Hans H Sathler

Bernardo Cougo
email: bernardo.cogo@irt-saintexupery.com

Gilles Segond

Tianyu Zhao
email: tianyuz@vt.edu

Rolando Burgos
email: rolando@vt.edu

François Costa
email: francois.costa@satie.ens-cachan.fr

Denis Labrousse
email: denis.labrousse@satie.ens-cachan.fr

Design of three-level flying-capacitor commutation cells with four paralleled 650V/60A GaN HEMTs

Keywords: Flying capacitor, GaN HEMT, low inductance loop design, PCB Layout

This paper presents two improved solutions (Vertical and Horizontal commutation cells) for three-level flying capacitor (FC) topologies with four 650 V/ 60 A (25 mΩ) GS66516T GaN (Gallium Nitride) HEMTs (High Electron Mobility Transistors) in parallel per switch intended to be used in a 540 V / 70 kVA power drive system. Both solutions were built and experimentally tested, as a buck converter, regarding switching speed (15.14 kV / µs max) and overvoltage (6 % max) for different current levels (7 A to 53 A).

I. INTRODUCTION

The concept of More Electrical Aircraft (MEA) was raised with the idea to replace conventional non-electrical systems such as pneumatic hydraulic and mechanical with electrical systems in the aircraft. These changes are intended to reduce CO2 emissions by reducing fuel consumption, which is achieved by increasing the power density and efficiency of these systems [START_REF]ACARE. Aeronautics and air transport: beyond vision 2020[END_REF]. Many of these systems are power drives which are usually composed of an inverter, cables, EMC filters and a motor. Multilevel inverters are one promising technology in this field since the high number of output voltage levels allows the reduction of switching frequency (consequently reducing switching losses), decreases input and output filter size due to lower dV/ dt. Authors in [START_REF] Sathler | Impact of multilevel converters on EMC filter weight of a 70 kVA power drive system for More Electrical Aircraft[END_REF] show a promising topology for a MEA 70 kVA power drive system achieving power density and efficiency superior to 10 kW / kg and 98.5 % respectively, in a global optimization comprising cooling system and EMI filters among others. The structure presented in the referred paper is composed of three paralleled legs connected through non-coupled inductors. Each leg is a three-level FC (flying capacitor [START_REF] Meynard | Multi-level conversion: high voltage choppers and voltage-source inverters[END_REF]) topology with four 650 V / 60 A GaN HEMT transistors (ref. GS66516T) in parallel per switch, as presented in Fig. 1, totalizing 16 semiconductors per leg.

The present paper proposes two solutions for three-level FC topology by optimizing power loops considering the complexity of paralleling four GS66516T devices per switch, as presented in [START_REF]ACARE. Aeronautics and air transport: beyond vision 2020[END_REF]. The state of art of paralleling GaN HEMTs and the use in multilevel topologies is presented in Section II. The design characteristics of two solutions, that comprises gate driver and power loop is presented in Section III. Experimental verification concerning loop inductance, proper functioning, switching speed and overvoltage, as well as the comparison between the two proposed solutions are presented in Section IV. Finally, the conclusions of the study are presented in Section V.

II. STATE OF ART OF PARALLELED GAN HEMTS

The state of the art of GaN HEMT applications presented in [START_REF] Ma | Review of GaN HEMT Applications in Power Converters over 500[END_REF] compile more than 162 technical reports and research papers and highlight the main advantages of GaN HEMT in relation to Silicon based power switches: excellent temperature capabilities allowing much higher power density (9.5 kW /in 3 ) and efficiency (up to 99.8 %) on power converters. One of the main characteristics of a Wide bandgap (WBG) semiconductors, as Silicon Carbide (SiC) MOSFETs and GaN HEMTs, is to present lower output capacitance. This implies a high switching speed as high as 100 V/ ns for GaN HEMTs [START_REF] Sun | Research of Low Inductance Loop Design in GaN HEMT Application[END_REF]. Due to high dV/ dt and dI/ dt, parasitic inductances of printed circuit board traces (PCB) are critical to safe operation of GaN HEMTs, by reducing voltage spikes on drain and source voltage, EMI noise and switching losses [START_REF] Lu | Design consideration of gate driver circuits and PCB parasitic parameters of paralleled E-mode GaN HEMTs in zero-voltageswitching applications[END_REF][7] [START_REF] Sun | Active dv/dt control of 600V GaN transistors[END_REF].

Reducing commutation loop inductance is probably the most affordable solution to reduce voltage across stray loop

Q1 | Q2 Q3 | Q4 Q9 | Q10 Q11 | Q12 Q13 | Q14 Q15 | Q16 Q5 | Q6 Q7 | Q8 CFC CDM Internal power loop External power loop Output V cc(+) V cc(-) V FC(-) V FC(+)
inductances due to high dI/ dt [START_REF] Lu | Paralleling GaN E-HEMTs in 10kW-100kW systems[END_REF], if packaged devices are used, since it only depends on PCB routing. The challenges of paralleling four GaN HEMTs 650 V/ 60 A (GS66516T) are presented in [START_REF] Lu | Paralleling GaN E-HEMTs in 10kW-100kW systems[END_REF] and [START_REF] Sun | Design, characteristics and application of pluggable low-inductance switching power cell of paralleled GaN HEMTs[END_REF] for a half-bridge topology. Beyond power loop reduction (0.7 nH [START_REF] Lu | Paralleling GaN E-HEMTs in 10kW-100kW systems[END_REF] and 0.81 nH [START_REF] Sun | Design, characteristics and application of pluggable low-inductance switching power cell of paralleled GaN HEMTs[END_REF]), gate driver loop must also be optimized (4.2 nH [9] and 2.5 nH / 3.6 nH [START_REF] Sun | Design, characteristics and application of pluggable low-inductance switching power cell of paralleled GaN HEMTs[END_REF]) to reduce VGS ringing and overshoot. An additional degree of complexity is inserted in power loop optimization for multilevel series topologies, since the power loop is defined by more than one commutation cell. In the three-level FC topology presented in Figure 1, the external power loop (in blue) is formed by all switches, CDM and CFC. While the internal power loop (in red) is formed by only inner switches and CFC. By cascading FC commutation cells, the number of power loops is defined by the number of output voltage levels -1, and the flying capacitor of the innermost commutation cell will be part of the preceding commutation cell power loop.

The use of 650 V/ 30 A GaN HEMTs in multilevel series topologies, with no paralleled devices, is presented in a 1 kW three-level Active Neutral Point Clamped (ANPC) [START_REF] Gurpinar | Ultra-low inductance design for a GaN HEMT based 3L-ANPC inverter[END_REF] achieving 3.51 nH and 5.37 nH for power loops. On the other hand, the use of FC topology is presented in [START_REF] Modeer | Design of a GaN-Based Interleaved Nine-Level Flying Capacitor Multilevel Inverter for Electric Aircraft Applications[END_REF] and [START_REF] Pallo | A 10-Level Flying Capacitor Multi-Level Dual-Interleaved Power Module for Scalable and Power-Dense Electric Drives[END_REF] for a nine-level and a ten-level topology, respectively, with no paralleled devices. To reduce power loop inductances (2.32 nH [START_REF] Modeer | Design of a GaN-Based Interleaved Nine-Level Flying Capacitor Multilevel Inverter for Electric Aircraft Applications[END_REF] and 2.7 nH [START_REF] Pallo | A 10-Level Flying Capacitor Multi-Level Dual-Interleaved Power Module for Scalable and Power-Dense Electric Drives[END_REF]), the complementary switches are oppositely placed on PCB top and bottom layers with ceramic flying capacitors placed in the input and output of each commutation cell.

Among the reviewed papers, despite the parallelism of GaN HEMTs and their use in multilevel series converters, the two problematics have not been addressed together: parallelism of GaN HEMTs in multilevel series converters. This is the main subject addressed in the present paper.

III. DESIGN OF A THREE-LEVEL FLYING CAPACITOR

As presented in the previous section, the key to parallel multiple devices is to guarantee a symmetric layout. In [START_REF] Lu | Paralleling GaN E-HEMTs in 10kW-100kW systems[END_REF] and [START_REF] Sun | Design, characteristics and application of pluggable low-inductance switching power cell of paralleled GaN HEMTs[END_REF], a similar schematic for a two level with two paralleled GaN is presented as shown in Fig. 2. Current sharing between the devices is strongly dependent of gate loop inductances, in green, LG1~LG4, power loop inductances LP1~LP5 and the mutual M1 and M2. In the case of four devices in parallel, the same reasoning can be done with the double of parasitic elements. To guarantee paralleled devices equal current sharing, the layout must consider, not only to minimize these inductances but LG1 = LG2 and LG3 = LG4 , as LP1 = LP3 , LP2 = LP4 and LP5 = LP6. Next, the details of the gate and loop layout developed in this work will be explored.

A. Gate Driver Design

Gate loop inductances (LG1~LG4) presented in Fig. 2 have a strong impact on VGS ringing and overshoot that can be increased if a strong unbalance between these loops is verified [START_REF] Lu | Paralleling GaN E-HEMTs in 10kW-100kW systems[END_REF]. To reduce VGS ringing, it is interesting to split RG1 [START_REF] Sun | Design, characteristics and application of pluggable low-inductance switching power cell of paralleled GaN HEMTs[END_REF] between Ciss (input capacitance) of paralleled devices, resulting in a RG-on (turn-on resistance) equal to RG1 + RG2 per device. On the other hand, RG-off (turn-off resistance) is defined only by RG1. The same idea of splitting resistors is employed with four paralleled transistors, so, one RG1 = 1 Ω is connected to each transistor in series with RG2 = 5 Ω for the present application.

In this project the isolate gate driver Si8271BBD-IS is used to control the four GaN HEMTs in parallel, totalizing four gate driver circuits for the three-level FC leg. The gate driver circuit layout is presented in Fig. 3 with four paralleled devices (two on the top layer and the other two on the bottom layer) based on the disposition proposed by [START_REF] Sun | Design, characteristics and application of pluggable low-inductance switching power cell of paralleled GaN HEMTs[END_REF]. Turn-on gate loops were estimated by using Ansys Q3D Extractor as 2.3 nH (green path in Fig. 3) and 2.5 nH for top and bottom devices respectively. Turn off gate loops, on the other hand, reached 3.6nH for both top and bottom devices (red path in Fig. 3). These gate loops' symmetry and values are achieved due to decoupling capacitors placed close to each device on both the bottom and top layers (680 nF p/ transistor), as shown in Fig. 3. This capacitor split also guarantees the balance between the quasicommon-source inductance (L G3 and LG4 showed in Fig. 2) discussed in [START_REF] Lu | Paralleling GaN E-HEMTs in 10kW-100kW systems[END_REF], since the same distance between source connection of each GaN to respective decoupling is reached. Another important part of the gate driver design is the isolated power supply that will provide the 6 V (turn-on voltage) and -4 V (turn-off voltage) for each switch (four GaN HEMTs). This power supply must have an isolation capacitance as small as possible in order to attenuate the common-mode current passing through gate driver to auxiliary power supply [14] [7]. This common-mode current can distort the output of the isolated power supply and gate

C DS C DG C GS L P1 L G1 R G1 C DS C DG C GS L P3 L G2 R G1 L P2 L P4 Q1 Q2 R G2 L G6 L G5 Gate Driver Gate Driver L P5 C DM Gate Driver Gate Driver L G3 L P6 R S1 R S2 L G4 M 1 M 2 GaN 1
GaN 3

Turn-off loop G1 Turn-on loop G3

driver, resulting in a risk to the integrity of the gate drive circuit as well as the converter itself [START_REF] Sun | Ultralow Input-Output Capacitance PCB-Embedded Dual-Output Gate-Drive Power Supply for 650 V GaN-Based Half-Bridges[END_REF]. The isolated power supply used is the 2 W/ 12 V NXE2S1212MC from Murata that presents only 3 pF. In other to regulate 12 V output voltage from the power supply, two zener diodes are connected in series, TDZ6V2J (6.2 V) and TDZ3V9J (-3.9 V), with a 400 Ω resistor.

B. Power Loops Design

Reducing power loop parasitic inductances can be done in different ways, as reducing the length of bus bars and supply terminals of the commutation cell, as well as choosing capacitors with low ESL (equivalent series inductance). However, the most effective way is to place ceramic capacitors as close as possible to the commutation cell [START_REF] Hoene | Packaging Very Fast Switching Semiconductors[END_REF]. These ceramic capacitors must have enough capacitance to provide the necessary energy lost and stored during a commutation instant, which is almost the same as the turn-on energy [START_REF] Cougo | Characterization of low inductance SiC module with integrated capacitors for aircraft applications requiring low losses and low EMI issues[END_REF]. In the case of the designed commutation cell, in the worst case for CDM, there would be eight GS66516T (Q1-Q4 and Q9-Q12 or Q5-Q8 and Q13-Q16) turning on at 270 V (half of bus voltage -540 V) and 15 A for each transistor, totalizing 60 A, at the same time. The total energy lost at these conditions, according to GS66516T datasheet, would be Et = 8* 81 µJ = 648 µJ. In the case of CFC, there would be only four devices in the worst case (Q9-Q12 or Q13-Q16), totalizing 324 µJ. The final voltage level Vf at CDM (or CFC) providing the energy Et at initial voltage VDC is presented by [START_REF] Cougo | Characterization of low inductance SiC module with integrated capacitors for aircraft applications requiring low losses and low EMI issues[END_REF], as:

𝑉𝑉 𝑓𝑓 =�𝑉𝑉 𝐷𝐷𝐷𝐷 2 - 2 • 𝐸𝐸 𝑡𝑡 𝐶𝐶 𝐷𝐷𝐷𝐷 𝑜𝑜𝑜𝑜 𝐹𝐹𝐷𝐷 (1)
By considering the maximum accepted voltage drop (VDC -Vf) of 1 % (5.4 V) for CDM and 2 % (5.4 V) for CFC, a minimum value of 225 nF is needed for each one. Since these capacitors will be part of the power loops, the large the contact surface of these devices with the PCB tracks, the lower the loop inductance. Then, 8*225nF (CGA8M1X7T2J224K) of 630 V were used for CDM and 6*225 nF (C3225X7T2W224K) of 450 V were used for CFC, being half on the bottom layer and the other half on top layer to ensure symmetry.

According to Fig. 1, CFC is part of external power loop with CDM, Therefore, its positioning has direct impact on current high frequency path during the commutation of Q1-Q4 and Q5-Q8. The challenge consists on placing CFC as close as possible to CDM in order to reduce external power loop. However, it must take into account a safe distance between them, since they will be in different electric potential and it should not compromise internal power loop inductance. The proposed solution on this paper is to split CFC in two parts. The first part is placed so the internal power loop is reduced, so, as close as possible between Q9-Q12 and Q13-Q16. The second part, named as CFC' and CFC'' (for the Horizontal version), is electrically connected in parallel with CFC but they are placed close to CDM, as presented in Fig. 4 , allowing the reduction of the external power loop. They will provide a preferable path during Q1-Q4 and Q5-Q8 commutation regarding CFC. On the other hand, since they are "electrically" far from Q9-Q12 and Q13-Q16, their contribution to the internal loop is way inferior compared to CFC Power loop reduction is directly related to devices choice and positioning. In this paper it was defined that film capacitors (DC Link and flying capacitor) and output connectors should be placed on an external PCB (named mother board), in the way that the three-level FC commutation cell work as pluggable PCB. The present application consists of an input voltage of 540 V and output current of 43 A, working as a buck converter. To prevent flying capacitor voltage oscillation, some film capacitors were placed on the mother board. Therefore, the interface between the two PCBs should be done by a power connector with, at least, five power connections: V CC(+), VCC(-),VFC(+),VFC(-) and output connection. The connector 46114-1032 from Molex have the necessary power connections to handle the desired current and also signal connections that were used to transfer PWM signals, 12 V for the isolated power supplies and 5 V for the gate driver not isolated side.

By positioning the different devices (driver circuits, power connectors, ceramic capacitors and the 16 GaN HEMTs) different solutions were evaluated through simulations with Ansys Q3D extractor. Two solutions were selected to be build and tested, based on external and internal power loop inductances as other characteristics that are presented in Table I: Vertical and Horizontal versions. V CC (-) Output differences of loop inductances, dimensions and the number of layers, three other characteristics must be pointed out: 1) Symmetry: This characteristic implies that high frequency current path for both top and bottom paralleled devices are identical. This characteristic may suggest that all paralleled devices will see the same impedance and consequently will switch with the same characteristics.

2) Increase of the number of output voltage levels: despite the objective of a three-level FC commutation cell, there is the interest of a PCB routing that can be generalized, in the way that flying capacitors cells can be easily cascaded in series to increase converter output voltage levels.

3) Signal and Power paths: Separate signal and power tracks is a good design rule for PCBs, it prevents noise propagation on signal tacks that can disrupt circuit proper functioning. Normally, the circuit is divided into regions with only signals and others for high current circulation.

The two different solutions shown in Table I are presented in detail in the following sections, highlighting the main differences about PCB layer stack, symmetry and power and signal separation.

C. Vertical commutation cell

The Vertical solution shown in Fig. 5 is formed by six layers (70µm copper thickness) and power flow between external and internal cell is in a vertical sense. Signals tracks (green arrows) are placed on the borders of the PCB, while power (yellow arrows) remains in the center. Half of all devices are placed on the bottom layer and measurement points were placed close to the power connector to evaluate commutation cell performance.

External power loop (represented in red) inductance was estimated by simulation as 6.4 nH with C FC' (0.9µF) and 10.8nH without it. This consists in 40.7 % reduction of loop inductance. On the other hand, the internal power loop achieved 5.7 nH with CFC' and 5.8 nH without it. These results confirm the idea of splitting ceramic flying capacitors can significantly reduce external power loop inductance and has minor impact on internal power loop. Vertical version PCB has 112 mm x 94 mm and the layers were selected to ensure symmetry. In Fig. 6, it is presented two transversal cuts that show the high-frequency current path during commutation for the external (in red) and internal (in blue) power loops. Due to the presence of ceramic capacitors on both sides of the PCB, current is equally divided between top and bottom paralleled devices. However, this disposition with vertical power flow complicates cascading other flying capacitors cells in series without increasing the number of PCB layers. This is mainly due to the power connector location down on the board, since all flying capacitor connections must be connected to external film capacitors on the mother board.

D. Horizontal commutation cell

The Horizontal solution shown in Fig. 7 is formed by eight layers (70 µm copper thickness) and power flow between external and internal cell is in a horizontal sense. Signals and power tracks are superposed for Q5-Q8 and Q13-Q16. In that case, the layer stack has a different disposition to distance power and signals close to the power connector. The three bottom layers were reserved to signal and to solder bottom GaN HEMTs. This allows a shield to be made between PWM signals and power supply passing through the seventh layer between the sixth and eighth (bottom), as shown in Fig. 8. Horizontal version PCB has 132 mm x 103 mm. Due to signal shielding and signal mix with power tracks, it is not possible to accomplish current symmetry between top and bottom devices as presented in Fig. 9, for both external and internal loops. Nevertheless, this version is easier to cascade flying capacitors cells in series with the same number of PCB layers as presented in Fig. 10. In that case, CFC'' can be used only for the first cell.

IV. EXPERIMENTAL RESULTS

This section is dedicated to the experimental evaluation of the two versions presented in the previous sections. Firstly, PCBs power loop inductances are verified before soldering all devices. Then, switching speed and overvoltage at output voltage were verified for different current levels for the converter working as a buck converter with Vbus = 540 V, Duty cycle = 0.3 and switching frequency of 20 kHz.

A. Power Loop Inductance

Power Loop inductance measurement was performed to verify the results from simulation with Ansys Q3D Extractor. For this test, an impedance analyzer was used with an impedance pin probe presented in Fig. 11. All drain to source connections were shorted with conductor copper tape soldered to the respective connections. In the case of internal power loop, the probe was placed between CFC connections as presented in Fig. 11. For external power loop measurement, CFC / CFC' / CFC'' were shorted and pin probe was placed between CDM connections. A good approximation is verified between the measurements and simulated values according to Table II

B. Switching speed and Overvoltage

The test setup to measure switching speed and overvoltage is shown in Fig. 11, where the Horizontal version is placed above a mother board where film capacitors are placed (CDMfilm = 60 µF and CFC-film = 100 µF) as well as input and output connectors. Voltage measurements were taken in the FC leg (after power connector) with the intent to verify waveforms as close as possible of the GaN HEMTs. Two heat sinks (3-100707U from CoolInnovations) were considered for the four paralleled GaNs, as presented in Fig. 13, totalizing eight (total of 50 g) per board. These two heat sinks were connected trhough four screws guaranteeing the same potential. A thin layer of thermal paste was considered between GaNs and heat sink. Since heat sinks are electrically connected to transistors source terminal, tests were carried out without connecting all eight heat sinks between them. Waveforms in Fig. 14 show the output voltage in green for 42 A (in yellow), and the flying capacitor voltage with an unbalance around of 5.5 V (2 % of 270 V) at 20 kHz. The output voltage has an equivalent switching frequency of 40 kHz, and each voltage step is related with an external loop device switching and the following step with a device of the the interenal loop. The difference between the commutation of one device from the internal or the external loop is also identified by the overvoltage during commutation. The larger the loop inductance, the larger the oscillations as the overvoltage observed. In Fig. 15, it is verified that external loop causes higher overshoot than internal power loop when a device is switched, reaching maximum 6 % of output steady-state voltage (around 270 V) at 43 A. Fig. [START_REF] Cougo | Characterization of low inductance SiC module with integrated capacitors for aircraft applications requiring low losses and low EMI issues[END_REF]. In this picture overvoltage is highlighted the according to oscillation frequency of 33MHz (HF) due to power loop inductance and 1.5MHz (LF) due to the resonance with the film capacitors placed on mother board Fig. 16 shows the overvoltage due to commutation, called HF (High frequency) and dependence on power loop inductance, but also (in red) the overvoltage due to resonance of film capacitors placed in the mother board, named LF (Low frequency). These two resonances are shown in Fig. 16, where 33 MHz was measured for HF and 1.5 MHz for LF. It is noticed that overvoltage caused by HF reduces with the current level, whereas the one caused by LF increases. But still, for the highest tested current level, it is under 6% (16.2 V).

Switching speed is presented in Fig. 17 for turn-on and turn-off, from 7 A to 53 A. Turn-off speed is dependent of output current and Coss as presented in [START_REF] Rossignol | Behavioral model of gallium nitride normally ON power HEMT dedicated to inverter simulation and test of driving strategies[END_REF] and [START_REF] Rossignol | Switching Optimization of WBG Power Devices on Inverter Leg[END_REF]. This behavior is valid if channel current (IOutput) does not reach the minimum value to guarantee VGS > VTH, then the channel is not formed characterizing the absence of miller effect. The high the current, the miller effect takes place and dV /dt is less dependend of IOutput. Similar results (turn off) are verified for both versions, except for low current (7 A). At this current, each GaN device is switching around 1.75 A, then the switch speed is basically due to the charge and discharge of Coss and parallel parasitic capacitances of PCB.. The difference between the two versions (3.58 kV/µs Horizontal version and 2.26 kV/µs Vertical version) suggests that the Vertical version may have higher parasitic PCB parasitic capacitances, if all GaN devices presents the same Coss. Turn-on switching speed, on the other hand mainly controlled by gate charge current, depending on VGS (VDR-VGSP), Rgon and CGD [START_REF] Bau | CMOS Active Gate Driver for Closed-Loop dv/dt Control of GaN Transistors[END_REF]. Gate drive voltage VDR is defined by the isolated power supply and VGSP is the Miller plateau voltage that increases with the current [START_REF] Rossignol | Switching Optimization of WBG Power Devices on Inverter Leg[END_REF]. Therefore, a small decay is verified in turn on switching speed with current increase. The fact of having lower loop indutances corroborates that Vertical version presents higher switching speed than Horizontal version, reaching 15.14 kV/µs at 22 A against 13.42 kV/µs, a difference of 12.5 %. In general, dV/dt is relatively low compared to other GaN applications, this is due to the limitation of the gate driver output current (2 Apk) to charge 4 x CGS and gate turn on resistance of 5.25 Ω to guarantee converter proper functioning.

V. CONCLUSIONS

This paper presents two design solutions for three level FC with four devices 650V/60A (25 mΩ) GS66516T in parallel per switch. The Horizontal version presents lower loop inductances (4.4 nH / 3.6 nH) but 8 layers and asymmetry between top and bottom paralleled devices. It can reach 15.14 kV/µs during turn-on and may present lower PCB stray capacitances due to higher turn-off (3.58 kV/µs) speed at low current. The main advantage of this solution is the possibility to easily cascade FC cells in series, while the proximity of the signal and power tracks goes against PCB good design practices. The Vertical version, on the other hand, is composed of 6 layers and a symmetrical arrangement of the components that compromises loop inductance (6.4 nH / 5.7 nH). This solution reaches 13.42 kV/µs during turn-on and 2.26 kV/µs during turn-off at a low current. This solution allows the separation between signal and power tracks, reaching a completely symmetrical circuit between the top and bottom devices as well as layers disposition.

It is worth to point out that power connectors have a big impact on routing and device disposition. So, the conclusions presented in this paper can differ if another power connector is used. Both solutions present the same gate circuit design and consequently the same gate loop (around 2.5 nH). Gate drivers have a strong influence on devices switching speed due to output current limitation that must charge all gate to source capacitances of paralleled devices.

The key to reduce power loop inductances in flying capacitor topologies with paralleled GaN devices is to split ceramic capacitors C FC and consider a portion close to precedent capacitors (close to CDM in the case of a three-level topology). This insight reduced power loop inductance by around 41 %.

Fig. 1 .

 1 Fig. 1. Three level FC commutation cell with four 650 V / 60 A GaN HEMTs (GS66516T) in parallel.

Fig. 2 .

 2 Fig. 2. Two paralleled GaN HEMTs in a two level commutation cell and their respective gate and power loops parasitic inductancs.

Fig. 3 .

 3 Fig. 3. Gate driver circuit responsible to drive four paralleled 650 V/ 60 A GaN HEMTs (GS66516T) in parallel with equivalent RG-on = 6 Ω (1 Ω + 5 Ω) and RG-off = 1 Ω per transistor. Turn-on gate loop path of GaN 3 is presented shown in green and turn-off loop of GaN 1 is presented in red.

Fig. 4 .

 4 Fig. 4. CFC' and CFC'' positioning three level FC commutation cell.

Fig. 5 .

 5 Fig.5. Vertical version commutation cell top layer view. External power loop is presented in red and the internal power loop in blue, the green and yellow arrows presents signal and power separated path on the PCB.

Fig. 6 .

 6 Fig. 6. PCB transversal view representing part of high frequencies current symmetric path for the Vertical version through PCB layers. The figure on the top represents the external power loop path (6.4 nH) and the figure on the bottom represents the internal power loop path (5.7 nH).

Fig. 7 .

 7 Fig.6 transversal view

Fig. 8 .

 8 Fig. 8. Signal and auxialy power supply shielding idea used both in Horizontal and Vertical versions to prevent coupling between power and signal tracks.

Fig. 9 .

 9 Fig. 9. PCB transversal view representing part of high frequencies current asymmetric path for the Vertical version through PCB layers.The figure on the top represents external power loop path (4.4 nH) and the figure on the bottom the internal power loop path (3.3 nH).

Fig. 10 .

 10 Fig. 10. Four FC cells connected (five level FC topology) in series based on the Horizontal version routing with 8 layers (5 layers used for power tracks).

Fig. 11 .

 11 Fig.8 transversal view Fig.8 transversal view

Fig. 12 .

 12 Fig. 12. Test setup with the Horizontal version connected in the mother board. Power supply (540 V) and an inductive load is also connected to the motherboard, as well as PWM signals generated by a DSP (TMDSDOCK28379D) with an optical interface.

Fig. 13 .

 13 Fig. 13. Cooling system. Two heat sinks were used per four devices, that are electrically connected trrough four screws. Thermal paste was considered between devices and heat sinks. Additionally, two fans were placed as close as possible of the PCB.

Fig. 14 .Fig. 15 .

 1415 Fig. 14. Measured waveforms of Horizontal version working as a buck converter with Vbus = 540 V, Duty cyle = 0.3, Output Current (IOutput) = 42.08 A at 20 kHz. The output voltage VOutput is shown in green.

Fig. 17 .

 17 Fig. 17. Measured dV/ dt for Horizontal and Vertical versions during the turn-on and turn-off for different current levels.

  at 43A Vertical Version (External loop commutation). Low Frequency (LF) OV. 1.5 MHz resonance due mother board film capacitors. High Frequency (HF) OV. 33 MHz due power loop inductance.

Gate 2 Gate 1 Decoupling Cap. Top Layer Gate 1 Gate 2 Driver GaN 2 GaN 4 Gate 1 Gate 2 Decoupling Cap. Decoupling Cap. Bottom Layer Gate 2 Gate 1 Decoupling Cap.

TABLE I .

 I THREE-LEVEL FLYING CAPACITOR COMMUTATION CELL

	V CC (+)
	External
	power loop
	CFC''	CDM	CFC'	CFC	Internal
					power loop
	VERSIONS CHARACTERISTICS
		Commutation cells characteristics
		Item		Vertical	Horizontal
	External Power Loop Inductance	6.4nH	4.4nH
	Internal Power Loop Inductance	5.7nH	3.6nH
	Dimensions			112mm x 94mm	132mm x 103mm
	# Layers			6		8
	Power Sense			Vertical	Horizontal
	Gate Signal			Shielded	Shielded
	Symmetry			Symmetric	Asymmetric
	Increase # of output levels		Difficult	Easy
	Signal and Power		Separated	Mixed
	Both solutions present positive and negative points that can
	be used to justify the use of one over the other. Beyond the

  at 100 MHz. A maximum error of 16 % was verified for the Internal loop of Horizontal Version.

TABLE II .

 II SIMULATED AND MEASURED POWER LOOP INDUCTANCES FOR VERTICAL AND HORIZONTAL VERSIONS.

	Parasitic Inductance
	Inductance	calculated (100 MHz)	Measured (100 MHz)
	External Loop Horizontal V.	4.4 nH	4.9 nH
	Internal Loop Horizontal V.	3.6 nH	4.2 nH
	External Loop Vertical V.	6.4nH	6.8 nH
	Internal Loop Vertical V.	5.7nH	6.4 nH

ACKNOWLEDGMENT

This paper is part of the EPOWERDRIVE Project, which is sponsored by Airbus, AKKA, Apsi3D, ELVIA, Liebherr, Meggit, Nidec, Safran and the French National Research Agency (ANR).


Cyprien Ruffino
email: cyprien.ruffino@insa-rouen.fr

Rachel Blin

Samia Ainouz

Gilles Gasso

Romain Hérault

Fabrice Meriaudeau

Stéphane Canu

Physically-admissible polarimetric data augmentation for road-scene analysis

Introduction

Road scene analysis is a vital task for driving assistance systems. Robust road users detection including vehicles, pedestrians, animals, is a precondition for a secure navigation. The complexity and variability of outdoor illumination conditions make RGB-based detection algorithms poor and limited, especially when they are faced to strong reflections, noises or bad weather conditions. In contrast, polarization encoded images are a non-conventional modality where each reflected wave light from a pixel is strongly linked to the physical properties of the surface. The significant interest resides in the fact that polarimetric imaging is a rich modality that enables to characterize an object by its reflective properties. In a polarimetric image, each pixel encodes information regarding the object's roughness, its orientation and its reflection [START_REF] Lawrence | Polarization camera sensors[END_REF]. Applications of polarimetric imaging range from indoor autonomous navigation [START_REF] Berger | Depth from stereo polarization in specular scenes for urban robotics[END_REF], depth map estimation [START_REF] Zhu | Depth From a Polarisation + RGB Stereo Pair[END_REF], 3D objects reconstruction [START_REF] Morel | Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging[END_REF] to differentiation of healthy and unhealthy cervical tissues in order to detect cancer at an early stage [START_REF] Rehbinder | Ex vivo Mueller polarimetric imaging of the uterine cervix: a first statistical evaluation[END_REF]. Also, recently, polarization imaging was exploited in autonomous driving applications either to enhance car detection [START_REF] Fan | Polarization-Based Car Detection[END_REF], road mapping and perception [START_REF] Todd M Aycock | Polarization-based mapping and perception method and system[END_REF] or to detect road objects in adverse weather conditions [START_REF] Blin | Road scenes analysis in adverse weather conditions by polarization-encoded images and adapted deep learning[END_REF].

The key element of this work comes from the combination of polarization richness and deep learning-1 arXiv:2206.07431v1 [cs.CV] 15 Jun 2022 based detection. However, these applications are still characterized by the reduced size of the available training databases which restrains them from using deep neural networks. To overcome this limitation, one possible solution is the generation of polarimetric encoded images.

Generative Adversarial Networks (GAN) [Goo+14; WSW19] are powerful deep generative models used to implicitly learn complex data distributions and to generate realistic samples from them. In its standard form, a GAN consists of two models: a generator which maps samples drawn from a latent lowdimensional distribution (usually uniform or Gaussian distributions) to high-dimensional points expected to follow the sought data distribution, and a discrimination model which discriminates the real samples from the generated ones [START_REF] Goodfellow | Generative Adversarial Nets[END_REF]. GAN have proven remarkable for various application domains including image generation [START_REF] Arjovsky | Wasserstein Generative Adversarial Networks[END_REF], image-toimage translation [Iso+17; Zhu+17; Hof+18] or image attribute manipulation [START_REF] Antipov | Face aging with conditional generative adversarial networks[END_REF] to name a few.

Arguably most of the impressive achievements of GAN were obtained for color images. A body of work attempted to extend GAN architectures to other uncommon imaging domains. For instance, some existing methods rely on CycleGAN [START_REF] Zhu | Unpaired image-toimage translation using cycle-consistent adversarial networks[END_REF], an image-to-image translation network, to generate infrared road scenes from RGB counterpart images [START_REF] Zhang | Synthetic data generation for end-to-end thermal infrared tracking[END_REF], to produce thermal images for person reidentification [START_REF] Kniaz | ThermalGAN: Multimodal Color-to-Thermal Image Translation for Person Re-Identification in Multispectral Dataset[END_REF] or for infrared image colorization [START_REF] Mehri | Colorizing Near Infrared Images through a Cyclic Adversarial Approach of Unpaired Samples[END_REF]. In the same vein, data augmentation in the field of medical imaging was achieved [START_REF] Nie | Medical image synthesis with context-aware generative adversarial networks[END_REF] by transforming MRI inputs into pseudo-CT images. Following the previous stream of work, this paper contributes to generative models for non-conventional imaging techniques. Specifically we propose a generative model framework to produce realistic polarimetric images.

Unlike RGB, LiDAR or infrared image generation which mostly responded to visual qualitative constraints, unless some learnable knowledge constraints are enforced (see [START_REF] Hu | Deep generative models with learnable knowledge constraints[END_REF] for pose conditional person image generation), sampling polarization images is more challenging. Indeed, this imaging technique comes with physical admissibility constraints on the pixels of an image [START_REF] Bass | Handbook of optics[END_REF]. To be physically fea-sible, each pixel entry of such an image should satisfy some physical constraints related to light polarization principle and to the calibration setup of the acquisition devices. Therefore, we propose a set of constraints that, when satisfied, ensure that the generated images are physically feasible. Based on these constraints, we formulate our problem of polarimetric image generation as a domain-transfer learning problem under physical constraints to ensure that the generated images are valid. We propose a learning framework based on CycleGAN [START_REF] Zhu | Unpaired image-toimage translation using cycle-consistent adversarial networks[END_REF], which enables unpaired image-to-image translation with relatively few images, to which we add constraints for handling the physical polarization constraints during training. This allows for circumventing the expensive labelling step by transferring a source labelled RGB dataset to the polarimetric domain while keeping the shapes and contents of the source image unchanged.

We demonstrate the effectiveness of our constrained-output CycleGAN on the KITTI dataset [START_REF] Geiger | Are we ready for autonomous driving? the kitti vision benchmark suite[END_REF] as well as on the Berkeley Deep Drive dataset (BDD100K) [START_REF] Xu | End-to-end learning of driving models from large-scale video datasets[END_REF], which are commonly used for object detection in road scenes. Using the generated polarization-encoded images to train a deep object detector, we witness an improvement of the detection performances of cars and pedestrians which are of great interest for autonomous driving applications.

To summarize, the contributions of this paper are:

• as far as our knowledge goes, we propose the first framework for generating physically admissible polarization-encoded images starting from RGB images,

• we propose an extension of CycleGAN which allows to generate polarimetric-encoded images while handling the physical constraints that should be satisfied by the pixels of the generated images,

• we show that, when plugged into the training procedure of an object detector for pre-training, the generated images help improving the detection performances.

The remainder of the paper is organized as follows: the polarization formalism and the physical constraints it involves are first presented. Then, the CycleGAN approach is described and a way to take into account these physical constraints during the training process of the CycleGAN for generating polarimetric images is investigated. Experimental evaluations are conducted; they aim to translate RGB images from KITTI and BDD100K datasets into polarimetric images. Finally, the generated images are exploited to boost the performances of an object detection network. The code for the experiments and the trained models are available at: https://anonymous.4open.science/ r/4a83820e-9c65-417c-af3a-ab2979d6e2e8/

Framework

This section introduces the polarization formalism, the GAN approach and the CycleGAN principles. It focuses on the formulation of the proposed modelling framework, namely the learning of a CycleGAN with output constraints. A solution approach and the related learning principle are presented.

Physical principles of polarimetry

Polarization is the direction of propagation of the electrical field of the light wave. The electrical field of the wave propagating in z direction at time t is characterized by the scalar components of the vector

E =   E x (z, t) E y (z, t) E z (t)   =   E 0x cos(ωt -kz) E 0y cos(ωt -kz + φ) 0   , (1)
with E 0x and E 0y the maximum amplitudes of each component. φ is the phase shift between the two components E x (z, t) and E y (z, t), ω is the angular frequency, k represents the wave number, directly related to the wavelength λ. Figure 1 depicts the different parameters.

The two components of the electrical field could be combined by eliminating the temporal term ωt in the following way :

E x (t) E 0x 2 + E y (t) E 0y 2 -2 E x (t)E y (t) E 0x E 0y cos(φ) = sin 2 (φ)
(2) If φ is constant over the time, Equation 2 defines an ellipse meaning that the electrical field of a propagating wave has an elliptical behaviour as illustrated in Figure 2.

According to sin(φ), three states of polarization could be observed. If sin(φ) > 0 the polarization is said to be right-handed, otherwise, if sin(φ) < 0, the polarization is left handed. If sin(φ) = 0, the trajectory of the wave is linear and the polarization is known as linear polarization. If sin(φ) = ±1, the polarization of the wave is respectively right and left circular. For these three cases the wave is totally polarized, otherwise, if φ is not constant over the time, the wave is unpolarized [START_REF] Bass | Handbook of optics[END_REF]. Usually, in an uncontrolled environment, a light wave is a combination of a totally polarized part and an unpolarized one, this is the reason why the natural light is known to be partially polarized.

Mathematical representation

There are different ways to describe the polarization of the light and its interaction with the media. The Stokes parameters are one of them to describe the polarization state of an electromagnetic wave, especially in the general case of a partially polarized wave form. The Stokes vector [S 0 , S 1 , S 2 , S 3 ] contains four polarization parameters where each component brings us information on the reflected light wave as follows:

• S 0 represents the total intensity of the light which is not a polarization property. S 0 > 0 is always positive, which means that every natural object, when illuminated, reflects a light,

• S 1 defines the amount of the polarized light in the wave horizontally or vertically polarized, meaning that the electrical field represented by figure 2 moves in a horizontal or a vertical way,

• S 2 is the amount of the polarized light oriented at 45 • and -45 • and,

• S 3 represents the amount of the circular polarization.

In the case of outdoor scenes, the sunlight is unpolarized and it is physically proved that when an unpolarized light wave is being reflected, it becomes partially linearly polarized. Its polarization depends on the normal surface and the refractive index of the material it impinges [START_REF] Lawrence | Polarization camera sensors[END_REF]. The circular component is thus neglected and set to zero in all applications concerning non controlled environments [START_REF] Morel | Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging[END_REF]. We focused in our work on the only linearly polarized light represented by the three first components of the Stokes vector, namely [S 0 , S 1 , S 2 ].

In order to access to observable optical field, the two components E x (t) and E y (t) are averaged over the time. The polarization ellipse of Equation 2 average in case of linear polarization (sin(φ) = 0) is given by the following equation:

E x (t) 2 E 2 0x + E y (t) 2 E 2 0y -2 E x (t)E y (t) E 2 0x E 2 0y = 0. (3)
The averaged values of the electrical components are:

E 2 x (t) = 1 2 E 2 0x (4) E 2 y (t) = 1 2 E 2 0y
(5)

E x (t)E y (t) = 1 2 E 0x E 0y . (6)
By replacing the time averaged terms in Equation 3, the elliptical equation can be rewritten in the following form

(E 2 0x + E 2 0y ) -(E 2 0x -E 2 0y ) -(2E 0x E 0y ) = 0. (7)
The linear Stokes vector is finally derived from Equation 7 as

S =   S 0 S 1 S 2   =   E 2 0x + E 2 0y E 2 0x -E 2 0y 2E 0x E 0y .   (8)
One salient physical property, obtained from the Stokes parameters, is the degree of linear polarization (DOP) [START_REF] Ainouz | Adaptive processing of catadioptric images using polarization imaging: towards a pola-catadioptric model[END_REF] defined by:

DOP = S 2 1 + S 2 2 S 0 . (9)
The DOP ∈ [0, 1] (or equivalently between 0 and 100% ). It refers to the rate of the polarized part in the light wave. It is equal to 1 for a totally polarized light, 0 for unpolarized light and between 0 and 1 for partially polarized light. Equation 7 describes the totally polarized wave. By construction of the Stokes vector, the total energy S 2 0 in this case is equal to the partial energy S 2 1 + S 2 2 meaning that the light is totally composed of polarized waves linearly oriented. In case of partially polarized wave, the wave contains an unpolarized part, the equality is transformed to an inequality meaning that S 2 0 S 2 1 + S 2 2 . From this property, the Stokes vector is said to be physically admissible if and only if the two following conditions are met:

S 0 > 0 and S 2 0 S 2 1 + S 2 2 . ( 10
)
This condition means also that the DOP never exceeds 1, otherwise the partial energy of the wave may exceed its total energy, which is, physically, an impossible phenomenon.

Stokes imaging

I αi = 1 2 1 cos(2α i ) sin(2α i )   S 0 S 1 S 2   , ∀i = 1, 4
(11) that is:

I = AS , (12)
where I = I 0 I 45 I 90 I 135 refers to the four intensities according to each angle of the polarizer (α i ) i=1:4 and A ∈ R 4×3 , to the calibration matrix of the polarization camera, defined as:

A = 1 2     1 cos(2α 1 ) sin(2α 1 ) 1 cos(2α 2 ) sin(2α 2 ) 1 cos(2α 3 ) sin(2α 3 ) 1 cos(2α 4 ) sin(2α 4 )     = 1 2     1 1 0 1 0 1 1 -1 0 1 0 -1     . (13)
An example of the different intensities for the same scene is shown1 in Figure 3. To get the unknown Stokes parameters from the measured intensities (see Equation ( 12)), we require à = (A A) -1 A ∈ R 3×4 the pseudoinverse of the matrix A. The relationship between S and I is then defined by:

S = ÃI =   1 0 1 0 1 0 -1 0 0 1 0 -1       I 0 I 45 I 90 I 135     =   I 0 + I 90 I 0 -I 90 I 45 -I 135   . (14)
Combining Equations ( 12) and ( 14), we have:

I = A ÃI , (15)
which is satisfied if and only if:

I 0 + I 90 = I 45 + I 135 . (16)
Stokes images should then satisfy two main conditions: the physical admissibility constraints in (10) and the calibration constraint given by (16). The generated polarimetric images have to comply with these essential constraints.

Unpaired image-to-image translation with CycleGAN

Given two domains X and Y , unpaired image-toimage translation is the task of learning the mapping functions M XY : X → Y and M Y X : Y → X using unpaired samples x i ∈ X with i ∈ [1.

.N ] and y j ∈ Y with j ∈ [1..M ]. An effective approach to achieve the task is CycleGAN [START_REF] Zhu | Unpaired image-toimage translation using cycle-consistent adversarial networks[END_REF]. It consists in learning the two mapping models M XY and M Y X by combining the objective function of the standard Generative Adversarial Network (GAN) [START_REF] Goodfellow | Generative Adversarial Nets[END_REF] with a Cycle-Consistency loss function. The adversarial cost related to the GAN serves for training the models to generate samples that will match the target domain distribution, while the Cycle-Consistency cost ensures that the learned models are able to correctly reconstruct an original image (of the source domain) from a generated one.

Formally a GAN is composed of a generative model G : Z → X which maps a known distribution p Z , usually normal or uniform, to the unknown distribution p X of the samples and a discrimination model D : X → [0, 1]. The generator G attempts to fool the discriminator D, which in turn tries to distinguish a real sample from a sample generated by the model G. Learning a GAN amounts to solve the following problem:

G * , D * = arg min G max D L GAN (D, G) , (17)
with:

L GAN (D, G) = E x∼p X log(D(x)) (18) + E z∼p Z log(1 -D(G(z))) , (19)
.

CycleGAN trains the two models M XY and M Y X by using solely unpaired real samples x ∈ X and y ∈ Y respectively drawn according to the (unknown) distributions p X and p Y as input. It also learns two discrimination networks D X : X → [0, 1] and D Y : Y → [0, 1] able to detect generated samples from real ones in the domains X and Y respectively. CycleGAN relies on the Least-Squares variant of GAN [START_REF] Mao | Least squares generative adversarial networks[END_REF] and considers the following adver-sarial costs:

L GAN (D Y ,M XY ) = E y∼p Y (D Y (y) -1) 2 + E x∼p X D Y (M XY (x)) 2 , (20) L GAN (D X ,M Y X ) = E x∼p X (D X (x) -1) 2 + E y∼p Y D X (M Y X (y)) 2 . ( 21
)
In order to ensure the cyclic consistency, i.e. both the compositions M XY • M Y X and M Y X • M XY are identity functions, a 1 reconstruction error term is devised for the mapping models:

L reco (M XY , M Y X ) = E y∼p Y ||y -M XY (M Y X (y))|| 1 + E x∼p X ||x -M Y X (M XY (x))|| 1 . (22)
Gathering all these elements leads to the objective function:

L CycleGAN (D X ,D Y , M XY , M Y X ) = L GAN (D Y , M XY )+ L GAN (D X , M Y X ) + λL reco (M XY , M Y X ) , (23)
where λ > 0 is an hyper-parameter that controls the reconstruction term. Training a CycleGAN consists in solving, via alternate gradient descent, the following minmax problem:

M * XY ,M * Y X , D * X , D * Y = arg min M XY M Y X max D X D Y L CycleGAN (D X , D Y , M XY , M Y X ) .
(24)

Proposed approach

As discussed above, our main goal is to learn a generative model able to produce realistic polarizationbased images starting from RGB images. As our proposed solution to this problem, we adopt the imageto-image translation framework and extend it to account for the constraints a polarimetric image must fulfill.

To generate a polarimetric image from an RGB one, we propose to use the CycleGAN approach to learn the translation models M XY and M Y X between X the domain of the polarimetric images and Y the RGB image domain. Let Î ∈ R 4 be the intensity vector associated to a pixel of a generated polarimetric image. To be physically admissible, each pixel has to satisfy the admissibility (10) and the calibration (16) constraints. We refer to these polarimetric constraints by C 1 , C 2 and C 3 as follows:

C 1 : I = AS , C 2 : S 2 0 S 2 1 + S 2 2 , (25)
C 3 : S 0 > 0 .
By design, S 0 is always positive as it represents the total intensity reflected from an object. As the last layer of the generation models customary uses the hyperbolic tangent as activation function, each output intensity Î is within the range ] -1, 1[ which we scale to ]0, 255[. Hence Ŝ0 = Î0 + Î90 (see ( 14)) is ensured to be strictly positive. Therefore, constraint C 3 can be deemed satisfied for the generated polarimetric images. To handle the remaining constraints C 1 and C 2 , one could resort to the Lagrangian dual of CycleGAN optimization problem (24) subject to these constraints. However, this may be computationally expensive, as it requires to optimize four neural networks (respectively the discrimination and the mapping network models) in an inner loop of a dual ascent algorithm. Moreover the overall optimization procedure may not be stable because of the minmax game involved in the CycleGAN learning.

In order to derive an efficient algorithm to learn CycleGAN under output constraints, we introduce a relaxation of the problem. Instead of strictly enforcing the constraints, we measure how far the generated image pixels are from the feasibility domain through additional cost functions we attempt to minimize. For the constraint C 1 , we propose to use a 2 distance between the generated image M Y X and A Ŝ as:

L C1 = E y∼p Y ||M Y X (y) -A Ŝ|| 2 , (26)
with Ŝ = Ŝ0 Ŝ1 Ŝ2 the Stokes vector computed from the generated image by M Y X (y) using equation ( 14). Similarly, to enforce the constraint C 2 , a rectified linear penalty L C2 is considered, defined as:

L C2 = E y∼p Y max Ŝ2 1 + Ŝ2 2 -Ŝ2 0 , 0 . (27)
The loss L C1 enforces the respect of the acquisition conditions according to the calibration matrix A while L C2 pushes the generated images towards respecting the physical admissibility constraint on the Stokes vectors obtained from the generated image. Gathering all these elements, we train our Cycle-GAN by optimizing the following objective function:

L f inal = L CycleGAN + µL C1 + νL C2 . ( 28
)
The full training algorithm is summed up in Algorithm 1

Algorithm 1 Constrained CycleGAN training algorithm Require: X and Y two unpaired datasets, M XY and M Y X the mapping networks, D X and D Y the discrimination models, m the mini-batch size repeat sample a mini-batch {x i } m i=1 from X sample a mini-batch

{y i } m i=1 from Y update D X by stochastic gradient descent of m i=1 (D X (x i ) -1) 2 + (D X (M Y X (y i ))) 2 update D Y by stochastic gradient descent of m i=1 (D Y (y i ) -1) 2 + (D Y (M XY (x i ))) 2 sample a mini-batch {x i } m i=1 from X sample a mini-batch {y i } m i=1 from Y update M XY by stochastic gradient descent of n i=1 (D Y (M XY (x i )) -1) 2 + λ(||x i - M Y X (M XY (x i ))|| 1 +||y i -M XY (M Y X (y i ))|| 1 ) update M Y X by stochastic gradient descent of n i=1 (D X (M Y X (y i )) -1) 2 + λ ||x i - M Y X (M XY (x i )|| 1 +||y i -M XY (M Y X (y i ))|| 1 +µ(||M Y X (y i ) -A Ŝ|| 2 ) + ν max( Ŝ2 1 + Ŝ2 2 -Ŝ2 0 , 0) until a stopping condition is met
The non-negative hyper-parameters µ and ν ∈ R + control respectively the balance of calibration and admissibility constraints according to the CycleGAN loss L CycleGAN (see ( 23)). As the values of L C1 and L C2 are computed pixel-wisely, we consider their averages over the whole image in the objective function. The training principle of the proposed generative model is illustrated in Figure 5.

Experimental evaluation

Hereafter, the experimental setup, including the image generation procedure and its evaluation, is presented.

Polarimetric images generation using CycleGAN

To conduct the experiments, we rely on a combination of two datasets presented in [Bli+20; Bli+19], composed of polarimetric and RGB images. From these datasets we select 2485 unpaired images from each domain (RGB and polarimetry). Example instances are shown in Figure 6. The polarimetric images are of dimension 500 × 500 × 4. The latter dimension is due to the four intensities acquired by the camera, namely I 0 , I 45 , I 90 and I 135 . The RGB images are of dimension 906 × 945 × 3. Our constrained CycleGAN is trained for 400 epochs on randomly cropped patches of size 200×200. As for the constraints, we found experimentally that setting the hyper-parameters µ = 1 and ν = 1 in (28) provides the best performances. The hyperparameter λ, controlling the reconstruction cost, is set to λ = 10. The learning rate is decreased linearly from 2 × 10 -4 to 2 × 10 -6 during the 400 training epochs.

To evaluate the effectiveness of our trained generative model, we consider KITTI and BDD100K (only using daytime images since polarimetry fails to characterize objects during nighttime) which often serve as testbed in applications related to road scene object detection. The constrained-output Cy-cleGANs we train, are used to transfer RGB images from KITTI and BDD100K to the polarimetric do-main. The resulting datasets are denoted respectively as Polar-KITTI and Polar-BDD100K. Since the Cy-cleGAN architecture is fully convolutional, it has no requirement on the input image's size. Therefore, even though the model was trained on 200 × 200 patches, it scales straightforwardly to the images of size 1250 × 375 from KITTI and of size 1280 × 720 from BDD100K datasets. We also randomly horizontally flip, as flipping the image does not alter the physical properties of a polarimetric image [START_REF] Blanchon | Polarimetric Image Augmentation[END_REF].

To assess whether or not fulfilling the physical constraints is paramount, we investigate a variant of Polar-KITTI and Polar-BDD100K: we learn a standard unconstrained CycleGAN based on the same unpaired RGB/polarimetric images.

Evaluation of the generated images

In order to assert the ability of the generated Polar-KITTI and Polar-BDD100K datasets to preserve the relevant features for road scene applications, we train a detection network following the setup in Figure 4. For this experiment, a RetinaNet-50 [START_REF] Lin | Focal loss for dense object detection[END_REF] pretrained on the MS COCO dataset [START_REF] Lin | Microsoft coco: Common objects in context[END_REF] is finetuned in three different settings. In the first setup, the detection model is fine-tuned on the original RGB KITTI (or BDD100K) while the second one considers fine-tuning on the generated polarimetric images from KITTI (Polar-KITTI) or BDD100K (Polar-BDD100K) datasets. The third one uses the unconstrained variant of the generated images from KITTI or BDD100K datasets. After the first fine-tuning, the three final detection models are fine-tuned one last time on the real polarimetric dataset (described in Table 1). Overall, the trained CycleGAN and detection networks under these settings are evaluated in qualitative and quantitative ways. The end goal is to check: (i) the ability of the generated images to help learning polarimetry-based features for object detection, and (ii) the influence of respecting the polarimetric feasibility constraints on detection performances.

We measure the visual quality of the generated images by computing the classical Fréchet Inception Distance [START_REF] Heusel | Gans trained by a two time-scale update rule converge to a local Nash equilibrium[END_REF]. Computing this distance requires Figure 4: Full setup of our approach. For each experiment, we 1): train the generator M Y X ; 2): pre-train the RetinaNet-50 detection network on the MS-COCO dataset; 3): generate a full polarimetric dataset using the trained generator M Y X ; 4): perform a first training on the generated polarimetric KITTI dataset; 5): fine-tune on the real polarimetric dataset; and 6): evaluate the detection model on the test set of the real polarimetric dataset. Note that while the procedure is illustrated with the KITTI dataset, it straightforwardly extends to the BDD100K dataset. to extract visual features from each set of images (real and generated) using a pre-trained deep neural network (usually an Inception v3 [START_REF] Szegedy | Rethinking the inception architecture for computer vision[END_REF] network pretrained on ImageNet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF]) and to evaluate the Fréchet (or Wasserstein) distance between the distributions of these features, which are assumed to be Gaussian distributions. We evaluate this distance using 509 images from each generated polarimetric dataset and from the test set as described in Table 1. As for the feature extractor, since the classical Inception v3 network is not adapted to polarimetric images, we use the convolutional part of a polarimetry-adapted RetinaNet detection network [START_REF] Blin | Road scenes analysis in adverse weather conditions by polarization-encoded images and adapted deep learning[END_REF], which has been trained on the MS-COCO dataset and fine-tuned on the same real polarimetric dataset that we used in our CycleGAN experiments. In order to evaluate the improvements in the detection, we compute the error rate evolution ER o . The improvement ER o on the detection of the object o is Table 1: Polarimetric dataset features. From the third row, the total number of instances within each class are indicated.

given by:

ER o = (1 -AP p o ) -(1 -AP RGB o ) 1 -AP RGB o , (29)

Results and discussion

First the qualitative coherence of the generated images is evaluated. The polarimetric images generated from their RGB equivalent are shown in Figure 7.

As for the constraints, Table 3 shows how including them to the CycleGAN's loss helps generating images which better fulfill the physical polarimetric properties at the pixel scale. The errors related to the constraints C 1 and C 2 on generated images are consistent with the observed measurement errors on the real images, whereas the unconstrained approach yields poor results. Constraint C 3 is met for all generated images thanks to the use of hyperbolic tangent as activation function for the last layer of the generative models. Additionally, the obtained Fréchet Inception Distances are of 6022.7 for the unconstrained CycleGAN and 4485.1 for our approach. Note that the scale of the FID scores computed with the pretrained RetinaNet is larger than when using a pretrained Inception v3 network. Thus, these scores are to be interpreted as relative metrics. This indicates that taking the constraints into account improves visual and physical quality of the generated samples. ). The mean and the median of the percentage of pixels in an image that do not fulfill the constraints C 2 is computed. Regarding the constraint C 1 , we normalize it by computing the mean and the median of ||I -AS||/(||I|| + ||AS||). Next, as an example, we show the benefits of the generated images through an object detection task. This enables to check if objects contained in the generated scene are physically coherent. A RetinaNetbased detection model is learned according to the setups described in Section 3.2 and the obtained detection performances in term of mean average precision (mAP ) are summarized in Table 2. We chose not to evaluate the bike and motorbike detection performances as the polarimetric dataset does not contain enough objects of those two classes.

As we can see in Table 2, using the generated polarimetric images improves the detection performance in real polarimetric images. The improvement is substantial for car and pedestrian detection. We achieve a 4% improvement of the error rate (see Equation 29) for car detection and of 12% for pedestrian detection which leads to a global improvement of 9% in the detection, using Polar-KITTI with constraints. Similarly, for the Polar-BDD100K dataset, we notice an improvement of 10% for pedestrian detection which leads to an increased mAP of 5% (pedestrians and cars). However, we notice that, for BDD100K, similar detection performances are obtained either for RGB or polarimetric images. This is due to the fact that generated images using CycleGAN tends to lack precision on small objects. In order to asses the impact of this effect, we compare the evolution of the detection scores when varying the minimal area of the bounding boxes from which objects are taken into account for the detection task. The obtained results are shown for the Polar-BDD100K and the RGB BDD100K datasets are shown in Figure 8 and illustrate that, when the minimal area of bounding boxes increases, the AP of car regarding the training including Polar-BDD100K, overcomes the one including RGB BDD100K. Even if the results are limited by the lower quality of the small objects in the generated images in this specific case, we can conclude that the generated polarimetric images help improving the overall detection results.

Conclusion and future work

In this work, we propose an efficient way to generate realistic polarimetric images subject to physical admissibility constraints. We design two loss terms that help enforcing these constraints and propose to adapt the CycleGAN algorithm to achieve the generation of physically admissible images. To train the proposed output-constrained CycleGAN, we combine the standard CycleGAN's objective function with two designed cost functions in order to handle the feasibility constraints related to each polarization-encoded pixel in the image. With the proposed generative model, we successfully translate RGB images from road scenes to polarimetric images showing an enhancement of the detection performances. Future work would consist in improving the quality of the small objects in generated images. Another extension could be the generation of polarimetric images to other domains such as medical and Synthetic-Aperture Radar (SAR) imaging. Additionally, to strictly ensure the physical feasibility constraints, solutions such as directly addressing the optimization task involved by the genuine constrained CycleGAN problem, instead of its proposed relaxation, using methods such as proximal gradient descent method could be explored. Finally, since the constraints we develop are not limited to the generative methods, they might be applicable to some other augmentation techniques, such as rotation or noise.

Acknowledgements

This work is supported by the ICUB project 2017 ANRprogram : ANR-17-CE22-0011. We also thank our colleagues at CRIANN, who provided us with the computation resources necessary for our experiments.

Figure 1 :

 1 Figure 1: Propagation principle of the electromagnetic wave, with E 0x and E 0y the amplitudes, λ the wavelength and φ the phase shift of the wave

  Polarimetric images are obtained by computing the Stokes vector related to each pixel. The acquisition principle is based on a device composed of a polarizer oriented at an angle α between the object and the sensor[START_REF] Wang | Polarimetric Camera Calibration Using an LCD Monitor[END_REF]. At least three acquisitions with three different angles are required to get the Stokes parameters. The reflected light from the object, represented by the unknown Stokes vector, passes through the polarizer before reaching the camera. In our work, we use a Polarcam 4D Technology polarimetric camera, providing simultaneously four images respectively obtained with four different linear polarizers oriented at (α i ) i=1:4 = (0°, 45°, 90°, 135°). The polarimetric camera measures an intensity I αi of the scene for each angle α i . The relationship between the Stokes vector S and the intensities I(α i ) i=1:4 reaching the camera is given by:

Figure 3 :

 3 Figure 3: Example of a polarimetric image. From left to right, the intensities corresponding to the polarizer rotation angles 0°, 45°, 90°and 135°.

Figure 5 :

 5 Figure 5: Training of the M Y X model using the Cy-cleGAN training process extended to our constraints. The full training objective consists of a GAN cost using the discriminator network D Y , a cycle-consistent reconstruction cost which ensures that M XY (X) remains consistent with X, and our custom objectives L C1 and L C2 (in red).

Figure 6 :

 6 Figure 6: Examples of images used to train the adapted CycleGAN. First row are the I 0 channels of some polarimetric images and second row are some RGB ones.

Figure 7 :

 7 Figure 7: Examples of generated polarimetric images. From left to right: I 0 , I 45 , I 90 , I 135 (ground truth), RGB image and I 0 , I 45 , I 90 , I 135 generated from RGB.

Figure 8 :

 8 Figure 8: Evolution of the average precision (AP) when setting a minimal area of the detected bounding boxes. Here green, blue and red lines respectively refer to the evolution of cars', mAP and person's detections. The dashed and the solid lines respectively refer to the use of BDD100K RGB and Polar-BDD100K during training.

Table 2 :

 2 Comparison of the detection performances for the person and car classes, as well as the mean average precision (mAP). RetinaNet-50 pre-trained on MS COCO is the baseline of all the experiments.

	Dataset	Class	Test	ER o	Dataset	Class	Test	ER o
	KITTI RGB	person 0.663 N/A BDD100K RGB person 0.736 N/A
		car	0.785 N/A		car	0.821 N/A
		mAP	0.724 N/A		mAP	0.778 N/A
	unconstrained person 0.673 -0.03	unconstrained	person 0.720	0.06
	Polar-KITTI	car	0.786 -0.01 Polar-BDD100K	car	0.816	0.03
		mAP	0.730 -0.02		mAP	0.768	0.05
	Polar-KITTI	person 0.704 -0.12 Polar-BDD100K person 0.762 -0.10
	with contraints	car	0.794 -0.04 with constraints	car	0.815	0.03
		mAP	0.749 -0.09		mAP	0.789 -0.05
	The first row is the fine-tunning on the original RGB KITTI/BDD100K. The second row is the fine-tuning
	on Polar-KITTI/Polar-BDD100K without constraints and the bottom row is the detection models fine-tuned
	on Polar-KITTI/Polar-BDD100K with the constraints. For a given dataset (KITTI or BDD100K), the best
	results overall are shown in boldface

Table 3 :

 3 Evaluation of the constraint fulfillment using the designed losses L C1 and L C2 at the pixel scale. Here, we compare the real polarimetric dataset to the generated polarimetric KITTI datasets, constrained and unconstrained. The column C indicates the evaluated constraint, with C 1 referring to the calibration constraint (I = AS) and C 2 to the optical constraint

	(S 2 0	S 2 1 + S 2 2

Since the polarimetric images can be very dark, we manually adjust their brightness in the Figures3, 4, 5, 6 and 7


Mohammad Esmail Alikhani
email: esmail.alikhani@sorbonne-universite.fr

Benjamin G Janesko
email: b.janesko@tcu.edu

Keywords: Solvated electron precursor, electron delocalization range, ELF topology, tetrahedral aluminum center, carbon dioxide reduction, Oxalate ion TOC

Two electron reducing reaction of CO 2 to oxalate anion: theoretical study of delocalized (presolvated) electrons in Al(CH 3 ) n (NH 3 ) m , n = 0 -2 and m = 1 -6, clusters

I. Introduction

It has been well known, since the early years of the nineteenth century, that the alkali metal solvation in ammonia significantly displaces the alkali valence electron from the metal center, which leads to confined electrons by the solvent shell. [START_REF] Thomas | Boundless Chemist, Physicist, Poet and Man of Action[END_REF][START_REF] Zurek | A Molecular Perspective on Lithium-Ammonia Solutions[END_REF][START_REF] Feng | Theoretical models for solvated electrons[END_REF] Over the last few decades, numerous experimental [START_REF] Catterall | Metal-ammonia solutions[END_REF][START_REF] Elg | Recent advances in understanding the role of solvated electrons at the plasma-liquid interface of solutionbased gas discharges[END_REF][START_REF] Seel | Low energy structural dynamics and constrained libration of Li(NH 3 ) 4 , the lowest melting point metal[END_REF][START_REF] Sacchetti | Complex dynamics in low electron density lithium ammonia solutions. Interacting modes revealed by comparing neutron and x-ray inelastic scattering[END_REF] and theoretical [START_REF] Glover | Simulating the Formation of Sodium:Electron Tight-Contact Pairs: Watching the Solvation of Atoms in Liquids One Molecule at a Time[END_REF][START_REF] Glover | Nature of Sodium Atoms/(Na + , e -) Contact Pairs in Liquid Tetrahydrofuran[END_REF][START_REF] Zurek | Barely) Solid Li(NH 3 ) 4 : The Electronics of an Expanded Metal[END_REF][START_REF] Chaban | Electron Solvation in Liquid Ammonia: Lithium, Sodium, Magnesium, and Calcium as Electron Sources[END_REF][START_REF] Baranyi | Excess electron solvation in ammonia clusters[END_REF][START_REF] Ariyarathna | Aufbau Principle for Diffuse Electrons of Double-Shell Metal Ammonia Complexes: The Case of M(NH 3 ) 4 @12NH 3 , M = Li, Be + , B 2+[END_REF][START_REF] Ariyarathna | Dative bonds versus electron solvation in tri-coordinated beryllium complexes: Be(CX) 3 [X = O, S, Se, Te, Po] and Be(PH 3 ) 3 versus Be(NH 3 ) 3[END_REF][START_REF] Ariyarathna | Aufbau Rules for Solvated Electron Precursors: Be(NH 3 ) 4 0,± Complexes and Beyond[END_REF][START_REF] Ariyarathna | Electronic and geometric structure analysis of neutral and anionic alkali metal complexes of the CX series (X = O, S, Se, Te, Po): The case of M(CX) n = 1-4 (M = Li, Na) and their dimers[END_REF] studies have addressed not only the solubility of different metals [START_REF] Catterall | Metal-ammonia solutions[END_REF][START_REF] Ariyarathna | Aufbau Rules for Solvated Electron Precursors: Be(NH 3 ) 4 0,± Complexes and Beyond[END_REF][START_REF] Ariyarathna | Stability and Electronic Features of Calcium Hexa-, Hepta-, and Octa-Coordinated Ammonia Complexes: A First-Principles Study[END_REF][START_REF] Ariyarathna | Superatomic nature of alkaline earth metal-water complexes: the cases of Be(H 2 O)0,+4 and Mg(H 2 O)0,+6[END_REF][START_REF] Jordan | Can boron form coordination complexes with diffuse electrons? Evidence for linked solvated electron precursors[END_REF][START_REF] Jackson | The nature of supermolecular bonds: Investigating hydrocarbon linked beryllium solvated electron precursors[END_REF][START_REF] Almeida | Transition-metal solvated-electron precursors: diffuse and 3d electrons in V(NH 3 )0,±6[END_REF][START_REF] Almeida | Electronic and structural features of octa-coordinated yttriumammonia complexes: the first neutral solvated electron precursor with eight ligands and three outer electrons[END_REF][START_REF] Khan | Scandium in Neutral and Positively Charged Ammonia Complexes: Balancing between Sc 2+ and Sc 3+[END_REF][START_REF] Jackson | Electronic and geometric structure of cationic and neutral chromium and molybdenum ammonia complexes[END_REF] in NH3 but also the role of different solvents in the formation of solvated electrons. [START_REF] Furukawa | Dissolution behavior of lithium compounds in ethanol[END_REF][START_REF] Mozhzhukhina | A conductivity study of preferential solvation of lithium ion in acetonitrile-dimethyl sulfoxide mixtures[END_REF][START_REF] Seng Tan | Synthesis and Characterization of Biphenyl-Based Lithium Solvated Electron Solutions[END_REF][START_REF] Ariyarathna | Geometric and electronic structure analysis of calcium water complexes with one and two solvation shells[END_REF] The study of solvated metals in NH3 has recently been applied to some transition metals. [START_REF] Almeida | Transition-metal solvated-electron precursors: diffuse and 3d electrons in V(NH 3 )0,±6[END_REF][START_REF] Almeida | Electronic and structural features of octa-coordinated yttriumammonia complexes: the first neutral solvated electron precursor with eight ligands and three outer electrons[END_REF][START_REF] Khan | Scandium in Neutral and Positively Charged Ammonia Complexes: Balancing between Sc 2+ and Sc 3+[END_REF] Experimental and theoretical works have evidenced that the solvation of metals in different solvents (ammonia, water, and tetrahydrofuran (THF), for example) give rise to the three categories of solvated species: tetracoordinated complexes such as Li(NH 3 ) 4 , [START_REF] Seel | Low energy structural dynamics and constrained libration of Li(NH 3 ) 4 , the lowest melting point metal[END_REF][START_REF] Ariyarathna | Superatomic nature of alkaline earth metal-water complexes: the cases of Be(H 2 O)0,+4 and Mg(H 2 O)0,+6[END_REF][START_REF] Seel | Electron Solvation and the Unique Liquid Structure of a Mixed-Amine Expanded Metal: The Saturated Li-NH 3 -MeNH 2 System[END_REF] hexacoordinated species such as Na(NH3)6, [START_REF] Ariyarathna | Stability and Electronic Features of Calcium Hexa-, Hepta-, and Octa-Coordinated Ammonia Complexes: A First-Principles Study[END_REF][START_REF] Dong | Electron Presolvation in Tetrahydrofuran-Incorporated Supramolecular Sodium Entities[END_REF] and octacoordinated compounds such as Y(NH3)8. [START_REF] Almeida | Electronic and structural features of octa-coordinated yttriumammonia complexes: the first neutral solvated electron precursor with eight ligands and three outer electrons[END_REF] Several works have been devoted to understanding and rationalizing the process of solvated electron formation during alkali metal solvation in ammonia. [START_REF] Glover | Simulating the Formation of Sodium:Electron Tight-Contact Pairs: Watching the Solvation of Atoms in Liquids One Molecule at a Time[END_REF][START_REF] Zurek | Barely) Solid Li(NH 3 ) 4 : The Electronics of an Expanded Metal[END_REF][START_REF] Jackson | The nature of supermolecular bonds: Investigating hydrocarbon linked beryllium solvated electron precursors[END_REF][START_REF] Dong | Electron Presolvation in Tetrahydrofuran-Incorporated Supramolecular Sodium Entities[END_REF][START_REF] Cavanagh | Watching Na Atoms Solvate into (Na + , e -) Contact Pairs: Untangling the Ultrafast Charge-Transfer-to-Solvent Dynamics of Nain Tetrahydrofuran (THF)[END_REF] In the case of Na(THF)6, [START_REF] Dong | Electron Presolvation in Tetrahydrofuran-Incorporated Supramolecular Sodium Entities[END_REF][START_REF] Cavanagh | Watching Na Atoms Solvate into (Na + , e -) Contact Pairs: Untangling the Ultrafast Charge-Transfer-to-Solvent Dynamics of Nain Tetrahydrofuran (THF)[END_REF] the valence electron 3s loses its spherical symmetry as the number of THFs encircling the neutral Na atom increases from 1 to 6. The main cause of this deformation is indeed the Pauli repulsion between the valence electron of Na and the O lone pair electrons.

Thanks to this repulsion, the valence electron of Na is pushed back in the opposite direction to the Na-O dative bond formation. The valence electron exhibits a three-dimensional fan shape (skullcap) located on the opposite side of four tetrahydrofuran molecules in Na(THF) 4 , which again gets its spherical shape in Na(THF)6. [START_REF] Cavanagh | Watching Na Atoms Solvate into (Na + , e -) Contact Pairs: Untangling the Ultrafast Charge-Transfer-to-Solvent Dynamics of Nain Tetrahydrofuran (THF)[END_REF] Miliordos and co-workers, in a systematic investigation on the nature of supermolecular bonds, showed that such a molecular complex mimics a super-atom whose valence electron is distracted in a super-orbital (outer orbital). [START_REF] Ariyarathna | Aufbau Principle for Diffuse Electrons of Double-Shell Metal Ammonia Complexes: The Case of M(NH 3 ) 4 @12NH 3 , M = Li, Be + , B 2+[END_REF][START_REF] Jackson | The nature of supermolecular bonds: Investigating hydrocarbon linked beryllium solvated electron precursors[END_REF][START_REF] Khan | Scandium in Neutral and Positively Charged Ammonia Complexes: Balancing between Sc 2+ and Sc 3+[END_REF][START_REF] Ariyarathna | Molecules mimicking atoms: monomers and dimers of alkali metal solvated electron precursors[END_REF] For instance, the neutral Mg(H2O)6 complex in its ground electronic state should be considered as a dication Mg(H2O)6 2+ core in tight contact with two electrons accommodated near the coordinated water molecules. [START_REF] Ariyarathna | Superatomic nature of alkaline earth metal-water complexes: the cases of Be(H 2 O)0,+4 and Mg(H 2 O)0,+6[END_REF] In this sense, a solvated electron distributed in the periphery of ligands in dative interaction with the cationic core differs from the proton encircled within a molecular building bloc. 33,[START_REF] Klein | Dynamic NMR Study of the Mechanisms of Double, Triple, and Quadruple Proton and Deuteron Transfer in Cyclic Hydrogen Bonded Solids of Pyrazole Derivatives[END_REF] Such molecular clusters were called "solvated electron precursors" (SEPs) for the first time by Miliordos and co-workers. [START_REF] Ariyarathna | Aufbau Rules for Solvated Electron Precursors: Be(NH 3 ) 4 0,± Complexes and Beyond[END_REF] To the best of our knowledge, dissolving aluminum in ammonia by electrolysis has only been mentioned once in a communication to the editor. [START_REF] Mcelroy | Metallic Aluminum in Solution in Liquid Ammonia[END_REF][START_REF] Dye | The alkali metals: 200 years of surprises[END_REF] We found no other experimental or theoretical work in the literature about the solvation of Al in NH3 forming solvated cations and solvated electrons.

The purpose of the present work is to extend the study of the solvated electrons to the molecular compounds containing Al, Al(CH3), and Al(CH3)2 in interaction with ammonia. Our study will focus on three questions: 1) for each of the three systems, what is the maximum number of NH3 in the first coordination shell? In other words, can we have hexacoordinated aluminum? 2) For the most stable structure of each system, do the valence electrons of Al remain localized on the metallic center, or delocalized to the periphery of NH3? 3) Can we use these delocalized electrons as reducing agent to capture and activate CO2 molecule?

II. Computational details

All first principles calculations were performed using the Gaussian 09 or Gaussian 16 quantum chemical packages. [START_REF] Frisch | Gaussian-09 Revision D.01[END_REF][START_REF] Frisch | Gaussian-16 Revision C.01[END_REF] Optimization of the stationary points on the potential energy surface was performed using density functional theory with Handy's long-range corrected version of B3LYP exchange-correlation functional [START_REF] Becke | Density-functional thermochemistry. III. The role of exact exchange[END_REF][START_REF] Lee | Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density[END_REF][START_REF] Vosko | Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis[END_REF][START_REF] Stephens | Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields[END_REF][START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulombattenuating method (CAM-B3LYP)[END_REF] including the D3 version of Grimme's dispersion with Becke-Johnson damping (GD3BJ) [START_REF] Grimme | Effect of the damping function in dispersion corrected density functional theory[END_REF] , noted as CAM-B3LYP-GD3BJ in this paper. The Pople triple-ζ quality basis set extended with polarization and diffuse functions, 6-311++G (2d, 2p) has been used for all atoms. [START_REF] Mclean | Contracted Gaussian basis sets for molecular calculations. I. Second row atoms, Z =11-18[END_REF][START_REF] Krishnan | Self-consistent molecular orbital methods. XX. A basis set for correlated wave functions[END_REF] Selected calculations use Hartree-Fock theory (HF), second-order many-body perturbation theory (MP2), or complete active space self-consistent field (CASSCF) methods at CAM-B3LYP geometries.

To analyze the structure of solvated electrons in real space, we used two topological approaches: the EDR (electron delocalization range) [START_REF] Janesko | How far do electrons delocalize?[END_REF][START_REF] Janesko | Quantifying solvated electrons' delocalization[END_REF][START_REF] Janesko | Topological analysis of the electron delocalization range[END_REF] and the ELF (electron localization function). [START_REF] Silvi | Classification of chemical bonds based on topological analysis of electron localization functions[END_REF] The EDR and ELF are two complementary techniques that allow us to quantify the diffuse character of solvated electron (EDR), and measure the degree of electronic localization (ELF) within a chemical compound. The ELF is large (near 1) in regions that have single-electron character, and small (near zero) in regions such as core-valence overlap regions.

The EDR(r;d) is the expectation value of a nonlocal one-electron operator, namely, the projection of the system's wavefunction onto a normalized one-electron Gaussian function centered at point r with Gaussian exponent d -2 . In regions where this overlap is near 1, the electronic structure about point r is dominated by orbitals delocalized over distance d. We quantify this using the orbital overlap distance D(r), defined as the distance d maximizing EDR(r;d). Plots of electron density isosurfaces, colored with the value of D(r), highlight regions of space containing delocalized electrons. For the ELF topological analysis, the partition of the molecular space in terms of nonoverlapping space-filling domains has been performed using the TopMod package. [START_REF] Noury | Computational tools for the electron localization function topological analysis[END_REF] The EDR and overlap distance are evaluated using the Multiwfn package [START_REF] Lu | Multiwfn: A multifunctional wavefunction analyzer[END_REF] or the Gaussian 16 package. 38

III. Results and discussion

Recently, by pure analogy with the results obtained for the Be(NH3)4 complex, Miliordos and coworkers suggested that Al(NH3)x should be hexacoordinated which could host three outer electrons. [START_REF] Ariyarathna | Aufbau Rules for Solvated Electron Precursors: Be(NH 3 ) 4 0,± Complexes and Beyond[END_REF] Therefore, we focus our investigation on three types of systems in which aluminum is surrounded by six molecules, but with different degrees of oxidation: Al(NH3)6, Al(CH3)(NH3)5, and Al(CH3)2(NH3)4. Before starting the discussion, let us note here the color code used in this paper to distinguish different atoms illustrated in different figures, as follows: the pink spheres represent an aluminum atom, blue spheres a nitrogen atom, white spheres a hydrogen atom, red spheres an oxygen atom, and gray spheres a carbon atom.

A. Al(0) in interaction with six NH3 molecules

All the optimized structures and corresponding relative energies of the neutral Al(NH3)6 are shown in Figure 1. with neat NH3 gave only an olive-green deposit assigned to the di-coordinated species. [START_REF] Howard | Electron paramagnetic resonance study of the reaction of ground-state aluminum atoms with ammonia in a rotating cryostat[END_REF] Our conclusion is also in agreement with the fact that only two adduct species Al(NH3) and Al(NH3)2 have been observed during the thermal reaction between Al and NH3 using matrix infrared spectroscopy. [START_REF] Gaertner | Structure and Bonding in the Aluminum Radical Species Al•NH 3 , HAlNH 2 , HAlNH 2 •NH 3 , and Al(NH 2 ) 2 Studied by Means of Matrix IR Spectroscopy and Quantum Chemical Calculations[END_REF] To analyze the fate of the three valence electrons of Al within the Al(0)-24 complex, the contours of the two highest occupied molecular orbitals (HOMO and SOMO), as well as the ELF topological domains, are depicted in Figure 2.

C. Al(II) in interaction with four NH3 molecules

In the previous two sections, we showed that the dative interactions Al-N are not strong enough to overcome the attachment energy of the non-bonding electrons of the metal center not to ionize the complex but to push them to the peripheral of the ligands so that the (Al + , e Is the single non-bonding electron always located on the metal center? As illustrated by the contour of the singly occupied molecular orbital (see Figure 7 left panel), it is no longer localized on Al but is distributed on peripheral NH3s. The ELF topology (see Figure 7 right panel) also found a peripheral basin placed near the hydrogen atoms of the two NH3s of the second solvation shell. However, it should be emphasized that because of the diffuse (delocalized) nature of this electron, the ELF topology is not able to well describe the topological domain attributed to this solvated electron. Indeed, we find by the ELF topology a basin dedicated to this delocalized electron, but the synaptic order as well as the population of this basin remains very poorly defined. In contrast, the EDR analysis is perfectly suited to quantify the diffuse character of such an electron.

D. Some ideas for extension and application

Inspired by a recent work on (CH3)B(NH3)3 and its extension following the aliphatic chain (NH3)3B(CH2)3-6B(NH3)3, [START_REF] Jordan | Can boron form coordination complexes with diffuse electrons? Evidence for linked solvated electron precursors[END_REF] we consider here Al2(CH3)2(CH2)2(NH3)4 resulting from the linkage of two Al(CH3)2(NH3)2 monomers via an aliphatic hydrocarbon bridge.

For this system containing two metal centers, the singlet state is calculated to be more stable than the triplet one by 8.1 kcal/mol (see SI). As illustrated in Figure 9, two valence electrons of aluminum coupled into a singlet state are delocalized and distributed on the outer parts of NH3s, forming a SEP compound. [START_REF] Jordan | Can boron form coordination complexes with diffuse electrons? Evidence for linked solvated electron precursors[END_REF] Figure 9. Contour of the highest occupied molecular orbital of the global minimum (singlet spin state) of Al2(CH3)2(CH2)2(NH3)4.

Figure 10 shows the orbital overlap distance computed for the singlet and triplet states of the two-metal-center Al2(CH3)2(CH2)2(NH3)4 system. The figure depicts the 0.001 electrons/bohr 3 density isosurface, with each point r colored by the value of the overlap distance D(r), ranging from 2.5 au (red) to 8 au (blue). Singlet and triplet states both show a delocalized electron density region on the amine groups' outer parts. However, the singlet-coupled species displays substantially larger delocalization lengths, corresponding to a deep blue color, and consistent with a delocalized bonding-type orbital. (Spin symmetry breaking stabilizes the HF wavefunction by 7 kcal/mol). Both DFT and Hartree-Fock theory predict that the solvated electron pair is relatively delocalized, visible as a dark blue color in the solvated electron region. MP2 and CASSCF calculations both predict more compact structures for the solvated electron pair. However, both CASSCF and MP2 predict relatively weak binding energies compared to the DFT calculations. This differs from the density-driven error visible in Figure 11, and motivates further studies using correlated multireference wavefunction approaches. atmosphere by CO2 capture and conversion into value-added products is currently a highly attractive hot topic for all scientists. A very high number of publications in 2023 testifies to the high level of activity of researchers in this field. [START_REF] Song | Recent progress in CO 2 conversion into organic chemicals by molecular catalysis[END_REF][START_REF] Detz | Electrochemical CO 2 conversion technologies: state-of-the-art and future perspectives[END_REF][START_REF] Jackson | Introducing Novel Materials with Diffuse Electrons for Applications in Redox Catalysis and Quantum Computing via Theoretical Calculations[END_REF][START_REF] Huang | Multifunctional Roles of Clathrate Hydrate Nanoreactors for CO 2 Reduction[END_REF][START_REF] Meng | Recent advances in the theoretical studies on the electrocatalytic CO2 reduction based on single and double atoms[END_REF][START_REF] Davison | A room-temperature-stable electride and its reactivity: Reductive benzene/pyridine couplings and solvent-free Birch reductions[END_REF] The approach of two CO2 molecules to the Al2(CH3)2(CH2)2(NH3)4 compound could lead to a twoelectron reduction reaction of CO 2 in the singlet state and thus the formation of the C-C bond.

The chemical compound containing two activated CO2 at the triplet state is calculated to be less stable than the product in the singlet state by 56.8 kcal/mol (see Figure 13, and SI for structural information).

Figure 13. Schematic one electron and two electrons reduction reaction of two CO2.

As shown in Figure 13, two CO2 molecules are activated in the triplet state without forming an oxalate ion, and system is actually composed of three subunits: two bent CO2 -ion-molecules and one [Al2(CH3)2(CH2)2(NH3)4] 2+ dication. Energetic stability between the three subunits is essentially ensured by hydrogen bonds formed between each CO2 -ion-molecule and the hydrogen atoms of amine groups of the dicationic core.

In the case of singlet state, the ELF topological analysis clearly shows that the population of the V(C, C) disynaptic basin rises to 2.3 e, characteristic of a single C-C bond (see Figure 14).

A

  two-electron reducing reaction of CO2 to an oxalate anion: a theoretical study of delocalized (presolvated) electrons in Al(CH3)n(NH3)m, n = 0-2 and m = 1-6, clusters esmail alikhani, Benjamin Janesko To cite this version: esmail alikhani, Benjamin Janesko. A two-electron reducing reaction of CO2 to an oxalate anion: a theoretical study of delocalized (presolvated) electrons in Al(CH3)n(NH3)m, n = 0-2 and m = 1-6, clusters. Physical Chemistry Chemical Physics, In press, 10.1039/D3CP06096A. hal-04461619

Figure 1 .Figure 1 .

 11 Figure 1. Optimized structures and relative energies (in kcal/mol) of the Al(NH3)6 compound.

Figure 2 .

 2 Figure 2. Contours of the two highest occupied molecular orbital (left panel) and the ELF domains (right panel) of the Al(0)-24 structure.

Figure 3

 3 Figure3shows the orbital overlap distance D(r) computed for Al(0)-24. The figure depicts the 0.001 electrons/bohr 3 density isosurface of the neutral Al(0)-24 complex, with each point r on the surface colored by the value of the overlap distance D(r), ranging from 2.5 au (red) to 5 au (blue). The overlap distance is around 2.5 bohr on the ammonia molecules, increasing to >5 bohr in the region corresponding to the weakly bound electrons. For comparison, the right panel of Figure3shows the overlap distance computed for isolated NH3 and an isolated neutral non-spherical doublet aluminum atom. From this perspective, the valence electrons of the Al(0)-24 complex are localized in diffuse orbitals on the aluminum. The electron density in the region of the frontier orbitals largely arises from diffuse and weakly bound electrons.

Figure 3 .

 3 Figure 3. Orbital overlap distance of the Al(0)-24 structure (left) compared to isolated neutral singlet nonspherical Al atom (right top) and isolated NH3 (right bottom). Blue regions denote delocalized electron density.

Figure 4 .

 4 Figure 4. Optimized structures and relative energies (in kcal/mol) of two relevant isomers of Al(CH3)(NH3)5.

Figure 5 .

 5 Figure 5. Contours of the highest occupied molecular orbital (left panel) and the ELF domains (right panel) of the Al(I)-23 structure.

  Figure 6.

Figure 6 .

 6 Figure 6. Optimized structures and relative energies (in kcal/mol) of the relevant isomers of Al(CH3)2(NH3)4.

Figure 7 .

 7 Figure 7. Contours of the singly occupied molecular orbital (left) and the ELF domains (right) with ELF=0.65 of the Al(II)-22 structure.

Figure 8

 8 Figure 8 shows the orbital overlap distance D(r) of the Al(II)-22 structure. The left panel depicts the 0.001 electrons/bohr 3 density isosurface, with each point r on the surface colored by the value of the overlap distance D(r), ranging from 2.5 au (red) to 7 au (blue). The right panel displays the isosurfaces EDR(r;d)=0.7, computed for d=2.5 au (blue surface) and d=6 au (red

Figure 8 .

 8 Figure 8. Orbital caption: Orbital overlap distance (left) and electron delocalization range at two representative distances d(right) of the Al(II)-22 structure.

Figure 10 .

 10 Figure 10. Orbital overlap distance computed for the singlet and triplet states of the two-metalcenter Al2(CH3)2(CH2)2(NH3)4 system.

Figure 11

 11 Figure11shows the orbital overlap distance computed for Al(0)-24 at three different levels of theory: CAM-B3LYP DFT (left), Hartree-Fock theory (center), and second-order many-body perturbation theory (MP2) (right). The figure depicts the 0.001 electrons/bohr 3 density isosurface of the neutral Al(0)-24 complex, with each point r on the surface colored by the value of the overlap distance D(r), ranging from 3 au (red) to 5 au (blue). Calculations use the DFT geometry from above. Each figure is labeled with the computed vertical electron binding energy. CAM-B3LYP DFT predicts the strongest electron binding, but also predicts that the solvated electron is relatively delocalized, visible as a darker blue color in the solvated electron region. Hartree-Fock and MP2 calculations predict more compact structures for the solvated electron. This is consistent with significant density-driven error in DFT calculations on solvated electrons.[START_REF] Johnson | Extreme density-driven delocalization error for a model solvated-electron system[END_REF]

Figure 11 .

 11 Figure 11. Orbital overlap distance computed for Al(0)-24 at three different levels of theory: CAM-B3LYP DFT (left), Hartree-Fock theory (center), and second-order many-body perturbation theory (MP2) (right).

Figure 12

 12 Figure 12 show the orbital overlap distance for singlet Al2(CH3)2(CH2)2(NH3)4 at four different levels of theory: CAM-B3LYP DFT, Hartree-Fock theory, second-order many-body perturbation theory (MP2), and multireference (two-electron two-orbital) CASSCF. The figure depicts the 0.001 electrons/bohr 3 density isosurface of the neutral complex, with each point r on the surface colored by the value of the overlap distance D(r), ranging from 3 au (red) to 8.5 au (blue). Each figure is labeled with the computed average vertical electron binding energy of the two solvated electrons. The bound electron pair is significantly more delocalized than the single solvated electron in Figure 11, with a large overlap distance around 8 bohr. In another contrast to Figure 11, the bound electron pair experiences a degree of "strong" electron correlation.

Figure 12 .

 12 Figure 12. Orbital overlap distance computed for singlet Al2(CH3)2(CH2)2(NH3)4 at four different levels of theory.

Figure 14 .

 14 Figure 14. The ELF topological domains (ELF=0.85) of the Al2(CH3)2(CH2)2(NH3)4(CO2)2 product at the singlet spin state. In addition, the net charge of the new species O2C-CO2 calculated within the ELF formalism being equal to -1.8 e, indicates the formation of an oxalate ion, C2O4 2-. Therefore, we should

Author Contributions

Mohammad Esmaïl Alikhani: Optimization calculations, ELF analysis, and writing the original draft Benjamin G. Janesko: Electronic and EDR analysis, and writing the original draft

Conflicts of interest

There are no conflicts to declare.

Electronic Supplementary Information (ESI)

An ESI file available