond the ultrafast sub-picosecond regime can hence now be ascribed to the presence of energy barriers, found on the triplet PESs, connecting the Franck-Condon and the singlet-triplet crossing region. Note however, that the values of ~0.5 eV found for the barriers from the 3 MC minima to the corresponding 3 MC/S 0 singlet-triplet crossing points represent upper bounds could be significantly lowered by the explicit calculations of the minimum energy paths. Furthermore, the non-adiabatic 3 MC→S 0 intersystem crossings could take place at regions with triplet-singlet energy differences of ~0.1-0.2 eV, and not necessarily at the crossing points shown in Figure 8 (~2.73-2.74 Å). As a matter of fact, we have considered C1 as a representative example to compute the T 1 /S 0 spin-orbit coupling (SOC) along the PESs shown in Figure 8a. The values at each Fe-N distance are shown in Table 3, indicating a strong coupling already at ~2.5-2.6 Å distances, supporting intersystem crossing events at shorter iron-nitrogen elongation and hence with a lower energy cost. The adiabatic evolution of T 1 from 3 MLCT to 3 MC at Fe-N distances of ca. 2.3 Å is also evidenced by the sudden increase in the SOC value of about 100 cm -1 between the 2.238 and 2.338 Å Fe-N distances. Regarding the role of the lowest-lying quintet [START_REF] Hagfeldt | [END_REF] MC state, the energy levels shown in Figure 8 suggest a minor participation of this state in the excited-state decay since no 5 MC/S 0 crossings are predicted along the explored Fe-N distances. This consideration is coherent with the behaviour observed in bidentate Fe-NHC. 16,17 Figure 9 shows the adiabatic profile of the lowest-lying 20 triplet states for the pyrazine-based complexes C1 and C4. As in the case of Fe-NHC bidentate complexes, 16 only the three lowest-energy triplet states lead to singlet-triplet crossing points, whereas the higher states are extremely destabilized by the Fe-N stretching. These results reveal the crucial role of these low-energy triplet states, since after light absorption an initial ultrafast intersystem crossing followed by vibrational cooling of the hot MLCT states is observed (see above). The upper bounds for the triplet energy barriers found in the present TD-DFT calculations (<0.5 eV for the Fe-N distances from ~2.2 to ~2.5-2.7 Å, see Figure 8) are in agreement with the experimental estimations based on the Arrhenius equation (~0.1 eV), as depicted in the discussion section.

Discussion

The four new compounds C1-C4 investigated in this work extend our previous sets of Fe(II) complexes with bidentate and tridentate ligands bearing respectively three and four carbene bonds. In all cases, the ligand field enhancement is large enough to prevent excited state relaxation into the 5 T 2 high spin manifold. 13 According to reaction scheme (1), ultrafast spectroscopy monitors the sub-100 fs 1 MLCT 3 MLCT intersystem crossing, and, most importantly, the effect of the ligand's nature on the 3 MLCT/ 3 MC lifetime. The present combined experimental and theoretical study rationalises the combined effect of the electronic deficiency of the pyrazine and benzimidazolylidene (BIm) moieties in lowering the 3 MLCT energy, in analogy to the strategy described previously. 14 We find that BIm is again most effective in this respect allowing to achieve, for C4, an excited state lifetime of  3 = 32±2 ps, meaning a ≈30% increase with respect to C1 and C2.

As indicated in Figures 8 and9, the ISC and relaxation into the singlet ground state involves Fe-N stretching and thus a MLCT MC conversion, but at the cost of an activation energy <0.5 eV according to the PESs. Alternatively, if we express  3 by the Arrhenius law, , and supposing v 0 = 3-4 ps -1 for the excited state Fe-N stretch frequency, one may estimate the excited state barrier  to be in the range of 3.5 to 5.0 k B T, meaning 90 to 130 meV (Figure 10), in agreement with the TD-DFT computations. Slight modulations of the excited state energy landscape lead to the variation of a factor of ≈2.7 in the excited-state lifetime (12 to 32 ps). A tentative explanation for the different excited-state lifetimes measured for the C1-C4 series can be formulated on the basis of possible out-of-plane (OOP) distortions in the 3 MLCT state located on the electron accepting nitrogen of the diazine ring, not considered in the present computations. In the 3 MLCT state, the extra charge on the ring breaks the aromaticity, since the number of  electrons is now 4n+3, which can lead to an OOP deformation, located on the electron accepting nitrogen. The higher the charge density over the Fe-coordinating nitrogen, the larger the out-of-plane amplitude and thus the driving force for increasing the Fe-N distance. Comparing pyrazine-based complexes (C1 and C2) and pyrimidine-based one (C3), that will have an asymmetric excited state charge distribution, one may expect, for the latter, a higher electron density on the Fe-coordinating nitrogen and thus a lower E, consistent with the shorter excited-state lifetime observed. C4, on the other hand, will have a smaller excess electron density on the pyrazine ring since the BIm moieties have a larger electron accepting character. 14 The OOP driving force is thus minimal for C4, consistent with its longer excited-state lifetime. Further calculations are required to validate these hypotheses. In addition to the excited-state lifetime, we report above a hitherto overlooked excited state relaxation process acting during  2 . The effect manifested by the ESA blue-shift is most pronounced in C1 and C3, while virtually absent in C2 and C4. The latter are bearing lateral hexyl chains instead of methyl groups in the former. The long hexyl chains can interact with the diazine rings of the other tridentate ligand. We argue that this interaction hinders the above out-of-plane (OOP) motion, and thus the additional structural relaxation. Consequently, and since OOP deformation increases the Fe-N and thus helps reaching the T 1 /S 0 crossing points (see Figure 10), the general trend of "more pronounced spectral relaxation effects go along with shorter excited state lifetimes" (C1 and C3) is expected. An alternative explanation for the  2 process would refer to dynamic solvation, i.e. re-orientation of the polar acetonitrile solvent molecules as a response to the electron density increase on the ligands in the 3 MLCT states. 26 Here again, the hexyl chains would decrease the effect due to a reduced ligandsolvent interaction.

Conclusions

The present combined experimental and computational study investigates the effect of replacing pyridine by diazines in tridentate C^N^C heterocyclic ligands in Fe(II) complexes. It reveals several factors, which are beneficial in stabilising the excited-state, leading to an improved > 30 ps excited-state lifetime for this class of four-carbene coordinating ligands. These ligands, as much as carboxylic acid groups used in our previous studues, 20 lower the MLCT energies in the Franck-Condon region, in addition to the BIm moieties. In addition, the present study confirms, from a computational viewpoint, the role of the Fe-N distance increase required for the excited state decay. Experiments indicated that an energy barrier must be overcome to reach the corresponding 3 MC/S 0 crossing points, which mediate the decay to the ground state. The possible role of the electronic charge distribution over the C^N^C ligands and the propensity of the diazine ring to undergo out-of-plane motion is also discussed. The present work provides useful synthetic tools to design Fe(II) complexes with long-lived excited-state lifetimes, and represents a step further in the comprehension of the intricate photo processes taking place in first-row transition metal complexes.

Experimental and computational details

Synthesis

Solvents and commercially available reagents were used as received. Thin layer chromatography (TLC) was performed by using silica gel 60 F-254 (Merck) plates and visualized under UV light. Chromatographic purification was performed by using silica gel 60 (0.063-0.2 mm/70-230 mesh). 1 H (400 MHz) and 13 C NMR (100 MHz) spectra were taken on a DRX400 Bruker spectrometer at ambient temperature. The chemical shifts (δ), were calibrated by using either tetramethylsilane (TMS) or signals from the residual protons of the deuterated solvents and are reported in parts per million (ppm) from low to high field. High-resolution mass spectrometry (HRMS) data was obtained by using Bruker micrOTOF-Q spectrometer. UV-vis spectra were recorded in a 1 cm path length quartz cell on a LAMBDA 1050 (Perkin Elmer), spectrophotometer. Cyclic voltammetry was performed on a Radiometer PST006 potentiostat using a conventional three-electrode cell. The saturated calomel electrode (SCE) was separated from the test compartment using a bridge tube. The solutions of studied complexes (0.2 mM) were purged with argon before each measurement. The test solution was acetonitrile containing 0.1 M Bu 4 NPF 6 as supporting electrolyte. The working electrode was a vitreous carbon rod (1cm2) wire and the counter-electrode was a 1 cm 2 Pt disc. After the measurement, ferrocene was added as the internal reference for calibration. All potentials were quoted versus SCE. In these conditions the redox potential of the couple Fc+/Fc was found at 0.39V. In all the experiments the scan rate was 100mV/s.

Synthesis of ligands

L1 27 : To a solution of 2,6-dichloropyrazine (300 mg, 2.01 mmol) in 40mL of dioxane, was added an excess of 1-methyl-1H-imidazole (0.64 mL, 8.06 mmol) and the mixture was refluxed overnight. After cooling at room temperature, a saturated solution of KPF 6 was added (20 mL), the precipitate was washed with distilled water and dried under vacuum affording L1 (750 mg, 70%). 1

L2:

To a solution of 2,6-dichloropyrazine (180 mg, 1.21 mmol) in 20mL of dioxane, was added an excess of 1-hexyl-1Himidazole (736 mg, 4.83 mmol) and the mixture was refluxed overnight. After cooling at room temperature, a saturated solution of KPF 6 was added (10 ml), the precipitate was washed with distilled water and dried under vacuum affording L2 (600 mg, 74%): 1 H NMR (400 MHz, CD 3 CN) δ 9.48 (s, 2H), 9.23 (s, 2H), 8.26 (t, J = 1.9 Hz, 2H), 7.72 (t, J = 1.9 Hz, 2H), 4.33 (t, J = 7.4 Hz, 4H), 2.02 -1.95 (m, 4H), 1.45 -1.29 (m, 12H), 0.91 (t, J = 7.1 Hz, 6H). 13

L3:

To a solution of 2,4-dichloropyrimidine (250 mg, 1.68 mmol) in 30mL of dioxane, was added an excess of 1-methyl-1H-imidazole (530 mg, 6.71 mmol) and the mixture was refluxed overnight. After cooling at room temperature, 40mL of diethyl ether was added and the precipitate was filtered and dried under vacuum affording L3 (405 mg, 77%): 1 H NMR (400 MHz, CD 3 CN) δ 9.58 (s, 2H), 9.18 (d, J = 5.6 Hz, 1H), 8.30 (d, J = 17.1 Hz, 2H), 7.91 (d, J = 5.5 Hz, 1H), 7.64 (d, J = 8.7 Hz, 2H), 4.03 (s, 6H). 13 Found: 341.1509. To a solution of BisBIm-Pz (150 mg, 0.48 mmol) in 2mL DMF was added 0.18mL of methyl iodide (1.93mmol) and the mixture was refluxed overnight. After cooling at room temperature, a saturated solution of KPF 6 was added (10 ml), the precipitate was washed with distilled water and dried under vacuum affording L4 (255 mg, 84%) 1 H NMR (400 MHz, DMSO-d6) δ 10.77 (s, 2H), 9.61 (s, 2H), 8.45 (d, J = 8.2 Hz, 2H), 8.24 (d, J = 8.1 Hz, 2H), 7.83 (dt, J = 24.0, 7.7 Hz, 4H), 4.31 (s, 6H). 13

Synthesis of complexes

General procedure of the synthesis of C1, C2 and C3 : To a solution of the appropriate ligand L (1 eq) in 2 mL of anhydrous DMF were added simultaneously a solution of KHMDS (4 eq) in 2 mL anhydrous DMF and a solution of FeBr 2 (0.5 eq) in anhydrous 1mL DMF, the resulting dark red mixture was stirred at room temperature for 10 min. A saturated solution of KPF 6 was added (10 ml), and the precipitate was collected by filtration. The solid was purified by column chromatography eluting with an acetone/H 2 O/KNO 3 (sat) = 10: 3: 1 mixture. The yellow fraction was collected and after the evaporation of acetone, the left solution was treated with a saturated solution of KPF 6 . Affording the precipitation of the complex, it was then filtered, washed with distilled water and dried under vacuum. C1: The general procedure starting from L1 (148 mg, 0.28mmol), KHMDS (222 mg, 1.11 mmol) and FeBr 2 (30 mg, 0.14 mmol) gave C1 (23 mg, 20% yield). 1 H NMR (400 MHz, CD 3 CN) δ 9.04 (s, 4H), 8.13 (d, J = 2.2 Hz, 4H), 7.04 (d, J = 2.2 Hz, 4H), 2.51 (s, 12H). 13

6 - 1 .

 61 Scheme 1. Synthesis of ligands L1-L5

Scheme 2 .

 2 Scheme 2. Synthesis of Complexes C1-C4

Figure 2 .

 2 Figure 2. UV-Vis spectra of C0-C4 in acetonitrile. The portion of the spectrum below 350 nm is not shown due to intense UV bands in case of C4.

Figure 3 .

 3 Figure 3. Selected transient absorption spectra of C4 (A), C2 (B), C1 (C) in AcN excited at 480 nm. Delay times are from 0.1 ps (blue) to 50 ps (red). Spectra of C4 and C2 display mainly intensity/exc. state population decay, while the spectra of C1 are subject to shifts and changes in shape until ≈20 ps. Inverted steady state absorption (SSA) spectra are shown for comparison (dot-dashed pink line). Note different scale for positive and negative parts of y-axes. In B, grey shaded area denotes excessive pump beam scatter.

Figure 4 .

 4 Figure 4. Selected kinetics of C4 plotted in semi-log scale. The kinetics are normalised and multiplied for an arbitrary factor as reported in the legend. For the GSB a negative factor is used. Dashed lines lines indicate the ~30 ps decay. Small deviations from the single exponential behaviour at early times relate to the minor spectral evolutions discussed in the text.

Figure 5 .

 5 Figure 5. Decay-Associated difference Spectra of C4 (A) and C2 (B) obtained with Glotaran (see ref.22). The DAS associated with τ1 have contributions from ESA and solvent response. The amplitude and shape of the 1.9 and 4.4 ps DAS are consistent with the minor spectral evolutions on this time scale

Figure 6 .

 6 Figure 6. Kinetic traces of TAS for C1, on a semi-log scale. The selected wavelengths cover the minor ESA band (420nm), the main GSB (447 and 490nm) and ESA (519 and 630 nm). The GSB traces are shown with inverted signs, and all traces except for 420nm are shifted on the log scale by arbitrary factors. Dashed lines represent a ≈20ps decay time, as guides-to-the-eye. Shaded areas in the 490 and 519 nm traces highlight the deviations from single exponential decay for t< 10 ps. For these wavelengths, they are due to the dynamic shift of the ESA spectrum, also seen at 630 nm, on a 2-6 ps time scale. These effects are smaller for the minor ESA band (< 450 nm).

Figure 7 .

 7 Figure 7. Results of 3-component fits for C1. Top: Lifetimes in ps; bottom: Amplitudes of the relevant components  2 and  3 . Dashed lines show the scaled and inverted steady-state absorption spectra (SS-OD). Data in the range 460-480 nm are not analysed due to pump laser scatter. The decay-associated amplitudes (bottom) follow approximately the shape of GSB and ESA (Figure 3), i.e. they are mostly associated with their decay times; except for the 490-530 nm region, where A2 represents an ESA rise time.

Figure 8 .

 8 Figure 8. Fe-N relaxed scan energy profiles of a) C1, b) C3 and c) C4. For the sake of simplicity, only the lowest-lying 3 MLCT, 3 MC and 5 MC states are shown.

Figure 1 :

 1 Figure 1: Kinetic traces of TAS for C1, on a semi-log scale. The selected wavelengths cover the minor ESA band (420nm), the main GSB (447 and 490nm) and ESA (519 and 630 nm). The GSB traces are shown with inverted signs, and all traces except for 420nm are shifted on the log scale by arbitrary factors. Dashed lines represent a ≈20ps decay time, as guides-to-the-eye. Shaded areas in the 490 and 519 nm traces highlight the deviations from single exponential decay for t< 10 ps. For these wavelengths, they are due to the dynamic shift of the ESA spectrum, also seen at 630 nm, on a 2-6 ps time scale. These effects are smaller for the minor ESA band (< 450 nm).

Figure 9 .

 9 Figure 9. Fe-N relaxed scan energy profiles of a) C1 and C4. The first 20 triplet states and the lowest-lying 5 MC states are shown.

Figure 10 .

 10 Figure 10. Sketch of the ultrafast excited state relaxation processes, along the PES of 1 MLCT (brown) and the triplet manifold (blue/red). In phase 1, vibrational relaxation brings the population into the 3 MLCT state (1). Structural and/or dielectric relaxation (red arrow) acts during phase 2, 2, and stabilises the populations in the red PES, leading to red-shift of ESA. Population decay into S0 takes place during both phases at

Table 1 .

 1 Photophysical and electrochemical properties of complexes

		(M -1 .cm -1 )) a abs-max (nm)	(V/SCE) b (Fe III /Fe II ) Eox		(V/SCE) Ered1	ΔE (eV) c
	C0	393 (9000)	0.80 (rev)		-1.95 (irrev.)	2.75
		460 (15900)
	C1	389 (6300)	0.98 (rev)		-1.39 (rev)	2.27
		487 (16100)
	C2	392 (6200)	1.09 (rev)		-1.32 (rev)	2.41
		493 (17300)
	C3	406 (8500)	0.93 (rev)		-1.70 (irrev)	2.63
		477 (12600)
	C4	350 (sh)(11000)	1.22 (rev)		-1.21 (rev)	2.43
		479 (16200)
	a Measured in CH3CN at 25 °C.	b First oxidation potential. Potentials are quoted vs SCE.
	Under these conditions, E1/2 (Fc+/Fc) = 0.39V/S.C.E. Recorded in CH3CN using Bu4N + PF6 -
	(0.1M) as supporting electrolyte at 100 mV. s	-1 .	c Electrochemical band gap (E =Eox -
	Ered1)

Table 2 :

 2 Time constants derived from the analysis of the TAS data.  3 is the excited state lifetime, and  1 and  2 describe excited state relaxation according to the sequential reaction scheme (see below). In the absence of global fitting, ranges of values are indicated for C1 and C3.

			a
	Complex	τ1 (ps)	τ2 (ps)	τ3 (ps)
	C1	0.10±0.05	2-6	21-25
	C2	0.10±0.05	4 ± 1	22±2
	C3	0.10±0.05	2-4	11-12
	C4	0.10±0.05	1.5 ±0.5	32±3

A

Table 3 .

 3 T 1 /S 0 SOCs for the C1 complex.

	Fe-N (Å)	2.138	2.238	2.338	2.438	2.538	2.638
	SOC (cm -1 )	47	88	189	228	247	259

  C NMR (101 MHz, CD 3 CN) δ 141.6, 137.1, 135.6, 124.9, 120.3, 51.4, 31.3, 29.8, 25.9, 22.7, 13.8. HRMS (ESI) calcd for C 22 H 34 N 6 PF 6 m/z = 381.2761 [M -PF 6 ]

+ . Found: 381.2765.

  2H), 9.19 (s, 2H), 8.29 (dd, J = 6.2, 2.6 Hz, 2H), 7.86 (dd, J = 6.2, 2.8 Hz, 2H), 7.49 -7.34 (m, 4H). HRMS (ESI) calcd for C 20 H 17 N 6 m/z = 313.1196 [M+H] + .

	C NMR (101 MHz, D 2 O) δ 163.9,
	154.8, 152.3, 125.6, 125.0, 119.4, 110.6, 36.7. HRMS (ESI)
	calcd for C 12 H 14 N 6 Cl 2 m/z = 121.0634 [M -2Cl] 2+ . Found:
	121.0632.
	L4 27 : A solution of Benzimidazole (642 mg, 5.44 mmol) in
	anhydrous DMF (3mL) was added dropwise to the suspension
	of NaH 95% (137 mg) in anhydrous DMF (3mL) over a period
	of 15 min at room temperature under inert atmosphere. Then a
	solution of 2,6-dichloropyrazine (300 mg, 2.01 mmol) in
	anhydrous DMF (3mL) was added, and the mixture was
	refluxed overnight. The reaction mass pH was then adjusted to
	pH 6.2 by adding 20mL of distilled water. The precipitate
	formed was subsequently filtered and washed with water and
	dried under vacuum affording 2,6-bis(1H-benzo[d]imidazol-1-
	yl)pyrazine (BisBIm-Pz) (533 mg, 84%) 1 H NMR (400 MHz,
	DMSO-d6) δ 9.32 (s,

  C NMR (101 MHz, DMSO-d6) δ 144.0, 142.6, 139.1, 132.6, 129.7, 128.7, 128.1, 115.8, 114.9, 34.7. HRMS (ESI) calcd for C 12 H 14 N 6 I m/z = 341.1507 [M -I] + . Found: 341.1509. L5. Br : To a solution of BisBIm-Pz (150 mg, 0.78 mmol) in 3mL DMF was added 0.46mL 1-bromohexane (1.93 mmol) and the mixture was refluxed overnight. After cooling at room temperature, 30mL diethyl ether was added and the precipitate was filtered and dried under vacuum affording L5.Br (225 mg, 73%) 1 H NMR (400 MHz, DMSO-d6) δ 10.80 (s, 2H), 9.62 (s, 2H), 8.45 (d, J = 8.3 Hz, 2H), 8.32 (d, J = 8.2 Hz, 2H), 7.81 (dt, J = 15.3, 7.4 Hz, 4H), 4.70 (t, J = 7.3 Hz, 4H), 2.14 -1.98 (m, 4H), 1.53 -1.43 (m, 4H), 1.41 -1.26 (m, 8H), 0.89 (t, J = 7.0 Hz, 6H). 13 C NMR (101 MHz, DMSO-d6) δ 142.9, 142.0, 138.6, 131.4, 129.4, 128.1, 127.6, 115.5, 114.4, 47.5, 30.6, 28.3, 25.3, 21.8, 13.8. HRMS (ESI) calcd for C 30 H 37 N 6 Br 2 m/z = 481.3074 [M-2Br] + . Found: 481.3044

  C NMR (101 MHz, CD 3 CN) δ 198.0, 150.9, 128.0, 126.6, 117.5, 35.2. HRMS (ESI) calcd for C 24 H 24 FeN 12 P 2 F 12 m/z =268.0793 [M-2PF 6 ] 2+ . Found: 268.0816 C2: The general procedure starting from L2 (187 mg, 0.28 mmol), KHMDS (222mg, 1.11 mmol) and FeBr 2 (46 mg, 0.14 mmol) gave C2 (23 mg, 30% yield). 1 H NMR (400 MHz, CD 3 CN): δ 9.12 (s, 4H), 8.23 (d, J=2.2 Hz, 4H), 7.18 (d, J=2.3Hz, 4H), 2.60 (t, J=2.2 Hz 8H), 1.17 -1.09 (m, 8H), 1.04 -0.95 (m, 8H), 0.82 (t, J= 7.3 Hz, 20H), 0.66 -0.56 (m, 8H). 13 C NMR (101 MHz, CD 3 CN) δ 196.6, 150.9, 127.1, 126.7, 49.7, 31.6, 31.0, 26.2, 22.7, 13.8. HRMS (ESI) calcd for C 44 H 64 N 12 P 2 F 12 m/z = 408.2358 [M -2PF 6 ] 2+ . Found: 408.2406. C3: The general procedure starting from L3 (101 mg, 0.32 mmol), KHMDS (238 mg, 1.30 mmol) and FeBr 2 (35 mg, 0.16

Acknowledgements

The project is funded by the French Agence Nationale de Recherche (ANR-16-CE07-0013-02). The L2CM team is grateful to F. Lachaud for mass spectrometry. The Strasbourg team thanks O. Crégut, J. Brazard and J. Léonard for help and fruitful discussions. Molecular modelling has been performed on the LPCT local computing resources. A.F.-M. acknowledges the Région Grand Est government (France) and the ANR agency for postdoctoral contracts, and the MINECO project CTQ2017-87054-C2-2-P (Spain). The Strasbourg group acknowledges financial support through the Labex NIE. This article is a proof of the LPCT team's recognition of Jean-François Hallet's accomplishments in the field of theoretical studies of transition metal compounds.

Please do not adjust margins mmol) gave C3 (20 mg, 15% yield). 1 H NMR (400 MHz, CD 3 CN): δ 8.91 (d, J=5.6 Hz, 2H), 8.08 (dd, J=15.1, 2.1, 4H), 7.79 (d, J=5.7 Hz, 2H), 7.06 (dd, J=9.7, 2.0 Hz, 4H), 2.56 (d, J=5.5 Hz, 12H). 13 C NMR (101 MHz, CD 3 CN) δ 202. 1, 199.4, 161.2, 159.4, 157.2, 128.4, 127.8, 116.8, 102.9, 35.5, 35.51. HRMS (ESI) calcd for C 24 H 24 FeN 12 P 2 F 12 m/z = 268.0793 [M -2PF 6 ] 2+ . Found: 268.0811. C4: A Schlenk tube was loaded with L5.Br (149 mg, 0.23 mmol) then 5mL of dry THF was added under inert atmosphere. The reaction mixture was cooled to -10 °C and LiHMDS (0.46 mL, 0.46 mmol, 1 M in THF) was added dropwise over 30min. The cooling bath was removed, and the temperature allowed to raise slowly up to 20°C. The reaction mixture was then stirred for 1h. A solution of anhydrous FeBr 2 (25 mg, 0.11 mmol) in dry THF (3 mL) was finally added via syringe to the Schlenk tube. The reaction mixture was stirred at room temperature for 24 h after which, the solvent was removed under reduced pressure. 3mL of distilled water was added followed by 10mL of a saturated solution of KPF 6 and the precipitate was collected by filtration. The solid was purified by column chromatography eluting with a acetone/H 2 O/KNO 3 (sat.) = 10: 3: 1 mixture. The yellow fraction was collected and after the evaporation of acetone, the residue was treated with a saturated solution of KPF 6 leading to precipitation of the complex that was then filtered, washed with distilled water and dried under vacuum giving C4 (20 mg, 13% yield). 1

Computational details

The ground-state equilibrium structures of all the complexes have been optimized using the standard DFT/B3LYP functional in combination with the 6-31+G(d,p), as implemented in the GAUSSIAN 09 (D.01 revision) software package. 28 This computational protocol has repeatedly proven its ability to describe the Franck-Condon geometry of many Fe(II) complexes. [29][30][31] Note that the hexyl groups of C4 have been substituted by ethyl groups in order to make the computations more affordable. The excited states at the Franck-Condon area have been computed by means of the TD-DFT method using the Tamm-Dancoff approximation, 32 in particular making use of the exchange correlation functional HCTH407 (hereafter, TD-HCTH), 33 according to previous studies on similar Fe-NHC systems as documented by our research group. 13,14,16,20,34 The theoretical spectra reported in the ESI have been obtained by sampling the nuclear density of the Franck-Condon region through a semiclassical Wigner distribution of 20 structures, and convoluting the excitation energies with Gaussian functions with a full-width at half-length (FWHL) of 0.3 eV. The first 40 singlet-singlet vertical absorptions have been computed for each geometry, making use of the larger 6-311G(d,p) basis set.

The nature of the excitations have been determined by computing the NTOs, 35,36 via post-processing of their density matrices obtained with the TD-HCTH method, using the NANCY_EX code. 36,37 For the readers' convenience, we remind that NTOs may be considered as the best orbitals to describe a given electronic transition, and in contrast to the Kohn-Sham representation, often one or two couples of occupied/virtual orbitals are enough to fully describe the electronic density rearrangement in the excited state. Whereas all triplet energies reported in the present work have been obtained with the TD-HCTH method and the 6-311G(d,p) basis set, different optimization techniques and basis sets have been employed to compute the different geometries that compose the PESs. The 3 MLCT equilibrium geometries have been obtained by means of TD-HCTH/6-31+G(d,p) optimizations using standard minimization algorithms. Starting from the latter structures, the Fe-N connectivity with the largest bond distance have been elongated through relaxed scan calculations defining increments of 0.1 Å. The smaller 6-31G* basis set have been used to make the scan calculations affordable. Final energies have been subsequently recomputed using the 6-311G(d,p) basis set, and the 3 MLCT or 3 MC nature of the state have been determined by computing the NTOs at each geometry. Scan calculations using the unrestricted (u)-DFT method leads to extremely similar results as reported in the ESI (see Figure S21). On the other hand, the quintet [START_REF] Hagfeldt | [END_REF] MC states have been calculated using the unrestricted DFT/HCTH ansatz.

The singlet-triplet SOCs

, where refers to the spin-orbit Hamiltonian, have been computed using the ADF 2018 modelling suite, 38 ,39,40 employing the TD-HCTH method and making use of the DZP basis set for all atoms except for iron, described with the TZP basis set.

Femtosecond transient absorption spectroscopy

The 800-nm output of a 5 kHz Ti:Sapphire amplifier (Amplitude Technologies) is sent into a commercial optical parametric amplifier (TOPAS, Light Conversion) to obtain the pump beam at 480 nm (pulse duration of ~60 fs). 0.4-0.5 mW average power (80-100 J per pulse) is used to work in a regime of linear signal response. In order to generate the broadband white-light, a small portion of the initial 800 nm beam is focused onto a CaF 2 crystal, mounted on a loudspeaker to prevent thermal degradation. The TA data and a reference beam are acquired in the range 350-680 nm using a Peltier-cooled charge-coupled device (CCD), which is operated at 220 Hz spectral read-out rate, and the exit of a 25-cm focal length spectrometer (resolution 3 nm). The polarization of pump and probe beams is set at the magic angle (54.7°) and both are focalised into the sample with a diameter of 60 and 40 µm respectively. The complexes are dissolved in AcN in a 1mm quartz cell (O.D. ≈ 0.3). The transient absorption spectra of the complexes and the solvent are acquired. All of them are treated to correct for Group Velocity Dispersion (+/-50 fs). The solvent data are subtracted (corrected with an appropriate factor in order to remove the Raman signals of CH 3 CN and the crossphase modulation artefacts (see ref. 41 ). A detailed qualitative analysis of the spectra and the kinetics is performed. This is followed, when indicated, by Single Value Decomposition (SVD) and Global Analysis (using Glotaran, www.glotaran.org) for C4 and C2 while, for the others, single wavelength fits are made. In this case, a multi-exponential is convoluted by the temporal instrument response function (IRF), assumed to be a Gaussian, centred at t=+/-50 fs, with 60 fs FWHM (see SI).

Conflicts of interest

There are no conflicts to declare