Artem Mikhailov
email: artem.mikhailov.a@gmail.com

Krzysztof A Konieczny

Maria Gladysheva

Pavel Plyusnin

Sébastien Pillet

Dominik Schaniel

Photogeneration of several linkage isomers and investigation of forward and backward nitro-nitrito isomerization processes in a palladium complex

Photoinduced linkage isomers (PLIs) of the nitro-ligand were generated and comprehensively characterized in a square planar unit [Pd(NH 3 ) 3 NO 2 ] + of the complex salts [Pd(NH 3 ) 4 ][Pd(NH 3 ) 3 NO 2 ][MOx 3 ]⋅yH 2 O (M = Cr (Cr), Rh (Rh), Co (Co), Ox = oxalate). The structural (XRD) and spectroscopic (IR, UV-vis) investigations at 10 and 150 K allowed to determine the structures of several photoinduced linkage isomers, endo-ONO (PLI1, 2) and exo-ONO (PLI3, 4) isomers generated by irradiation with 365 nm from the initial NO 2 (GS), along with the assignment of the IR-bands to each structural isomer. Based on a combination of these methods the photo-and thermally induced interplay of PLIs was investigated. Irradiation in the temperature range 10-80 K induces the formation of both endo-and exo-ONO isomers, while increasing the temperature up to 150 K results in the formation of only endo-ONO isomers. The structural arrangement of the endo-ONO and exo-ONO PLI is strongly influenced by intermolecular interactions due the partial occupation of a neighboring site by water molecules. The investigation of thermal dynamics of PLIs revealed that the thermal decay of the exo-ONO isomer occurs via two steps exo-ONO→endo-ONO→NO 2 . The kinetic parameters (E a , k 0 ) of both decay processes were determined together with the characteristic decay temperatures (T d ) by IR-spectroscopy. According to the photoinduced dynamics measured by IR-spectroscopy, the mechanism of PLIs formation in [Pd(NH 3 ) 3 NO 2 ] + could be described as NO 2 →endo-ONO→exo-ONO.

Introduction

The ability of a ligand to change its conformation under an external stimulus such as electric field, pH value or light irradiation has attracted a lot of attention in the last decades. For example, the works by J.P. Sauvage, J. F. Stoddart and B. L. Feringa on molecular machines were honored by the Nobel Prize in 2016. 1,2 There exist various families of switchable ligands in coordination compounds, amongst which the nitro-ligand NO 2 -that can undergo photoinduced isomerization M-NO 2 →M-ONO (M = metal). [START_REF] Hatcher | Photocrystallographic Studies on Transition Metal Nitrito Metastable Linkage Isomers: Manipulating the Metastable State[END_REF] Moreover, depending on the nature of the complex and the experimental conditions, the nitrito isomer ONO can be stabilized in several configurations, such as endo-ONO, exo-ONO or κ 2 -O,O-(ONO). [START_REF] Hatcher | Metastable Linkage Isomerism in [Ni(Et4dien)(NO 2)2]: A Combined Thermal and Photocrystallographic Structural Investigation of a Nitro/Nitrito Interconversion[END_REF][START_REF] Hatcher | Steady-State and Pseudo-Steady-State Photocrystallographic Studies on Linkage Isomers of [Ni(Et 4 Dien)(η 2 -O,ON)(η 1 -NO 2 )]: Identification of a New Linkage Isomer[END_REF][START_REF] Ahmed | Relating Excited States to the Dynamics of Macroscopic Strain in Photoresponsive Crystals[END_REF][START_REF] Schaniel | Wavelength-Selective Photoisomerisation of Nitric Oxide and Nitrite in a Rhodium Complex[END_REF][START_REF] Borowski | Exploring Photoswitchable Properties of Two Nitro Nickel(II) Complexes with ( N , N , O ) -Donor Ligands and Their Copper(II) Analogues[END_REF] Photoinduced linkage isomers (PLIs) of the nitrite ligand can be generated by light with different wavelengths in the range of 350-650 nm. [START_REF] Schaniel | Wavelength-Selective Photoisomerisation of Nitric Oxide and Nitrite in a Rhodium Complex[END_REF][START_REF] Warren | Photoactivated Linkage Isomerism in Single Crystals of Nickel, Palladium and Platinum Di-Nitro Complexes -a Photocrystallographic Investigation[END_REF][START_REF] Chizhik | Quantification of Photoinduced Bending of Dynamic Molecular Crystals: From Macroscopic Strain to Kinetic Constants and Activation Energies[END_REF] In general, in these systems a backward reaction M-ONO→M-NO 2 is accessible only by heating, i.e. nitrito PLIs are metastable states, however, there are two examples of partial light induced backward reaction in case of [Co(Me-dpt)(NO 2 ) 3 ] (Me-dpt = 3,3'-diamino-N-methylpropanediamine) and ruthenium complex trans-[Ru(py) 4 (NO 2 ) 2 ] (py = pyridine). [START_REF] Deresz | An Optically Reversible Room-Temperature Solid-State Cobalt( <scp>iii</Scp> ) Photoswitch Based on Nitro-to-Nitrito Linkage Isomerism[END_REF][START_REF] Schaniel | Reversible Photoswitching between Nitrito-N and Nitrito-O Isomers in Trans-[Ru(Py)4(NO2)2[END_REF] The degree of metastability is governed by the height of the activation barrier, [START_REF] Skelton | Thermal Isomerisation and Photochemistry of the Linkage-Isomer System [Ni(Et 4 Dien)(η 2 -O,ON)(η 1 -NO 2 )[END_REF][START_REF] Muya | Theoretical Investigation on the Ground State Properties of the Hexaamminecobalt( <scp>iii</Scp> ) and Nitro-Nitrito Linkage Isomerism in Pentaamminecobalt( <scp>iii</Scp> ) in Vacuo[END_REF][START_REF] Muya | A Theoretical Insight into the Role of Counter Anions and Their Interactions in Nitropentaamminecobalt(III) toward Linkage Isomerism-induced Photochemical Motion[END_REF] and in practice the majority of nitrito PLIs are stable at temperatures below room temperature, such as mentioned above cobalt complex or platinum compound trans-[Pt(PCy 3 ) 2 (NO 2 ) 2 ] (PCy 3 = tricyclohexylphosphine). [START_REF] Deresz | An Optically Reversible Room-Temperature Solid-State Cobalt( <scp>iii</Scp> ) Photoswitch Based on Nitro-to-Nitrito Linkage Isomerism[END_REF][START_REF] Warren | Solid-State Interconversions: Unique 100 % Reversible Transformations between the Ground and Metastable States in Single-Crystals of a Series of Nickel(II) Nitro Complexes[END_REF] The forward nitro-nitrito isomerization might be thermally induced at high temperature (≈400 K) as well, which was shown for nickel complexes. [START_REF] Sabbani | Reversible Nitro-Nitrito Inter-Conversion in a Simple Mono-Nuclear Nickel(II) Complex [NiII{C6H4(NH2)2}2(NO2)2] in the Solid State[END_REF][START_REF] Chao | Thermochromic Nitro-Nitrito Interconversion Mediated by Weak-Linked Amide in Nickel (II) Diaminodiamide Complexes in the Solid State[END_REF] The isomerization process is sensitive to the nearest surrounding of the NO 2 ligand. The reaction cavity shape and volume available for the nitro-nitrito ligand influence significantly the conversion of the M-NO 2 →M-ONO reaction. [START_REF] Nakamura | Nitro-Nitrito Photoisomerization of Platinum(II) Complexes with Pt(NO 2 ) 4 2-and (FSO 2 ) 2 N -Anions: Correlation between Isomerization Ratio and Reaction Cavity[END_REF][START_REF] Nakamura | Nitro-Nitrito Photoisomerization of Cationic Platinum(II) Complexes in the Solid State: Reactivity in Polymorphic Crystals and Glassy State[END_REF] For instance, in the [Pt(NO 2 ) 4 ] 2-complex the population of the endo-ONO isomers is varied for each ligand in the range of 0-65% depending on the size of the cavity around NO 2 (from 0.95 to 3.40 Å 3 ). [START_REF] Nakamura | Nitro-Nitrito Photoisomerization of Platinum(II) Complexes with Pt(NO 2 ) 4 2-and (FSO 2 ) 2 N -Anions: Correlation between Isomerization Ratio and Reaction Cavity[END_REF] The presence of a hydrogen bond between H-donor and NO 2 ligand also influences the population of PLIs. [START_REF] Hatcher | The Impact of Hydrogen Bonding on 100% Photo-Switching in Solid-State Nitro-Nitrito Linkage Isomers[END_REF] It was shown that the presence of hydrogen bonds can lead to a reduction of the population of PLI, and that the hydrogen bond lengths can be manipulated by temperature. As a result, higher population of PLIs are observed at higher temperatures (around 150 K) compared to the ones obtained at 100 K.

Quantum chemical calculations suggest that the mechanism of the nitro-nitrito isomerization in octahedral complexes consists of two stages NO 2 →endo-ONO→exo-ONO. [START_REF] Skelton | Thermal Isomerisation and Photochemistry of the Linkage-Isomer System [Ni(Et 4 Dien)(η 2 -O,ON)(η 1 -NO 2 )[END_REF][START_REF] Muya | Theoretical Investigation on the Ground State Properties of the Hexaamminecobalt( <scp>iii</Scp> ) and Nitro-Nitrito Linkage Isomerism in Pentaamminecobalt( <scp>iii</Scp> ) in Vacuo[END_REF][START_REF] Muya | A Theoretical Insight into the Role of Counter Anions and Their Interactions in Nitropentaamminecobalt(III) toward Linkage Isomerism-induced Photochemical Motion[END_REF] Although the structures of endo-and exoisomers were reported for different square planar and octahedral compounds, [START_REF] Hatcher | Steady-State and Pseudo-Steady-State Photocrystallographic Studies on Linkage Isomers of [Ni(Et 4 Dien)(η 2 -O,ON)(η 1 -NO 2 )]: Identification of a New Linkage Isomer[END_REF][START_REF] Warren | Solid-State Interconversions: Unique 100 % Reversible Transformations between the Ground and Metastable States in Single-Crystals of a Series of Nickel(II) Nitro Complexes[END_REF][START_REF] Kutniewska | Photocrystallographic and Spectroscopic Studies of a Model (N,N,O) -Donor Square-Planar Nickel(II) Nitro Complex: In Search of High-Conversion and Stable Photoswitchable Materials[END_REF] up to date there are no systematic experimental reports on the mechanism of the photoisomerization. This might be related to several restrictions: not every compound exhibits the formation of both endo-and exo-isomers due to the limitations induced by the reaction cavity around NO 2 , the absorption properties of the complex or the stability of the crystal against light irradiation [START_REF] Hatcher | Photocrystallographic Studies on Transition Metal Nitrito Metastable Linkage Isomers: Manipulating the Metastable State[END_REF] , and temperatures below 80 K may be required to stabilize PLIs.

The classic octahedral nitro complex [Co(NH 3 ) 5 NO 2 ] 2+ is the subject of investigation since more than 100 years. [START_REF] Jörgensen | Zur Konstitution Der Kobalt-, Chrom-Und Rhodiumbasen[END_REF][START_REF] Heyns | An Infrared Study of the Nitro-Nitrito Linkage Isomerization in Solid Nitro-and Nitritopentamminecobalt(III) Chloride[END_REF][START_REF] Boldyreva | Intramolecular Linkage Isomerization in the Crystals of Some Co(III) -Ammine Complexes -A Link Between Inorganic and Organic Solid State Chemistry[END_REF][START_REF] Boldyreva | Effect of High Pressure on the Infrared Spectra of Solid Nitro-and Nitrito-Cobalt(Iii) Ammine Complexes. Berichte der Bunsengesellschaft für[END_REF] The simple ligand composition of the complex allowed to investigate its nitro-nitrito isomerization experimentally and theoretically. [START_REF] Muya | A Theoretical Insight into the Role of Counter Anions and Their Interactions in Nitropentaamminecobalt(III) toward Linkage Isomerism-induced Photochemical Motion[END_REF][START_REF] Naumov | Dynamic Single Crystals: Kinematic Analysis of Photoinduced Crystal Jumping (The Photosalient Effect)[END_REF] However, according to the Cambridge Crystallographic Data Centre (CCDC) there are no known structures of square planar complexes with formula {M(NH 3 ) 3 NO 2 }. In the Inorganic Crystal Structure Database (ICSD) only one related structure [Pd(NH 3 ) 3 (NO 2 )][Rh(NH 3 ) 2 (NO 2 ) 4 ] was found. [START_REF] Khranenko | Crystal Structures of [Pd(NH3)3(NO2)][Rh(NH3)2(NO2)4] and[END_REF] Unfortunately, the pure bulk phase of the complex was not reported. Compounds with such formula could be convenient species for the theoretical modeling of nitro-nitrito isomerization in square planar compounds, as well as for the experimental investigation of possible PLIs.

Recently, a series of isostructural compounds with formula [Pd(NH 3 ) 4 ][Pd(NH 3 ) 3 NO 2 ][MOx 3 ]⋅yH 2 O (M = Cr (Cr), Rh (Rh), Co (Co), Ox = oxalate) was synthesized for the preparation of catalytic systems. 29,[START_REF] Gladysheva | COMPLEX SALT [Pd(NH3)4][Pd(NH3)3NO2][RhOx3]•H2O AS A PROSPECTIVE PRECURSOR OF Pd-Rh NANOALLOYS[END_REF] Interestingly, up to date the complex [Pd(NH 3 ) 3 NO 2 ] + could only be stabilized in such complex salt. Since, in the series of the compounds the anionic part [MOx 3 ] 3-differs, there is a possibility to investigate the influence of the anion on the properties of PLIs in [Pd(NH 3 ) 3 NO 2 ] + . Our current report is devoted to the experimental investigation of PLIs in [Pd(NH 3 ) 3 NO 2 ] + , especially to the study of forward -photoinduced and backward -thermally induced processes of nitro-nitrito isomerization.

Experimental

Synthesis

The compounds of general summary formula [Pd(NH 3 ) 4 ][Pd(NH 3 ) 3 NO 2 ][MOx 3 ]⋅yH 2 O (M = Cr (Cr), Rh (Rh), Co (Co), Ox = oxalate) were synthesized according to the previously published procedure. 29,30

SCXRD

The X-ray data were collected on Rigaku Supernova diffractometer with Excalibur CCD detector using microfocus MoKα X-ray source (λ = 0.71073 Å). CrysAlisPro (ver. 1.171.41.93a) software package was used both for data collection and reduction. [START_REF]CrysAlisPro Software System[END_REF] The low temperature during measurements was obtained using liquid nitrogen at 150 K and liquid helium at 10 K.

The crystal structures of ground states (GS) and photoinduced linkage isomers (PLIs) were solved using ShelXT. [START_REF] Sheldrick | SHELXT -Integrated Space-Group and Crystal-Structure Determination[END_REF] All structures were refined by means of ShelXL 33 implemented in Olex2. [START_REF] Dolomanov | OLEX2: A Complete Structure Solution, Refinement and Analysis Program[END_REF] The disorder in photoexcited crystals was refined based on Fourier difference map of given structure and photodifference map calculated between respective PLI and GS observed structure factors (F obsPLI -F obsGS ). [START_REF] Pillet | Time-Resolved Structural Analysis: Probing Condensed Matter in Motion[END_REF] The atoms were named according to the labels given in Fig. 1 and2. The structures of the disordered fragments in the determined structures are presented in Fig. 2 with corresponding photodifference maps in the plane of the nitro group in Fig. S1. The Pd-N/O distances for all observed NO 2 /ONO forms are gathered in Table 1. In all crystal structures all non-hydrogen atoms were refined anisotropically, except for the Cr structure measured at 10 K after light excitation where NO 2 /ONO disorder was refined over 5 occupation sites, in which minor components were refined isotropically. The hydrogen atoms positions in NH 3 groups were constrained and treated as riding rotating groups. Water molecules were treated as rigid groups, with positions of hydrogen atoms deduced from hydrogen bonds. In disordered crystal structures a set of restraints on bond lengths, valence angles and ADP's were applied, adjusted individually for each structure, but in an analogous manner within a measurement's series. In disordered structures the atoms N1, O1 and O2 forming ONO were additionally labeled with suffixes "A" (PLI1), "B" (PLI2), "C" (PLI3) and "D" (PLI4) depending on the conformation of PLI (see Fig. 2).

Selected crystallographic data are gathered in Table S1. CCDC numbers 2191057-2191064 for Co (150 K, GS), Co (150 K, GS+PLI1, 2), Cr (10 K, GS), Cr (150 K, GS), Cr (10 K, GS+PLI1, 2, 3, 4), Cr (150 K, GS+PLI1, 2), Rh (150 K, GS), Rh (150 K, GS+PLI1, 2), respectively, contain the supplementary crystallographic data for this paper. These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/structures. All single crystal irradiations were performed in-situ on the diffractometer by means of 365 nm Thorlabs LED (Light Emitting Diodes) with 150 mW optical power. All crystals studied at 150 K were irradiated for 60 min. Additionally, the crystal of Rh was irradiated for another 180 min (240 min in total). Cr crystal at 10 K was irradiated for 30 min. During irradiation the crystals were rotated constantly around the axis perpendicular to the light beam to ensure a homogenous distribution of the photoproduct in the whole crystal volume. After each irradiation time the XRD experiment was performed and the crystal structure was determined. Additionally, all structures were measured after thermal relaxation of PLIs to confirm the relaxation of PLIs back to GS.

IR-and UV-vis-spectroscopy

IR-spectroscopy measurements with irradiation were performed using a Nicolet 5700 FT-IR spectrometer with a resolution of 2 cm -1 in the range 400-4000 cm -1 . The sample (around 1-2 mg) was mixed with KBr (around 100 mg) and pressed into common pellets for IR measurements. The pellets were bonded onto the cold finger of a closed-cycle cryostat (Oxford Optistat) allowing to control the temperature in the range 10 -300 K. The irradiation procedures were performed by Thorlabs LEDs through KBr windows perpendicular to the samples with light of different wavelengths (365 -660 nm) with 50 -200 mW optical power. The highest population of PLIs was achieved by using 365 nm LED for all compounds. The assignment of IR bands of PLIs was made based on the populations of PLIs obtained from the XRD data measured at 150 and 10 K. The photogeneration of PLIs at 150 K results in the formation of endo-isomers PLI1, 2, while the photogeneration at 10 K additionally leads to the formation of exo-isomers PLI3, 4 (PLI4 was not directly observed in IR). Importantly, the populations of PLI1, 2 determined by XRD, and relative band intensities of PLI1, 2 measured by IR show the same trend for all three investigated compounds (higher population of PLI1 from XRD corresponds to higher band intensity of PLI1 from IR). The IR measurements performed at different temperatures confirm these assignments and allow following the evolution of the spectra: for example, thermal decay of PLI3 to PLI1 reveals the decrease of PLI3 bands, the increase of PLI1 bands and no intensity change in case of PLI2. To investigate the isothermal kinetics during the decay of PLIs the following protocol was applied: on the first stage the sample was irradiated at 150 or 10 K by 365 nm light in order to generate a sufficient amount of PLI; then the temperature was increased up to a chosen value (90-110 K for PLI3 and 230-250 K in case of PLI1, 2) and then repeated measurements (the delay between spectra was about 60 s) was performed in isothermal mode. The decay constants were calculated from the decrease of the intensity of the maximum of ν s (ONO) band of PLI1-3 in the range 1000-1100 cm -1 . The obtained experimental data were approximated using the equation of a first-order decay A=A 0 •exp(-k•t)+A ∞ . Further, using the decay constants (k) determined at each measured temperature (T), the activation energy (E a ) and frequency factor (k 0 ) were determined in Arrhenius coordinates (k=k 0 •exp(-E a /(R•T)), R -molar gas constant).

UV-vis spectra were recorded using transparent KBr pellets with the embedded complex on a Varian CARY 4000 spectrometer. The transparent pellets were prepared and irradiated as for the IR measurements. Lowtemperature measurements were performed using the same cryostat as that for the IR measurements, except that the KBr windows were exchanged for standard borosilicate glass windows. The baseline was measured using the same diaphragm with a KBr pellet, which can be mounted on the cryostat sample holder. The water content was approximately the same for all crystals, y ≈ 0.7, except for the crystal of Cr measured at 10 K for which y ≈ 0.5, since these measurements with the use of liquid helium were conducted a few weeks after data collection at 150 K that resulted in water loss in the crystal due to evaporation over time. In all investigated compounds the solvate water molecule is in the vicinity of the NO 2 ligand with the donor-acceptor distance O3(H 2 O)•••O2(NO 2 ) of about 2.7 Å (Fig. 2,a). Since the occupancy of the H 2 O site is about 0.7 (or 0.5 in case of Cr structure measured at 10 K), there are 70% (or 50%) of NO 2 ligands with the H 2 O molecule in the nearest environment, while the remaining 30% (or 50%) of NO 2 are sites without neighboring solvent H 2 O. This fact strongly influences the formation of linkage isomers in the investigated compounds, as discussed below.

Results and discussion

SCXRD

Irradiation at 150 and 10 K

The irradiation (light of wavelength 365 nm) of single-crystals at different temperatures leads to different observations. Irradiation at 150 K leads to the photoisomerization of GS nitro groups (NO 2 ) to endo-ONO nitrito groups having two possible conformations (photoproducts "A" (PLI1) and "B" (PLI2)) (see Figs. 2,S2). The structural difference between endo-ONO isomers PLI1 and PLI2 is shown in detail in Fig. 2, b-d. Due to the partial occupancy of solvate H 2 O molecules (about 0.7), two types of the endo-ONO isomers are formed after light irradiation of GS. For the [Pd(NH 3 ) 3 NO 2 ] + molecules where H 2 O is present (70%) the light excitation of GS leads to the formation of PLI1 (Fig. 2,c), whereas the sites without water (the remaining 30%) in the vicinity of [Pd(NH 3 ) 3 NO 2 ] + form PLI2, as shown in Fig. 2,d. Generally, the formation of PLI2 is sterically impossible if there is a water molecule nearby, since the intermolecular distance O3(H 2 O)•••O2B(ONO) in that case would be 1.4 Å (see Fig. 2,b), and would be even shorter in case of O3-H(H 2 O)•••O2B(ONO) contact (about 0.6 Å). Structurally, the isomers PLI1 and PLI2 in case of Cr, Rh and Co are almost identical (see Table 1), the corresponding representative structural models obtained for Cr are shown in Fig. 2, a-d.

In the crystal of Cr studied at 10 K two more exo-ONO nitrito groups of two conformations (photoproducts "C" (PLI3) and "D" (PLI4)) were detected (see Fig. 2, e, f and Fig. S2), which are stable below 80 K. Similar to PLI1 and PLI2, the exo-ONO isomer PLI3 is found in the case with a nearby water molecule, whereas PLI4 is related to the [Pd(NH 3 ) 3 NO 2 ] + without H 2 O in the closest environment. The Pd-N/O distances and populations of all isomers within the studied crystals, based on site occupation factors from determined X-ray structures, are gathered in Tables 1 and2. Comparison of crystals measured at 150 K shows that the highest population of endo-ONO occurs for Cr (82.6% of products in total) and the lowest for Rh (29.3% of products in total) even though the Rh single crystal was irradiated 4 times longer. This is most probably due to the limited light penetration depth, [START_REF] Hatcher | Monitoring Photo-Induced Population Dynamics in Metastable Linkage Isomer Crystals: A Crystallographic Kinetic Study of [Pd(Bu 4 Dien)NO 2 ]BPh 4[END_REF] since the crystal of Rh was significantly bigger (0.012 mm 3 for Cr vs 0.030 mm 3 for Rh). For Co 39.9% conversion was observed after 60 min of crystal irradiation, while the crystal size (0.014 mm 3 ) was comparable to the crystal of Cr. Significantly lower population of PLIs in this case could be possibly related to the sample degradation over irradiation time due to CO 2 release (see data quality monitoring and IR-spectroscopy parts). Another reason of the different population could be related to a difference in absorption properties of the compounds discussed below in the UV-vis-spectroscopy of PLIs part. The discrepancy in the population however, seems not to be related to the reaction cavity, which is identical for all studied crystals in terms of shape and volume, which is equal to 40(2), 39(2) and 41(2) Å 3 for Cr, Co and Rh, respectively (Fig. 3). The void calculations were performed for a spherical volume of 1.0 Å radius using 0.2 Å grid. 38 Fig. 3. The reaction cavity expressed as volume occupied by NO 2 group and the void surrounding it using ball of 1.0 Å radius as a probe with 0.1 Å grid. [START_REF] Macrae | Mercury 4.0: From Visualization to Analysis, Design and Prediction[END_REF] The investigated compounds offer the possibility to study the influence of hydrogen bonding on the efficiency of the PLI formation, since there is partial occupancy of water molecule. The water molecule plays a key role in the formation of the PLIs through the creation of a O3-H⋅⋅⋅O2 hydrogen bond with the nitro group of GS. The donor-acceptor distance in all GS structures is very similar and varies only slightly from 2.727(4) to 2.754(4) Å (see Table S2). Such a close contact affects photoisomerization in two ways:

1) The water molecule is a steric obstacle excluding the formation of endo-ONO "B" (PLI2) and exo-ONO "D" (PLI4) conformations. Hence, forms "B" (PLI2) and "D" (PLI4) are formed only if there is no water near the reaction center, and the total occupancy of both forms cannot exceed ca. 30% for crystals studied at 150 K and ca. 50% for crystal studied at 10 K, in agreement with data gathered in Table 2, assuming that the occupancy can be determined with an accuracy of 5-6 percentage points. The absence of water molecule results in a relatively large void volume equal to 46 (10), 47 (10) and 48(10) Å [START_REF] Hatcher | Photocrystallographic Studies on Transition Metal Nitrito Metastable Linkage Isomers: Manipulating the Metastable State[END_REF] for Cr, Co and Rh at 150 K, respectively, allowing the structural rearrangement of NO 2 during "B" and "D" PLI's formation (for probe/grid equal to 0.6/0.2Å).

2) Apparently, the conformations "A" (PLI1) and "C" (PLI3) are formed regardless of water presence. Based on the conducted experiments it is hard to determine if the PLI1,3-isomerization is hindered or favored by the hydrogen bond, however it can be assumed that this interaction stabilize s the position of the O2 atom in the nitro group. At first sight the presence of both "A" (PLI1) and "B" (PLI2) endo-conformations suggests that the formation of "B" (PLI2) is favorable, and "A" (PLI1) occurs only if there is a water molecule near the reaction center. Indeed, in case of Cr crystal studied at 150 K, the conformation "B" (PLI2) is formed whenever it is possible (for ca. 70% water occupancy ca. 30% of "B" (PLI2) was formed). However, at 10 K the total amount of forms "B" (PLI2) and "D" (PLI4) is equal to ca. 13%, even though the water occupies only half of the whole reaction centers, whilst forms "A" (PLI1) and "C" (PLI3) together occupy almost 50% of molecules (see Table 2). The directionality of the photoisomerization depends then, not only on water presence/absence but the temperature at which light excitation occurs setting up the photochemical equilibrium between all isomers and allowing the formation of the less stable exo-isomers PLI3, 4. The reaction route also might be tuned by the closest environment of the photoswitchable ligand, which is variable in function of temperature [START_REF] Hatcher | Photocrystallographic Studies on Transition Metal Nitrito Metastable Linkage Isomers: Manipulating the Metastable State[END_REF][START_REF] Borowski | Exploring Photoswitchable Properties of Two Nitro Nickel(II) Complexes with ( N , N , O ) -Donor Ligands and Their Copper(II) Analogues[END_REF][START_REF] Kutniewska | Photocrystallographic and Spectroscopic Studies of a Model (N,N,O) -Donor Square-Planar Nickel(II) Nitro Complex: In Search of High-Conversion and Stable Photoswitchable Materials[END_REF][START_REF] Kutniewska | Photo-A Nd Thermoswitchable Half-Sandwich Nickel(II) Complex: [Ni(Η5-C5H5)(IMes)(Η1-NO2)][END_REF] and product content, [START_REF] Zimmerman | Quantitative Cavities and Reactivity in Stages of Crystal Lattices: Mechanistic and Exploratory Organic Photochemistry[END_REF][START_REF] Zimmerman | Development of Experimental and Theoretical Crystal Lattice Organic Photochemistry: The Quantitative Cavity. Mechanistic and Exploratory Organic Photochemistry1[END_REF][START_REF] Konieczny | The Reasons for Different Kinetics of the Norrish-Yang Reaction in Crystals. Structural and Spectroscopic Studies[END_REF][START_REF] Konieczny | Structural Reasons for the Formation of Multicomponent Products and the Influence of High Pressure[END_REF] however, in current case no significant differences are observed between GS structures measured at 10 and 150 K (Table 1, S2).

Thus, the XRD measurements at 150 and 10 K allowed to determine the structures of four PLIs denoted as PLI1, 2 (endo-ONO) and PLI 3, 4 (exo-ONO). The information on the relative populations of PLIs and their thermal stability allowed to unambiguously assign the vibrational bands discussed below in the IR-spectroscopy section to the corresponding linkage isomer.

Data quality monitoring

The photoexcitation may have additional effects affecting crystal quality, such as crystal damage caused by internal stresses due to potentially significant displacements of atoms during photoisomerization. This can be prevented if the molecular movement during the reaction is relatively small and the atoms forming photoproduct occupy similar positions in a crystal, if there is a lot of free space around reaction center and/or if the crystal has a high elasticity. [START_REF] Hatcher | Photocrystallographic Studies on Transition Metal Nitrito Metastable Linkage Isomers: Manipulating the Metastable State[END_REF][START_REF] Li | Understanding Solid-State Photoswitching in [Re(OMe 2 -Bpy)(CO) 3 (η 1 -NO 2 )] Crystals via in Situ Photocrystallography[END_REF] In the studied cases the volume of the unit cell changed after irradiation by 1.3, 1.0, 1.1 and 0.5% for Cr at 10 K, Cr, Co and Rh at 150 K, respectively. This relatively small changes suggest rather little strain in the crystals when nitrito isomers are formed. [START_REF] Hatcher | Photocrystallographic Studies on Transition Metal Nitrito Metastable Linkage Isomers: Manipulating the Metastable State[END_REF][START_REF] Coulson | Exploring the Influence of Polymorphism and Chromophore Co-Ligands on Linkage Isomer Photoswitching in [PdIJbpy4dca)IJNO 2 ) 2 ] † ‡[END_REF] Crystal structures data gathered in Table S1 show that in case of Cr and Rh the quality of the crystal is maintained during irradiation, e.g. for crystal of Rh the R int value increased by 0.83 percentage point (from 5.34% to 6.17%) and the number of observed reflections decreased by about 0.65% (from 7502 to 7452 reflections) after irradiation for 240 min. In the case of Co however, strong decrease of the crystal quality was observed after crystal irradiation and relaxation back to the ground state, i.e. the R int value increased by 8.86 percentage point (from 5.42% through 7.23% after irradiation to 14.28% after relaxation) and the number of observed reflections decreased by 39% (from 7125 through 5526 to 4322 reflections). This effect is related to a CO 2 release from the [CoOx 3 ] 3-part, which is in accordance with IR-spectroscopy data of Co (see IR-spectroscopy part).

IR-spectroscopy

Since all the compounds with formula [Pd(NH 3 ) 4 ][Pd(NH 3 ) 3 NO 2 ][MOx 3 ] (M = Cr (Cr), Rh (Rh), Co (Co)) contain oxalate ligands (Ox), the comparison of IR-spectra of sodium or potassium salts [MOx 3 ] 3+ and [Pd(NH 3 ) 4 ][Pd(NH 3 ) 3 NO 2 ][MOx 3 ] has been done in order to reveal the position of the bands related to the NO 2 ligands. Clearly, the bands at around 1700, 1400, 1250, 900, 800, 600, 550, 480 and 420 cm -1 are characteristic for the vibrations of Ox, as illustrated on the example of Cr compound (see Fig. S3). Thus, the known bands of NO 2 ligand at around 1250 and 820 cm -1 are overlapped with those bands and can be observed as several spikes in the range 1250-1300 cm -1 shown in Fig. S3.

Irradiation at 150 K

The irradiation at 150 K with 365 nm light induces similar changes in the IR-spectra of all compounds Cr, Rh and Co . We illustrate the findings on the example of Cr, shown in Fig. 4, as it exhibits the highest population. The light excitation leads to a decrease of the intensity of the v(NO 2 ) bands at 1441 (v as ), 1424 sh (v as ), 1338 (v s ), 1309 (v s ) and 1255 (v s ) cm -1 and to the appearance of new bands at 1467 (v as ), 1097 (v s ), 1053 (v s ) cm -1 characteristic for the v(ONO) modes. [START_REF] Borowski | Exploring Photoswitchable Properties of Two Nitro Nickel(II) Complexes with ( N , N , O ) -Donor Ligands and Their Copper(II) Analogues[END_REF][START_REF] Schaniel | Reversible Photoswitching between Nitrito-N and Nitrito-O Isomers in Trans-[Ru(Py)4(NO2)2[END_REF][START_REF] Kutniewska | Photocrystallographic and Spectroscopic Studies of a Model (N,N,O) -Donor Square-Planar Nickel(II) Nitro Complex: In Search of High-Conversion and Stable Photoswitchable Materials[END_REF][START_REF] Boldyreva | Effect of High Pressure on the Infrared Spectra of Solid Nitro-and Nitrito-Cobalt(Iii) Ammine Complexes. Berichte der Bunsengesellschaft für[END_REF][START_REF] Brayshaw | Photocrystallographic Identification of Metastable Nitrito Linkage Isomers in a Series of Nickel( <scp>ii</Scp> ) Complexes[END_REF][START_REF] Warren | Reversible 100 % Linkage Isomerization in a Single-Crystal to Single-Crystal Transformation: Photocrystallographic Identification of the Metastable [Ni(Dppe)(η 1 -O NO)Cl] Isomer[END_REF] Importantly, the intensities of the bands at 1053 and 1097 cm - 1 are different and can be used for the estimation of the relative population of PLIs. Based on the populations of PLIs determined by XRD, the bands are assigned to the PLI1 and PLI2, respectively. Moreover, when keeping the KBr pellet with the compound overnight in the cryostat under vacuum, the relative intensities of the bands at 1053 and 1097 cm -1 after light irradiation change (see Fig. S4). This effect is due to the evaporation of the solvent water from the compounds in vacuum, leading to a redistribution of populations of PLI1 and PLI2 (more PLI2 at the cost of PLI1). This observation thus confirms the assignment of these bands and illustrates that the presence (or absence) of intermolecular contacts noticeably influence the population of PLIs in NO 2 -compounds. [START_REF] Hatcher | The Impact of Hydrogen Bonding on 100% Photo-Switching in Solid-State Nitro-Nitrito Linkage Isomers[END_REF] After nitro-nitrito photoisomerization other new bands appear at 3406 and 636 cm -1 at 150 K (Fig. 4, S5), which are discussed in detail below. Additionally, the difference spectra before and after generation of PLIs are shown in Fig. S6. The photogeneration of PLIs was probed at different temperatures in the range 10-200 K, the highest population of PLI1 and PLI2 was observed at 150 K, which is consistent with the results reported by Hatcher. [START_REF] Hatcher | The Impact of Hydrogen Bonding on 100% Photo-Switching in Solid-State Nitro-Nitrito Linkage Isomers[END_REF] It is important to mention that the light excitation at 340, 365, 385, 405, 455, 617 and 660 nm was tested, and the highest population of any PLI was achieved in case of 365 nm. The irradiation at 455, 617 and 660 nm does not induce the photoisomerization reaction.

Irradiation at 10 K

The irradiation of the complexes at 10 K by the same wavelength (365 nm) leads to the formation of additional new bands in the IR-spectra (see Fig. 5). Besides of PLI1 (1055 cm -1 ) and PLI2 (1097 cm -1 ), a band at 1012 (v s ) cm -1 has appeared and is assigned to the v(ONO) mode of the PLI3 isomer detected by XRD. Tentatively, a small shoulder at 1000 cm -1 could be assigned to PLI4, however due to its low intensity and overlap with other bands, further discussion is limited to PLI1-3. Importantly, the bands v s (NO 2 ) provide more complete information on the relative population and dynamics of PLIs compared to v as (NO 2 ), since they do not overlap with the bands of oxalate ligands. For example, the v as (NO 2 ) bands of PLI2 at 1440 cm -1 appear as low-intensity shoulders (see Table 3).

Based on the photogeneration measurements of PLIs at different temperatures, the main bands of all PLIs at 10 K were assigned and summarized in Table 3. At 10 K the v(NO 2 ) bands of GS exhibit several spikes, which are better resolved with respect to the spectra at 150 K. Tentatively, on the example of Cr compound, the bands at 1445 and 1428 cm -1 are assigned to the v as (NO 2 ) mode, the bands at 1343, 1336, 1311, 1285 and 1258 cm -1 reflect the v s (NO 2 ) bands. In case of the v(ONO) vibrations a larger split of (v as ) and (v s ) modes occur. Thus, for PLI1-3 the bands (v as ) and (v s ) are found at 1465 and 1055 cm -1 , 1442 sh and 1097 cm -1 , 1489 and 1012 cm -1 , respectively. The v(OH) bands of solvent water are also changed after photoisomerization. The bands of GS at 3591, 3515, 3449 cm -1 have shifted to 3516, 3406 and 3501, 3456 cm -1 for PLI1 and PLI3, respectively (see Fig. S7). The absence of these bands in case of PLI2 is expected and confirmed by IR-spectroscopy. Further, a change of the ρ(NO 2 ) mode of GS is observed at 606 cm -1 . The bands corresponding to the PLI1/2 and PLI3 are found at 646 and 624 cm -1 , respectively. Since the δ(NO 2 ) bands are strongly overlapped with the bands of oxalate ligands, the assignment of those bands is tentative (see Table 3). Likewise, the bands below 600 cm -1 cannot be explicitly assigned due to the overlapping with the bands of oxalate ligands. Similar changes are observed for Rh and Co compounds and summarized in Table 3.

Finally, the light-induced CO 2 release from the oxalate parts [MOx 3 ] 3-is observed for all investigated compounds indicated by the appearance of a band at 2340 cm -1 . In case of Cr and Rh the CO 2 release occurs after a long irradiation time (more than 200 min), whereas Co compound exhibits a high sensitivity towards CO 2 release -the band at 2340 cm -1 appears even after 2 min of light irradiation (see Fig. S8). Indeed, earlier investigation of CO 2 release in the oxalate compounds showed that [CoOx 3 ] 3+ is quite efficiently decomposed after irradiation compared to chromium oxalate.

Thermal stability of PLIs

The thermal stability of PLIs was investigated by IR-spectroscopy. The reaction constants (k) of the first order thermal decay of each PLI were measured in isothermal mode at several chosen temperatures . Further, activation energies (E a ) and frequency factors (lgk 0 ) were determined in Arrhenius coordinates (Fig. S9). Additionally, the decay temperatures (T d ) were calculated at k = 10 -3 s -1 . T d is a useful parameter to compare thermal stability of different linkage isomers. [START_REF] Morioka | Light-Induced Metastable States in Nitrosyl-Ruthenium Complexes Containing Ethylenediamine and Oxalate Ion Ligands[END_REF] The kinetic data of the thermal decay of PLI1-3 in all compounds are listed in Table 4. In order to investigate thermal decay of PLI1-3, on the first stage PLIs were photogenerated at 150 or 10 K, on the second stage the decay was measured at different temperatures in isothermal conditions. The decay of PLI3 in Cr at 100 K is shown in Fig. 6. From Fig. 6, one can observe that PLI3 thermally decays selectively to PLI1: the bands of PLI3 at 1489 and 1012 cm -1 decrease and the bands of PLI1 at 1467 and 1053 cm -1 grow. Since the bands of PLI2 and GS do not change their amplitudes this indicates that a single process PLI3→PLI1 occurs. The reason why the exo-isomer PLI3 decays only to the endo-isomer PLI1, but not to the endo-isomer PLI2, is that PLI3 and PLI1 are related to the same crystallographic site where the nearby solvate site is occupied by a water molecule (see SCXRD section). Similar decay of the exo-ONO isomer to the endo-ONO was reported recently for a nickel complex. [START_REF] Kutniewska | Photo-A Nd Thermoswitchable Half-Sandwich Nickel(II) Complex: [Ni(Η5-C5H5)(IMes)(Η1-NO2)][END_REF] As can be noted from Table 4, for different compounds the parameters E a and lgk 0 are different, however the parameter T d is the same within error (92-93 K). We suggest that T d is a more reliable parameter describing thermal decay by a single value, whereas E a and lgk 0 are different for each compound due to the mathematical correlation between E a and lgk 0 . It seems thus that for the present case small differences in the intermolecular interactions of exo-ONO ligand do not affect thermal stability of PLI3. Further heating leads to an efficient decay of PLI1 and PLI2 at temperatures around 240 K (see Fig. 7). Similar to PLI3, kinetic parameters of the decay were measured and shown in Table 4. The decay temperature of PLI1 is slightly lower compared to T d of PLI2. The difference is small but significant, indicating a higher stability for the PLI2 configuration of the endo-isomer, i.e. the one without neighboring water molecule. From the structural point of view, there is a difference in the intermolecular contacts of the endo-ONO in PLI1 and PLI2. Considering all the compounds, the average distance of the closest contact O1A-NH 3 in PLI1 is 2.88 Å, whereas the closest contact O2B-NH 3 in PLI2 is longer and reaches 3.44 Å. In addition, in the case of PLI1 the presence of the nearby water molecule results in a hydrogen bond of 2.8 Å (O2A-O3) between NO 2 and H 2 O, only slightly longer than the one for GS (2.7 Å) (Fig. 2). Thus, it seems that despite the stronger intermolecular interactions present in PLI1, its activation energy is slightly lower than one in PLI2. Overall, based on the thermal behavior of PLI1-3, the thermal decay process of the exo-ONO isomer in the investigated compounds can be schematically described as exo-ONO→endo-ONO→NO 2 .

According to the theoretical [START_REF] Skelton | Thermal Isomerisation and Photochemistry of the Linkage-Isomer System [Ni(Et 4 Dien)(η 2 -O,ON)(η 1 -NO 2 )[END_REF][START_REF] Muya | Theoretical Investigation on the Ground State Properties of the Hexaamminecobalt( <scp>iii</Scp> ) and Nitro-Nitrito Linkage Isomerism in Pentaamminecobalt( <scp>iii</Scp> ) in Vacuo[END_REF][START_REF] Muya | A Theoretical Insight into the Role of Counter Anions and Their Interactions in Nitropentaamminecobalt(III) toward Linkage Isomerism-induced Photochemical Motion[END_REF] and experimental [START_REF] Hatcher | The Impact of Hydrogen Bonding on 100% Photo-Switching in Solid-State Nitro-Nitrito Linkage Isomers[END_REF] investigations of related nitro-compounds, exo-isomers tend to exhibit a lower thermal stability compared to endo-ONO. For example, in [Pd(Et 4 dien)(NO 2 )]OTf complex (Et 4 dien = N,N,N′,N′-tetraethyldiethylene-triamine, OTf = trifluoromethanesulfonate) the endo-ONO isomer decays at a temperature of about 190 K, whereas the exo-ONO form relaxes at 120 K. [START_REF] Hatcher | The Impact of Hydrogen Bonding on 100% Photo-Switching in Solid-State Nitro-Nitrito Linkage Isomers[END_REF] Several theoretical works on octahedral nickel and cobalt compounds show that the nitro isomer (GS) is thermodynamically more stable and possesses lower electronic energy with respect to endo-and exo-nitrito isomers. [START_REF] Skelton | Thermal Isomerisation and Photochemistry of the Linkage-Isomer System [Ni(Et 4 Dien)(η 2 -O,ON)(η 1 -NO 2 )[END_REF][START_REF] Muya | Theoretical Investigation on the Ground State Properties of the Hexaamminecobalt( <scp>iii</Scp> ) and Nitro-Nitrito Linkage Isomerism in Pentaamminecobalt( <scp>iii</Scp> ) in Vacuo[END_REF][START_REF] Muya | A Theoretical Insight into the Role of Counter Anions and Their Interactions in Nitropentaamminecobalt(III) toward Linkage Isomerism-induced Photochemical Motion[END_REF] Regarding the activation energies of the decay processes of nitrito isomers, theoretical investigation s suggest that the activation barrier of endo-ONO→NO 2 reaction is 2-4 times higher than one for exo-ONO→endo-ONO reaction. The results obtained in the current investigation are in agreement with these theoretical conclusions made for octahedral compounds, thus, apparently, square planar nitro-nitrito compounds might exhibit similar patterns of PLI behavior.

Mechanistic aspects of PLIs photogeneration

A gradual irradiation of the compounds at 10 K induces a simultaneous formation of all isomers PLI1-3 (see Fig. 5), where the efficiency of the formation and population of PLI1, 3 is higher with respect to PLI2. This observation may give us the impression that each PLI could be generated directly from GS. However, the generation of PLI3 from the mixture of GS, PLI1 and PLI2 warns from drawing such a general conclusion. In the Fig. 8 the evolution of the spectra at 50 K (where PLI1 and PLI2 were previously photogenerated at 150 K) after 365 nm irradiation is shown. The irradiation of the mixture of GS, PLI1 and PLI2 at a temperature where PLI3 is stable (50 K) leads to a depletion of the intensities of PLI1, 2 and the appearance of the bands of PLI3, whereas the bands of GS only slightly increase the amplitudes. Thus, there is a route of the photoinduced formation of PLI3 (and apparently PLI4) from PLI1 and PLI2. It is important to mention, that PLI1 and PLI3 are related to the same site with H 2 O molecule. It is also possible that PLI4 has the same IR signature as PLI3, i.e. a band at 1012 cm -1 , which would explain naturally the transfer from PLI2 to this band. Otherwise, which is unlikely, the potential process PLI2→PLI3 is expected to include several stages through the formation of GS. The situation is simpler for the PLI1→PLI3 reaction, and we focus the further discussion on this case. Actually, the obtained results do not allow to exclude the GS→PLI3 process, however they suggest that the photoreaction PLI1→PLI3 is favorable.

Summarizing the obtained experimental results we can draw two possible mechanisms for the photoisomerization of the NO 2 ligand in the investigated system (see Fig. 9).

Scenario a:

This pathway shown in Fig. 9, panel a, implies photoinduced two-step process with endo-ONO isomer as an intermediate state towards exo-ONO (NO 2 →endo-ONO→exo-ONO). Thus, on the first step the light excitation of GS leads to the formation of endo-ONO, the following light absorption by endo-ONO leads the system to exo-ONO state. Knowing that the thermal stability of endo-PLI1 is much higher than that of exo-PLI3, the endo-PLI1 is stabilized at temperature below 200 K after being generated directly from GS. At these temperatures any potential population of exo-PLI3 immediately relaxes back to the endo-PLI1 due to the comparatively small activation barrier between PLI3 and PLI1. Below temperatures of 90 K the second step endo-PLI1→ exo-PLI3 can lead to a stable population of the exo-isomer PLI3, as now the activation barrier is high enough preventing thermal relaxation. In this low-temperature situation, the photostationnary equilibrium defining the relative populations of GS, PLI1 and PLI3 only depends on the forward and backward rates of the optical excitation . Note that this scheme does not consider or exclude the formation of other possible transient or metastable states. From the theoretical point of view, the photochemical route NO 2 →endo-ONO→exo-ONO is also favorable. [START_REF] Skelton | Thermal Isomerisation and Photochemistry of the Linkage-Isomer System [Ni(Et 4 Dien)(η 2 -O,ON)(η 1 -NO 2 )[END_REF][START_REF] Muya | Theoretical Investigation on the Ground State Properties of the Hexaamminecobalt( <scp>iii</Scp> ) and Nitro-Nitrito Linkage Isomerism in Pentaamminecobalt( <scp>iii</Scp> ) in Vacuo[END_REF][START_REF] Muya | A Theoretical Insight into the Role of Counter Anions and Their Interactions in Nitropentaamminecobalt(III) toward Linkage Isomerism-induced Photochemical Motion[END_REF] According to the calculations, the light irradiation of a NO 2 compound in GS excites the system to a transient singlet or triplet state followed by the relaxation to endo-ONO. The lowest LUMO orbitals of GS usually include a large contribution from antibonding orbitals of the M-NO 2 unit. As soon as the endo-ONO has appeared it can be excited to another transient state with further relaxation to exo-ONO. Importantly, the possibility of the formation of endo-or exo-isomer is also controlled by the free space required for the rotation of the NO 2 ligand, and by the activation energies of the exo-ONO→endo-ONO and endo-ONO→NO 2 reactions, which set the kinetic control of the thermal decay of PLIs at certain temperatures. However, one must be aware that above mentioned calculations have been done for octahedral systems, and for square planar species the mechanism might be different. It is important to mention that an evaluation of the mechanism of NO 2 ligand isomerization has been made for a model square planar complex [Ni(dmpe)Cl(NO 2 )] (dmpe = (CH 3 ) 2 PCH 2 CH 2 P(CH 3 ) 2 ). [START_REF] Warren | Solid-State Interconversions: Unique 100 % Reversible Transformations between the Ground and Metastable States in Single-Crystals of a Series of Nickel(II) Nitro Complexes[END_REF] According to DFT calculations, the excited transient state resembles exo-ONO configuration having tetrahedral arrangement, however, no similar square planar exo-ONO linkage isomer has been found on the ground state potential energy surface.

Scenario b:

Based on the obtained experimental results another mechanism can be proposed, as schematically shown in Fig. 9, panel b. In this case the exo-ONO is considered as an intermediate state towards the formation of endo-ONO. Thus, the light excitation of GS drives the system to the formation of exo-PLI3. Due to the low activation barrier between exo-PLI3 and endo-PLI1 (E a ≈ 0.2 eV), at higher temperature (150 K) exo-PLI3 decays directly to endo-PLI1, while at low temperature (10 K) exo-PLI3 can be thermally stabilized. At this low temperature the further transformation to endo-PLI1 occurs thus through light excitation. As in the previous scenario, the relative populations of the three states GS, endo-PLI1, and exo-PLI3 then depends on the optical forward and backward rates between the three states.

Finally, based on our reported experimental data and theoretical investigations performed for related octahedral systems the scenario a looks more favorable. However, the two scenarios presented based on our experimental results take into account only the metastable configurations on the ground state potential energy surface. In order to unravel the complicated mechanism of isomerization in this system either time resolved experiments that allow capturing a transient species or extensive DFT/TDDFT calculations are necessary. The solid state UV-vis-spectrum of Cr in KBr matrix at 150 K is shown in Fig. 10,a. The spectrum shows bands similar to those in the spectrum of water solution. Photogeneration of PLI1, 2 leads to an increase of absorption in the UV-range (280-450 nm, Fig. 9, a, insert), which supports proposed mechanisms of photoinduced reaction endo-ONO→exo-ONO, since the bands potentially allowing the reaction exist at the excitation wavelength. The difference spectrum of PLI3 measured at 10 K exhibit similar features and are shown in Fig. 10,b, permitting photoinduced transfer exo-ONO→ endo-ONO.

Conclusions

The properties of photoinduced linkage isomers (PLIs) in a square planar complex [Pd(NH 3 ) 3 NO 2 ] + being a part of the complex salts [Pd(NH 3 ) 4 ][Pd(NH 3 ) 3 NO 2 ][MOx 3 ]⋅yH 2 O (M = Cr (Cr), Rh (Rh), Co (Co), Ox = oxalate) were investigated. It was shown by XRD, that the irradiation at 365 nm at 150 K of the single crystals of Cr, Rh and Co leads to the formation of endo-ONO isomers PLI1 and PLI2. PLI1 and PLI2 differ in their intermolecular surrounding, where the partial occupation of a neighboring site by a solvate water molecule prevents the photoinduced formation of PLI2. Considering the fact that the water content could be adjusted, the structural tuning of the linkage isomers is possible in these systems. The total population of endo-ONO isomers in Cr, Co and Rh is 83, 40 and 29%, respectively. Further, on the example of Cr compound, PLIs were photogenerated at 10 K. Photocrystallographic analysis reveals the formation of exo-ONO isomers PLI3 and PLI4, which are not sufficiently stable at 80 K. Thus, four different PLIs can be photogenerated in [Pd(NH 3 ) 3 NO 2 ] + , moreover, the type and population of PLIs could be influenced by the amount of solvent H 2 O molecules, by the type of [MOx 3 ] 3- and by the temperature of the photogeneration.

Given the knowledge on the structures of PLI1-4 revealed by photocrystallography, the population/depopulation dynamics of PLIs was investigated in more detail using IR-spectroscopy in the temperature range of 10-300 K. As soon as a PLI appears its stability is governed by the kinetic parameters E a and lgk 0 . A gradual heating of the PLIs showed that endo-ONO and exo-ONO isomers in the complexes have a different thermal stability and different decay paths. Namely, the exo-ONO isomer decays to the endo-ONO, further heating of the endo-ONO leads to its decay to the GS nitro isomer NO 2 . Thus, the thermal decay of exo-ONO and endo-ONO isomers occurs via the scheme exo-ONO→endo-ONO→NO 2 (see Scheme 1). On the example of endo-ONO isomers PLI1 and PLI2, a small difference in the thermal stability of endo-ONO could be related to a different strength of the intermolecular interactions of endo-ONO or NO 2 isomers, however, no specific parameters affecting the stability have yet been found. The investigation of the forward photoinduced reaction revealed presumable routes of PLIs generation. Accordingly, the mechanism of the photogeneration of exo-ONO rather can be described as NO 2 →endo-ONO→exo-ONO process, thus likely two photons are needed to generate exo-ONO from GS. However, to prove the mechanism and to rule out the NO 2 →exo-ONO→endo-ONO scenario further investigations are needed. It is important to emphasize, that both forward and backward processes are strongly affected by the nearest surrounding of the photoswitchable ligand, since the parameters such as reaction cavity shape and volume can influence the thermal stability and a possibility of the generation of a certain PLI. The presented results show that the complex [Pd(NH 3 ) 3 NO 2 ] + can be used as a convenient and simple model compound for the theoretical investigation of the nitro-nitrito isomerization in square planar species.

  All determined crystal structures are isostructural, regardless of the central atom (M) in the oxalate part [MOx 3 ] 3-. The asymmetric unit in GS at 10 K is shown on the example of Cr in Fig. 1. The composition of the crystals is as follows: [Pd(NH 3 ) 4 ][Pd(NH 3 ) 3 NO 2 ][MOx 3 ]⋅yH 2 O, where M = Cr (Cr), Rh (Rh) or Co (Co).

Fig. 1 .

 1 Fig. 1. Asymmetric unit of Cr at 10 K. Atomic displacement parameters are drawn at 50% probability level.[START_REF] Macrae | Mercury 4.0: From Visualization to Analysis, Design and Prediction[END_REF]

Fig. 2 .

 2 Fig. 2. The photoswitchable fragment [Pd(NH 3 ) 3 NO 2 ] + / [Pd(NH 3 ) 3 ONO] + in the structure of Cr. Panel a: the fragment [Pd(NH 3 ) 3 NO 2 ] + (GS) with depicted intermolecular contact with H 2 O, the structure is measured at 150 K; panel b: the structures of endo-ONO isomers PLI1 and PLI2 in [Pd(NH 3 ) 3 ONO] + measured at 150 K after 365 nm irradiation, GS part (NO 2 ) is omitted for clarity; panel c: the endo-ONO PLI1 isomer having H 2 O molecule in the nearest surrounding, GS and PLI2 parts are omitted for clarity; panel d: the endo-ONO PLI2 isomer, which does not have H 2 O molecule in the vicinity, GS and PLI1 parts are omitted for clarity; panel e: the structure of exo-ONO isomer PLI3 in [Pd(NH 3 ) 3 ONO] + measured from another crystal at 10 K after 365 nm irradiation, GS,

Fig. 4 .

 4 Fig. 4. IR-spectra of Cr at 150 K before (GS) and after (PLI1 + PLI2) light irradiation at 365 nm.

48

 48

Fig. 5 .

 5 Fig. 5. Panel a: IR-spectra of Cr at 10 K before (GS) and after (PLI1 + PLI2 + PLI3) light irradiation at 365 nm; panel b: the evolution of intensities of the v s (NO 2 ) bands of GS and PLI1-3 during irradiation of Cr at 10 K with 365 nm.

Fig. 6 .

 6 Fig. 6. The thermal evolution of IR-spectra of Cr at 100 K after photogeneration of PLIs at 10 K.

Fig. 7 .

 7 Fig. 7. The thermal evolution of IR-spectra of Cr at 250 K after photogeneration of PLIs at 150 K.

Fig. 8 .

 8 Fig. 8. Panel a: IR-spectra of Cr at 50 K: on the first stage PLI1 and PLI2 were generated at 150 K until the saturation (PLI1 + PLI2), on the second stage the sample was cooled down to 50 K and further irradiated at 365

Fig. 9 .

 9 Fig. 9. Possible energetic diagrams of NO 2 ligand isomerization: panel a represents scenario with endo-ONO isomer as an intermediate state, panel b -the route with intermediate exo-ONO.

Fig. 10 .

 10 Fig. 10. UV-vis-spectra of Cr in KBr pellets: a) spectra of GS and PLI1, 2 at 150 K and corresponding difference spectrum in the insert; b) spectra of PLI1, 2 and PLI1-3 at 10 K and corresponding difference spectrum in the insert. Similar to Fig. 8, on the first stage (a) PLI1 and PLI2 were generated at 150 K until the saturation (PLI1 + PLI2), on the second stage (b) the sample was cooled down to 10 K and further irradiated by 365 nm (PLI1 + PLI2 + PLI3).

Scheme 1 .

 1 Scheme 1. The photo-and thermally induced transformations in Cr ([Pd(NH 3 ) 4 ][Pd(NH 3 ) 3 NO 2 ][MOx 3 ]⋅yH 2 O, M = Cr (Cr), Rh (Rh), Co (Co), Ox = oxalate). The coefficients roughly represent the populations of the linkage isomers, the populations of PLIs in case of Rh and Co are lower.

Fig. S1 .

 S1 Fig. S1. Photodifference maps between photoexcited (a) Co, (b) Rh and (c) Cr, and corresponding ground state data expressed as FobsPLI-FobsGS at given temperature and irradiation time. Contour level is equal to 0.5 e/Å 3 . [1] The 2D maps gives just approximate view of the disorder, since photoexcited ONO groups do not lie perfectly in O1-N1-O2 plane.

Fig. S2 .

 S2 Fig. S2. The X-ray structures of PLIs of Co (a), Rh (b) and Cr (c, d) at given temperature and irradiation time. ADP's were drawn at 50% probability level. For clarity only photoreactive units areshown.[2]

Fig. S3 .

 S3 Fig. S3. Comparison of IR-spectra of [Pd(NH 3 ) 4 ][Pd(NH 3 ) 3 NO 2 ][CrOx 3 ] (Cr) and its precursor K 3 [CrOx 3 ] at room temperature (RT).

Fig. S4 .

 S4 Fig. S4. IR-spectra of Cr at 150 K before (GS) and after (PLI1 + PLI2) light irradiation by 365 nm after vacuum pumping over the night.

Fig. S5 .

 S5 Fig. S5. IR-spectra of Cr at 150 K before (GS) and after (PLI1 + PLI2) light irradiation by 365 nm.

Fig. S6 .

 S6 Fig. S6. Difference IR-spectra of Cr at 150 K (panel a) and 10 K (panel b) before and after light irradiation at 365 nm.

Fig. S7 .

 S7 Fig. S7. IR-spectra of Cr at 10 K before (GS) and after (PLI1 + PLI2 + PLI3) light irradiation by 365 nm.

Fig. S8 .

 S8 Fig. S8. IR-spectra of Co at 10 K before and after light irradiation by 365 nm.

Fig. S9 .

 S9 Fig. S9. Arrhenius plots of the reactions of thermal decay of PLI1-3 in Cr (a), Rh (b) and Co (c).

Fig. S10 .

 S10 Fig. S10. UV-vis-spectra in water of ≈10 -3 M solutions of Cr, Rh, Co, corresponding oxalate salts and [Pd(NH 3 ) 4 ](NO 3 ) 2 .

Table 1 .

 1 List of distances between Pd1 and photoreactive ligand (NO 2 /ONO).

	Complex T, K	State	Pd1-N1 Pd1-O2A Pd1-O1B Pd1-O2C Pd1-O1D
	150	GS	2.000(2)	-	-	-	-
	Co
	150	GS, PLI1, 2	2.05(2)	1.95(3)	1.91(7)	-	-
	150	GS	1.997(2)	-	-	-	-
	Rh
	150	GS, PLI1, 2	2.03(1)	1.98(3)	1.88(9)	-	-
	150	GS	1.995(2)	-	-	-	-
	150	GS, PLI1, 2	1.965(9) 2.028(5)	2.007(6)	-	-
	Cr
	10	GS	1.999(2)	-	-	-	-
	10 GS, PLI1, 2, 3, 4 1.981(6)	2.05(1)	2.06(2)	1.981(6)	2.03(2)

Table 2 .

 2 Products' content in photoexcited structures.

				Component occupancy [%]
	Complex	T, K	GS NO 2	PLI1 "A" endo-ONO	PLI2 "B" endo-ONO	PLI3 "C" exo-ONO	PLI4 "D" exo-ONO
	Co	150 60.1	21.7	18.2	-	-
	Rh	150 70.7	18.3	11.0	-	-
	Cr	150 17.4 10 39.2	48.4 20.8	34.2 7.5	-26.8	-5.8

Table 3 .

 3 Assigned

	Complex Mode	GS	PLI 1	PLI2	PLI 3
		v(OH)	3591, 3515, 3449	3516, 3406	-	3501, 3456
		v(NO 2 )	1445, 1428, 1343, 1336, 1311,	1465, 1055	1442 sh , 1097 1489, 1012
	Cr	δ(NO 2 )	1285, 1258 824 sh	835	867	830
		ρ(NO 2 )	606	646	646	624
		Other	564, 547, 527 sh , 487, 470, 420	566, 474 sh , 418	-	481 sh
		v(OH)	3594, 3519, 3454	3517, 3403	-	3502, 3457
		v(NO 2 ) 1454, 1425 sh , 1347, 1335, 1310,	1468, 1053	1444 sh , 1096 1488, 1012
	Rh	δ(NO 2 )	1286, 1251 822	835	867 sh	830 sh
		ρ(NO 2 )	601	645	645	624
		Other	572, 555, 527 sh , 478, 469, 414	561 sh , 472	-	484 sh
		v(OH)	3574, 3501, 3446	3400	-	3480, 3451
		v(NO 2 )	1435, 1347, 1336, 1309, 1286,	1467, 1056	1440 sh , 1097 1486, 1013
	Co		1260
		δ(NO 2 )	822	840	864	820
		ρ(NO 2 )	606	650	650	628

IR-bands of GS, PLI1-3 at 10 K [cm

-1

].

Table 4 .

 4 Thermal stability parameters of PLIs: activation energies (E a ), frequency factors (lgk 0 ) and decay temperatures (T d ).

		PLI1	PLI2	PLI3
		E a , kJ mol -1 lgk 0	T d , K E a , kJ mol -1 lgk 0	T d , K E a , kJ mol -1 lgk 0	T d , K
	Cr	47.1±2.0 7.4±0.4 237±10 66.1±3.2 11.2±0.7 243±12 13.4±0.5 4.6±0.3 92±3
	Rh	55.6±7.2 9.2±1.6 238±30 62.7±14.1 10.4±3.0 245±55 18.3±2.8 7.4±1.5 92±14
	Co	60.9±11.4 10.3±2.4 239±45 85.2±14.4 15.5±3.1 241±41 21.1±0.4 8.9±0.2 93±2

Table S2 .

 S2 The hydrogen bond between water molecule and photoreactive ligand.

	Dx [g/cm 3 ]	2.215	2.192	2.231	2.200
	μ[mm -1 ]	2.268	2.245	2.278	2.278
	Reflections collected	54679	54065	19498	19784
	Reflections idependent	8643	8388	8552	8668
	Reflections observed [Fo > 4sig(Fo)]	7256	6717	7573	7220
	Completeness [%]	100		100	100	100
	Rint		0.0528	0.0612	0.0269	0.0326
	R [F 2 >2(F 2 )],wR, S	0.0270, 0.0617, 1.100	0.0328, 0.0747, 1.217	0.0313, 0.0782, 1.027	0.0446, 0.0895, 1.166
	Δρmax, Δρmin [eÅ -3 ]	1.186, -1.189	1.069, -1.578	1.914, -1.784	1.862, -3.021
	T, K	Structure		HB	D⋅⋅⋅A [°]	∡DHA [°]
		Co (GS)		O3-H3E⋅⋅⋅O2	2.727(4)	176.64(19)
	150	Co (GS+PLI1, 2)	O3-H3E⋅⋅⋅O2 2.720(14) O3-H3E⋅⋅⋅O2A 2.69(3)	169.5(6) 144.7(10)
		Rh (GS)		O3-H3E⋅⋅⋅O2	2.754(4)	164.05(19)
	150	Rh (GS+PLI1, 2)	O3-H3E⋅⋅⋅O2 2.722(12) O3-H3E⋅⋅⋅O2A 2.84(3)	172.1(5) 155.2(12)
		Cr (GS)		O3-H3E⋅⋅⋅O2	2.746(4)	167.35(19)
	a [Å] b [Å] 150 c [Å]	0.357 × 0.307 × 0.127 8.1322(2) 13.1555(5) O3-H3E⋅⋅⋅O2 8.0984(2) 13.0299(4) 2.72(3) 0.461 × 0.293 × 0.219 8.11747(15) 13.1250(3) 154.6(6) 0.461 × 0.293 × 0.219 8.1436(2) Cr (GS+PLI1, 2) O3-2.750(7) 174.7(3) H3E⋅⋅⋅O2A 13.1617(4) 19.9015(5) 19.8593(6) 20.0572(4) 20.0440(6) Cr (GS) O3-H3D⋅⋅⋅O2 2.735(4) 166.7(2)
	β [°]		95.784(2) O3-H3E⋅⋅⋅O2 2.422(12) 96.256(3) 96.2711(19) 166.0(4)	96.550(3)
	Volume [Å 3 ] Dx [g/cm 3 ] μ[mm -1 ] Reflections collected Reflections idependent 10 Cr (GS+PLI1, 2, 3, 2089.35(10) 2.270 2.575 54691 8504 H3E⋅⋅⋅O2A 2111.96(12) 2.245 2.574 50030 7332 O3-2.493(17) 2124.14(8) 2.370 2.529 56489 8648 156.9(8) 4) O3-H3E⋅⋅⋅O2C 3.037(9) 158.0(4)	2134.35(10) 2.358 2.517 56281 8695
	Reflections observed [Fo > 4sig(Fo)]	7125	5526	7610	7452
	Completeness [%]	100	100	100	100
	Rint		0.0542	0.0723	0.0534	0.0617
	R [F 2 >2(F 2 )],wR, S	0.0280, 0.0639, 1.104	0.0562, 0.1613, 1.135	0.0256, 0.0598, 1.102	0.0572, 0.1204, 1.343
	Δρmax, Δρmin [eÅ -3 ]	1.111, -1.433	3.543, -2.303	1.043, -1.553	2.603, -2.321
	Structure	Cr (GS)	Cr (GS+PLI1, 2)	Cr (GS)	Cr (GS+PLI1, 2, 3, 4)
	Irradiation time, 365 nm [min]	0		60	0	30
	Temperature [K]	150		150	10	10
	GS population [%]	100		17.4	100	39.2
	Mr		706.76	706.38	703.38	703.04
	Crystal dimensions [mm]	0.378 ×0.263×0.119	0.378 ×0.263×0.119	0.320 × 0.180 × 0.158	0.320 × 0.180 × 0.158
	a [Å]		8.1355(2)	8.2028(2)	8.1106(3)	8.1372(4)
	b [Å]		13.1028(3)	13.2208(3)	13.0066(7)	13.1598(8)
	c [Å]		19.9916(5)	19.8854(5)	19.9569(10)	19.9519(11)
	β [°]		95.969(2)	96.748(2)	95.933(4)	96.584(5)
	Volume [Å 3 ]	2119.50(9)	2141.58(9)	2093.99(17)	2122.43(19)

Acknowledgements

The work has been supported by grant of ANR (grant no. ANR-21-CE30-0045-01). Artem Mikhailov is grateful for financial support from the Metchnikov bourse program 2021. We thank the PMD2X platform of Université de Lorrain for granting access to XRD diffractometers.

ASSOCIATED CONTENT

Supporting Information available. IS contains crystallographic and structural data (Table S1, S2; Fig. S1,S2); additional IR and UV-vis spectroscopy data (Fig. S3-S10).

Supplementary Material: Photogeneration of several linkage isomers and investigation of forward and backward nitro-nitrito isomerization processes in a palladium complex

Artem Mikhailov a *, Krzysztof A. Konieczny a,b , Maria Gladysheva


A port-Hamiltonian formulation for the full von-Kármán plate model Andrea Brugnoli * , Denis Matignon * * ISAE-SUPAERO, Université de Toulouse, France Summary. In this contribution, a port-Hamiltonian reformulation of the full von-Kármán dynamical model for geometrically non- linear plates is detailed, including the collocated boundary control and observation. Starting from the canonical equations, a set of variables is chosen so as to make the total energy quadratic. The model, reformulated in these variables, highlights a port-Hamiltonian structure ruled by a state-modulated interconnection operator.

Classical model

The classical full von-Kármán dynamical model is presented in [START_REF] Bilbao | Conservative numerical methods for the full von kármán plate equations[END_REF]. The problem, defined on an open connected set Ω ⊂ R 2 , takes the dimensionless form

ü = Div N , ẅ = -div Div M + div (N grad w), N = Φ(ε), M = Φ(κ), ε = Grad u + 1/2 grad w ⊗ grad w, κ = Grad grad w, (1)
where u ∈ R 2 is the in-plane displacement, w is the vertical displacement, ε is the in-plane strain tensor, κ is the curvature tensor, N is the in-plane stress resultant and M is the bending stress resultant. The notation a ⊗ b = ab denotes the dyadic product of two vectors. The div operator is the divergence of a vector field, and grad the gradient of a scalar field. The operator Grad = 1 2 ∇ + ∇ designates the symmetric part of the gradient (i. e. the deformation gradient in continuum mechanics). For a tensor field U : Ω → R 2×2 , with components U ij , the divergence Div(U ) is a vector, defined column-wise as

Div(U ) := 2 i=1 ∂ xi U ij , ∀j = {1, 2}.
The linear tensor mapping Φ is positive and preserves symmetry:

Φ(A) = ν Tr(A)1 + (1 -ν)A, A = A =⇒ Φ(A) = Φ(A) , where 1 = Diag(1, 1).
The total energy of the model (Hamiltonian functional)

H = 1 2 Ω u 2 + ẇ2 + N . . ε + M . . κ dΩ, where A . . B = Tr(A B) (2)
consists of the kinetic energy and both membrane and bending deformation energies. This model proves conservative, see [START_REF] Bilbao | Conservative numerical methods for the full von kármán plate equations[END_REF]. Indeed, this implies that a port-Hamiltonian realization of the system exists. We shall demonstrate how to construct a port-Hamiltonian realization, equivalent to (1).

The equivalent port-Hamiltonian system (pHs)

To find a suitable port-Hamiltonian system, we first select a set of new energy variables to make the Hamiltonian functional quadratic. The selection is the same as for both the linear plate problems in Brugnoli et al. [2019a,b]:

α u = u, α w = ẇ, A ε = ε, A κ = κ. ( 3
)
The energy is quadratic in these variables

H = 1 2 Ω α u 2 + α 2 w + Φ(A ε ) . . A ε + Φ(A κ ) . . A κ . (4)
By computing the variational derivative of the Hamiltonian, one obtains the so-called co-energy variables:

e u := δ αu H = u, e w := δ αw H = ẇ, E ε := δ Aε H = Φ(A ε ), E κ := δ Aκ H = Φ(A κ ). (5)
Before stating the final formulation, consider the operator

C(w)(•) : L 2 (Ω, R 2×2 sym ) → L 2 (Ω) acting on symmetric tensors C(w)(T ) = div(T grad w). ( 6
)
Proposition 1 The formal adjoint of the C(w)(•) is given by

C(w) * (•) = - 1 2 [grad(•) ⊗ grad(w) + grad(w) ⊗ grad(•)] . ( 7
)
Proof 1 Consider a smooth scalar field v ∈ C ∞ 0 (Ω) and a smooth symmetric tensor field

U ∈ C ∞ 0 (Ω, R 2×2 sym ) with compact support. The formal adjoint of C(w)(•) satisfies the relation v, C(w)(U ) L 2 (Ω) = C(w)(v) * , U L 2 (Ω,R 2×2 sym ) . (8)
The proof follows from the computation

v, C(w)(U ) L 2 (Ω) = v, div(U grad w) L 2 (Ω) ,
Integration by parts,

= -grad v, U grad w L 2 (Ω,R 2 ) , Dyadic product properties, = -grad v ⊗ grad w, U L 2 (Ω,R 2×2 sym ) , Symmetry of U , = -1/2(grad v ⊗ grad w + grad w ⊗ grad v), U L 2 (Ω,R 2×2 sym ) . (9)
This means

C(w) * (•) = - 1 2 [grad(•) ⊗ grad(w) + grad(w) ⊗ grad(•)] , (10)
leading to the final result.

The pH realization is then given by the following system

∂ ∂t     α u A ε α w A κ     =     0 Div 0 0 Grad 0 -C(w) * 0 0 C(w) 0 -div Div 0 0 Grad grad 0         δ αu H δ Aε H δ αw H δ Aκ H     , (11)
The second line of system (11) represents the time derivative of the membrane strain tensor. To close the system, variable w has to be accessible. For this reason, its dynamics has to be included. The augmented system reads

∂ ∂t       α u A ε w α w A κ       =       0 Div 0 0 0 Grad 0 0 -C(w) * 0 0 0 0 1 0 0 C(w) -1 0 -div Div 0 0 0 Grad grad 0       J       δ αu H δ Aε H δ w H δ αw H δ Aκ H       . (12)
Given the results in Brugnoli et al. [2019a,b] and Proposition 1, the operator J is formally skew-adjoint. If only the kinetic and deformation energies are considered, it holds δ w H = 0. In general this terms allows accommodating other potentials, for example the gravitational one. Suitable boundary variables are then obtained considering the power balance

Ḣ = γ 0 e u , γ ⊥ E ε ∂Ω + γ 0 e w , γ ⊥⊥,1 E κ + γ 0 (E ε n • grad w) ∂Ω + γ 1 e w , γ ⊥⊥ E κ ∂Ω , (13)
where γ 0 e u = e u | ∂Ω is the Dirichlet trace, γ ⊥ E ε = E ε n| ∂Ω is the normal trace (n is the outward normal vector), γ ⊥⊥,1 E κ = -n • Div E κ -∂ s (n E κ s)| ∂Ω is the effective shear force at the boundary (s is the tangent versor at the boundary), γ 1 e w = ∂ n e w | ∂Ω is the normal derivative trace and γ ⊥⊥ E κ = n E κ n is the normal to normal trace. The boundary conditions are consistent with the ones assumed in [START_REF] Puel | Global existence for the full von kármán system[END_REF] for deriving a global existence result for this model.

Conclusions

We have presented a pHs formulation of the full von-Kármán model. The dynamics of the system exhibits a state modulated interconnection operator, while the energy remains quadratic in the chosen variables. Of particular interest is the discretization of such a model for simulation and control purposes. The Partitioned Finite Element Method (PFEM), an extension of mixed finite elements to pHs, seems to be particularly suitable to achieve a structure-preserving discretization of this model, as in Cardoso-Ribeiro et al. [2020] for the 2D Shallow Water Equation, which exhibits the same kind of polynomial nonlinearity.