-C Böttcher

S Pillet

D Schaniel

E.-E Bendeif

T Woike

H.-C Böttcher

Wavelength-selective photoisomerisation of nitric oxide and nitrite in a rhodium complex

Wavelength-selective photoisomerisation of nitric oxide

and nitrite in a rhodium complex

Introduction

The structural flexibility of the NO ligand in transition-metal complexes has been illustrated in many publications concerning photoinduced linkage isomers (PLI) [1][2][3][4][5][6]. Thereby the oxygen and nitrogen atom can adopt a variety of structural conformations (N-bound or O-bound) without the NO ligand being released. Some authors found indications that some of these PLI are implicated as precursor steps in the photorelease of NO [7]. PLI represents a rather strong modification of the molecular structure and hence we might expect significant influence of PLI on the structure and electronic properties of the whole molecule. However, most structural studies and DFT calculations show that the structural and electronic changes of the other ligands in the molecule are rather small [4], such that in good approximation the NO ligand can be treated in a partial M-NO description such as the Enemark-Feltham scheme {MNO} n , with n the total number of electrons associated with the metal d and π*(NO) orbitals [8]. This description is based on the donor-acceptor properties within MNO and especially the backbonding of the metal to the NO via the π*(NO). In this description the complex [Rh(NO)(NO 2 ) 2 (Bu t 2 PH) 2 ] has a {RhNO} 8 configuration, for example, formally Rh(d 8 )NO + . However, in view of the observed structural parameters of [Rh(NO)(NO 2 ) 2 (Bu t 2 PH) 2 ] with the NO being formally a bent NO - ligand, what is clearly indicated by the Rh-N-O angle of 125˚ in the ground state, the formal oxidation state of Rh should be discussed as Rh (III). If more than one photoswitchable ligand with the ability to form PLI is present, the situation gets more complicated and we might expect a more intricate interplay between the electron density distribution and the PLI properties. Also in this case, the simple correlation between the positive or negative shift of vibrational frequencies, e.g. of ν(NO), and the nature of the PLI [9,10] is no longer valid. We need to get a thorough crystallographic description of the structure in order to safely interpret the changes in the vibrational spectra [11][12][13]. In the present case the two photoswitchable ligands NO and NO 2  dispose of a plethora of potential PLI. Especially NO 2  with its increased degrees of freedom offers many different PLI as shown in Scheme 1 (see e.g. Ref. [12]). Some of these geometries were already characterised by photocrystallography in compounds with different central atoms [13,14]. Evidence for multiple PLI within one complex was given by Kovalevsky et al. [14] for the hexacoordinated ruthenium complex [Ru(NO)(NO 2 )(bpy) 2 ](PF 6 ). From Fourier difference maps the photoswitched ligands can be identified as a function of irradiation wavelength and temperature. We present a combined photocrystallographic and infrared spectroscopic study of PLI in [Rh(NO)(NO 2 ) 2 (Bu t 2 PH) 2 ]. We show that irradiation in the red spectral range generates the isonitrosyl configuration of the NO ligand while irradiation in the blue spectral range generates the (O,O')-chelating configuration of the NO 2  ligand. Both PLI influence the vibrations of the other photoswitchable ligand, such that an unambiguous PLI identification is only possible by photocrystallography. Long-time irradiation in the blue spectral range further leads to NO photorelease, which again can be quantified by photocrystallography.

Experimental

a. Université de Lorraine, CNRS, CRM2, F-54000 Nancy, France. b. Institut für Festkörper-und Materialphysik, TU Dresden, Zellescher Weg 16, Dresden, Germany. c. Departement Chemie der Ludwig-Maximilians-Universität,  81377 Munich, Germany. † Electronic address: Dominik.schaniel@univ-lorraine.fr.

Synthesis and preparation of [Rh(NO)(NO 2 ) 2 (Bu t 2 PH) 2 ] was performed according to the procedure described in Ref. [15].

Infrared spectroscopy

Infrared spectra were collected on a Nicolet 5700 FTIR spectrometer in the range 4000-360 cm -1 with a resolution of 2 cm -1 . The sample was mixed with KBr (spectroscopy grade), finely ground, and pressed to pellets. The pellets were glued with silver paste to a copper sample holder on the cold finger of an Oxford Optistat V01, allowing temperature regulation in the range 9-300 K. KBr windows allow for in-situ irradiation of the sample in the ultraviolet, visible and near infrared spectral range.

Photocrystallography

Single-crystal X-ray diffraction data were collected on a SuperNova Microfocus diffractometer equipped with a twodimensional ATLAS detector, using Mo Kα radiation (λ = 0.71073 Å) and a Helijet He open-flow cryosystem. The temperature was fixed at 10 K. A single crystal was mounted on a glass fiber using vacuum grease. Diffraction data were first collected at 10 K in the ground state (GS) using ω scans. The unit-cell determination and data reduction were performed using the CrysAlisPRO program suite [START_REF]CrysAlis CCD and CrysAlis RED[END_REF] on the full data set. 22314 reflections were measured up to a maximum resolution of sin(θ)/λ = 0.76 Å -1 and merged to 4104 unique reflections (R int = 0.0452). A numerical absorption correction was performed according to the crystal faces of the [Rh(NO)(NO 2 ) 2 (Bu t 2 PH) 2 ] crystal. The corresponding crystal structure was solved in the space group Pbca by direct methods with SHELXS software and refined on F 2 by weighted full matrix least-squares methods using the SHELXL program [START_REF] Sheldrick | [END_REF]. All non-H atoms were refined anisotropically. H atoms were located in difference Fourier maps and treated using a riding model, constraining the isotropic displacement parameters to 1.2U eq of the parent C atom. The crystal structure of [Rh(NO)(NO 2 ) 2 (Bu t 2 PH) 2 ] exhibits strong disorder which was taken into account using 2 positions for the phosphorus atoms of the di-tert.butylphosphine ligands (labelled P and P') with approximately 78% and 22% occupancies, and several configurations for the tert.butyl moieties. The Rh atom is furthermore disordered symmetrically around the position (0.5, 0.5, 0.5) (distance to the special position is approximately 0.29 Å) generating the crystallographic center of inversion in the Pbca space group. The nitrosyl ligand (N1-O1) is disordered about the center of inversion. Releasing all the symmetry constraints and refining the structure in space group P1 did not allow resolving the disorder. Figure 1 shows the complex and the disorder over the two positions (small ball-stick model for the second position). We note that the NO 2 -ligand is not disordered. 8) Å in the 635 nm photoinduced state. 20199 reflections were measured up to a maximum resolution of sin(θ)/λ = 0.76 Å -1 , and merged to 4056 unique reflections (R int = 0.0447). Empirical absorption correction was performed. All data are given in Table 1.

Photo-difference maps were calculated for visualization of the light-induced changes in electron density, and for identification of the related structural changes from the GS to the 635 nm photoinduced state. Common independent reflections between the GS and photo-irradiated state were used to compute the experimental X-ray photo-difference map by Fourier transform of the difference [F photo-irradiated obs ( hkl ) -F GS obs ( hkl )], using the structure factor phases from the GS structural refinement. 3955 common independent reflections were included in the calculation. The photodifference map is shown in Figure 2. In a second step, after having relaxed the PLI created by 635 nm by heating to room temperature, the same sample was irradiated at 10 K with a diode laser of 445 nm (P = 120 mW) for 20 min in the dark. Complete diffraction data were collected in the photo-stationary state; no space-group change occurs with respect to the ground state (space group Pbca). The unit cell parameters changed to a = 12.2256(15) Å, b = 12.5161 (11) Å and c = 15.2656 (15) Å in the 445 nm photo-induced state. 21397 reflections were measured up to a maximum resolution of sin(θ)/λ = 0.76 Å -1 , and merged to 4107 unique reflections (R int = 0.1063). Numerical absorption correction was performed. Photodifference maps were calculated from these data as shown in Figure 4. These data are of lower quality than the ground state and the 635 nm photoirradiated state, as highlighted by the high internal agreement factor R int .

R1 = ∑|Fo-Fc|/Fo. b wR2= {w(Fo 2 -Fc 2 ) 2 ]/wFo 2 ) 2 ]} 1/2 . c GoF= {w(Fo 2 - Fc 2 ) 2 ]/(Nobs-Nvar)} 1/2 .
Figure 2: Section of the 635nm photo-difference map in the RhN1N2 plane with isocontour of ±0.2 eÅ -3 (red-negative and blue positive electron density).

Results

Photocrystallography

The 635 nm photo-difference map (Figure 2) shows important structural changes between the ground state (GS) and the photo-irradiated state. A strong contribution occurs in the vicinity of Rh, with negative difference electron density at the Rh position in GS, and positive contribution on the center of inversion. This is most probably related to a displacement of the Rh atom in the photo-irradiated state into the direction of the center of inversion at (0.5, 0.5, 0.5). The two nitrosyl (NO) and nitrito (NO 2 -) ligands exhibit very different characteristics.

Strong negative contributions at the N1 and O1 positions together with positive contributions indicate a reorientation of the nitrosyl ligand. On the contrary, no significant features can be detected in the neighborhood of the nitrito ligand. The photo-difference map therefore evidences a selective photoisomerisation process of the nitrosyl ligand being triggered by the 635 nm excitation. According to the observations derived from the λ = 635nm photo-difference map, a structural model of the photoirradiated state was defined and refined against the experimental diffraction data. All the atoms, except N1 and O1 were considered as in the ground state configuration, and refined as a rigid group. Two configurations were considered for the nitrosyl ligand, one nitrosyl configuration (N1A-O1A) corresponding to the ground state with GS population P GS , and one nitrosyl configuration (N1B-O1B) corresponding to the photo-irradiated state detected in the photodifference map with population P PI =1-P GS . The refinement converged to a refined population of the photo-irradiated state of P PI =0.35 (2).

In order to define whether the coordination mode in the photoirradiated state is a nitrosyl (Rh-N1B-O1B) or an isonitrosyl (Rh-O1B-N1B) configuration, the two possibilities were separately refined. The results of the two refinements are given in Figure 3. The corresponding refinement agreement statistics are very similar for the two structural models (marginally better for the nitrosyl hypothesis), and therefore do not allow a decisive choice to be made. One can further inspect the thermal ellipsoids of the terminal O1B and N1B atoms. In the nitrosyl configuration, the equivalent atomic displacement parameter (ADP, Ueq) for O1B is higher than the ADP of O1A in the ground state. On the contrary, in the isonitrosyl configuration, the ADP of N1B is clearly too small (0.048(7) Å 2 ) with respect to O1A in the ground state. This comparison is very informative, but does not allow clearly discriminating between the nitrosyl or isonitrosyl hypotheses. The refined positions of N1B-O1B indicate a strong increase of the Rh-N1-O1 angle from 125.3(1)° in the ground state to 160.7(10)° in the 635nm photoirradiated state.

The 445 nm photodifference map shown in Figure 4 evidences very different features by comparison with the 635 nm photodifference map (Figure 2). At first, negative difference densities are detected at the positions of O1 and N1, but without any positive contributions which would have been the signature of a photoinduced metastable state of the N1-O1 ligand as detected with 635 nm irradiation. This could be rather attributed to NO release induced by the 445 nm irradiation. Negative contributions are also observed at the N2, O2 and O3 positions, with three positive peaks, evidencing a photoisomerisation of the nitrito ligand. The three positive peaks could correspond to a configuration with Rh coordinated to both O2 and O3, the binding mode to Rh is therefore modified by 445 nm irradiation. This hypothesis, corresponding to the (O,O')-chelating configuration of the NO 2 -ligand, is depicted as green dotted lines in In order to estimate the amount of NO release, evidenced by the photocrystallographic results, we have performed the same structural refinement as before, while varying the population of the nitrosyl ligand from 100 % to 40 % (varying the occupancy of the N1 and O1 atoms in the structural refinement). Since the atomic displacement parameters are strongly correlated to the occupancy factors in a structural refinement, decreasing the N1 and O1 occupancies result in decreased refined atomic displacement parameters. The results of the refinements are illustrated in Figure 5. As expected, as the fraction of NO release increases (and conversely the occupancy factors of O1 and N1 decrease in the refinement), the refined equivalent atomic displacement parameters decrease. At nearly 50 % NO release, the obtained values are similar to the ground state values. This is obviously a rough estimate, but we can nevertheless conclude that 445 nm irradiation induces a high contribution of NO release. Note that the NO release and the correspondingly reduced crystal quality explain the higher internal agreement factor R int for the 445nm irradiation compared to those obtained for GS and the irradiation with 635nm.

Infrared spectroscopy

Comparing the proposed structural changes based on the analysis of the photodifference maps to the infrared spectra, measured at 10K, one should be able to identify the corresponding decrease of the GS bands and assign the novel bands after photoirradiation to the different PLI. Inspecting first the changes in the NO stretching vibration ν(NO), we find that irradiation with 635 nm reduces the GS band at 1687 cm -1 by 37 % (Figure 6). A new band at 1781 cm -1 with a shoulder at 1768 cm -1 arises as well as a weaker band at 1865 cm -1 . In terms of frequency shift, this amounts to a positive shift of 94 and 178 cm -1 . The area of these new bands is considerably smaller than that of the GS. If we assign the δ(Rh-N-O) deformation mode to the GS band at 565 cm -1 , we find the corresponding new band at 587 cm -1 , i.e., a shift to higher energies by 22 cm -1 . The ν(Rh-N) band is at 445 cm -1 in GS and decreases upon irradiation, however the corresponding new band could not be unambiguously identified. Below 400 cm -1 the signal-to-noise ration becomes unfavourable so that weak bands arising in this region might not be detected. In this low-frequency region, we tentatively assign also the wagging mode of NO 2  , ρ(NO 2 ), at 597 cm -1 . It decreases upon irradiation and a new band is arising at 610 cm -1 , i.e. a shift to higher wavenumbers by 13 cm -1 . The assignment of the 565 and 597 cm -1 bands to δ(Rh-N-O) and ρ(NO 2 ) is not unambiguous, but based on available literature [18,19], we assign the higher energetic band to the wagging mode. Looking now to the other spectral ranges of the NO 2  vibrations, we observe reductions of the band-intensities at 1388 cm -1 (ν as (N=O 2 )) and 1308 cm -1 (ν s (N-O 2 )), without the appearance of clearly distinguishable new bands. Taking into account the overlap with neighboring bands, the reduction of the area of the central band at 1388 cm -1 necessarily induces a reduction of the band amplitude of neighboring bands. This is actually the case for the band at 1408 cm -1 . For the doublet at 1373/1366 cm -1 on the other hand we observe an increase in amplitude. Moreover, the main band at 1388 cm -1 exhibits after irradiation three maxima at 1389, 1386, and 1382 cm -1 , with the main maximum at 1382 cm -1 shifted by -6 cm -1 with respect to GS. One may thus conclude that the new band ν* as (NO 2 ) is shifted by about -6 cm -1 to 1382 cm -1 . Similar reasoning can be applied to the ν s (N-O 2 ) band at 1308 cm -1 . The area of this band is significantly reduced, while the intensity of the immediate and strongly overlapping neighboring band at 1319 cm -1 is hardly affected. Moreover, a shoulder at 1325 cm -1 appears after irradiation. Hence we conclude that the new ν* s (N-O 2 ) band is shifted to slightly higher energies (around 1315 cm -1 ).

The deformation mode δ(NO 2 ) of the GS is found at 836 cm -1 overlapping with the bands at 817/813 cm -1 . After irradiation the absorption between the two bands increases indicating the generation of a new band around 826 cm -1 , corresponding to a shift of -10 cm -1 to lower energies. Irradiation with blue light (λ = 445nm) leads to much more pronounced changes in the vibrational spectrum (see Figure 7). The area of the ν(NO) vibration at 1687 cm -1 is reduced by more than 80 %. The remainder exhibits a double peak at 1686 and 1671 cm -1 . New peaks with significantly smaller area arise at 1746, 1775, and 1859 cm -1 . Like in the case of 635 nm irradiation these bands are shifted to higher energies with respect to GS, however, they appear at different wavenumbers. Equally the areas of the δ(Rh-N-O) deformation mode and the ν(Rh-N) stretching mode at 565 cm -1 and 445 cm -1 decrease by more than 80 %. Since the band at 588 cm -1 increases we assign this band to the new δ*(Rh-N-O) vibration. Again the new ν*(Rh-N) band cannot be unambiguously identified in the spectrum. The NO 2  related vibrations are also changed significantly. The wagging mode ρ(NO 2 ) shifts from 597 cm -1 to 622 cm -1 , superposed to an original GS band at this wavenumber. The ν as (N=O 2 ) band at 1388 cm -1 is no longer visible. Three new bands at 1391, 1385, and 1380 cm -1 arise. The double band of GS at 1373/1367 cm -1 transforms into a single band at 1371 cm - 1 with two shoulders at 1365 cm -1 and 1358 cm -1 . We infer that the ν as (N=O 2 ) band does not simply shift but splits into several components corresponding to the new ν* as (N=O 2 ) band. The symmetric stretching vibration ν s (N-O 2 ) also vanishes almost completely. The area of the neighboring band at 1319 cm -1 is in consequence reduced also, but exhibits now a shoulder at about 1326 cm -1 , which might be assigned to the new ν* s (N-O 2 ). The deformation mode δ(NO 2 ) of the GS at 836 cm -1 is strongly modified. A shoulder at 845 cm -1 appears and a new maximum is formed at 828 cm -1 , whereas the area of the new bands is smaller than that of the GS. The neighboring doublet at 817/813 cm -1 is also modified, the new band exhibiting a maximum at 811 cm -1 with shoulders at 795 cm -1 and 820 cm -1 . mode cannot be unambiguously detected, probably it is found around 1470 cm -1 , since this band increases upon irradiation. interpretation is in accordance with the assignment of the two stretching vibrations in [Re(dmb)(CO) 3 (ONO)] based on 15 N isotope substitution [20] as well as studies on a [Cu(CH 3 CN) 4 (NO 2 ) 2 ](PF 6 ) 2 complex [21] and a thermally induced nitro-nitrito(O,O) isomerisation in a nickel complex [22]. Photoexcitation as a function of wavelength reveals that this ν(N-O) band at 1148 cm -1 arises by irradiation with 532 nm, has a maximum upon irradiation with 445 nm, and is less pronounced again upon irradiation with 405 nm. This wavelength dependence is reflected also in the behaviour of the ν(P-H) band. Since the hydrogen atoms are only indirectly influenced by the structural rearrangement of the NO and NO 2  ligands, the modification of the ν(P-H) is due to the overall change of the electron density distribution. Upon irradiation the GS band at 2351 cm -1 is shifted to lower wavenumbers: 2348 cm -1 for irradiation with 635 nm, 2340 cm -1 for irradiation with 445 nm, 2341 cm -1 for irradiation with 405 nm (see Fig. S1 of supplementary material). Upon heating, the PLI decays via intermediate states back to the GS. Figure 8 illustrates this behaviour for the case of irradiation with 445 nm at T = 10 K, for selected temperatures in the range 10 -280 K at which significant changes occur. For full temperature dependence see supplementary material (Figures S2-S6). The two bands at 1865 cm -1 and 1775 cm -1 relax upon heating to 130 K, but starting from around 80 K a new band arises at 1657 cm -1 , thus -30 cm -1 shifted with respect to the GS ν(NO) band at 1687 cm -1 . This band at 1657 cm -1 remains up to about 240 K, where it starts to relax back to the GS band. The NO 2  related vibrational bands evolve also upon heating. Up to about 80 K new broad bands arise at 1425/1408/1384/1350 cm -1 , which evolve back to GS upon heating above 240 K. Equally, the ν s (NO 2 ) band builds out a peak at 1316 cm -1 around 80 K, which at about 240 K shifts back to the GS position at 1308 cm -1 . A similar temperature evolution is observed for the δ(NO 2 ) mode at 836 cm 1 . Around 80 K new bands arise at 833 and 813 cm -1 , which evolve back to the GS positions at 836 cm -1 and 817/813 cm -1 upon heating to 240 K. The band at 1148 cm -1 shows a particularly interesting evolution. Around 80 K a band at 1121 cm -1 becomes clearly visible at the expense of the 1148 cm -1 band. Until 140 K this band grows and develops a shoulder at 1105 cm -1 and an additional band at 1056 cm -1 . These bands stay unchanged until about 240 K where they start to decrease and the GS is re-established. We can only speculate on the nature of this intermediate PLI. In the literature there is a report on a transient species in [Co(NH 3 ) 5 (NO 2 )]Cl 2 [19], where a broad band around 1100 cm -1 was tentatively assigned to the ν(N-O) vibration of a chelating (N,O) configuration. Interestingly the band at 1657 cm -1 as well as two bands at 1108 and 1127 cm -1 arise also upon heating after irradiation with red light (635 nm). These bands develop starting from 80 K to about 130 K, temperature where the corresponding PLI has completely relaxed, and then decrease upon heating to 260 K. Figure 9 illustrates the main evolutions, for a complete temperature dependence see Figures S7 andS8 of the supplementary material. These measurements indicate that the relaxation of both PLI, the one generated with red and the one generated with blue light, occur via intermediated steps, probably further isomeric structures. Moreover, they exhibit a similar infrared signature. In order to shine more light on this behaviour we performed irradiations with blue and red light at 100 K, temperature at which the relaxation of the different PLI has set in. Figure 10 shows the results. Indeed, we observe that at 100 K we find the same signature for both irradiation wavelengths and corresponding to that obtained during heating after irradiation at 10 K. This is a strong indication that both PLI have a similar relaxation channel.

Discussion

For the interpretation of the PLI structures we can rely on the combined photocrystallographic and infrared spectroscopic measurements. We note at first, that in the case of 635 nm irradiation the ν(NO) vibration shifts to higher energies. This is analogous to observations in octahedrally coordinated {PtNO} 8 complexes [10], where the red light irradiation led to a shift to higher wavenumbers. In these cases, DFT calculations showed that the PLI corresponds to an isonitrosyl (κON) configuration of NO, whereby the NO flips over to a Pt-O-N with a slightly larger angle (127°) compared to the GS Pt-N-O (118°) and a significantly longer Pt-O bond length of 2.72 Å compared to 2.18 Å in the GS configuration Pt-N-O. We suppose that in the RhNO complex we encounter the same situation. The photocrystallographic measurements show that the angle is increased from 125° to 161°. While these crystallographic measurements do not allow an unambiguous identification of an isonitrosyl configuration, since the analysis based on the ADPs is not conclusive, the analogy to the PtNO compounds together with the infrared results leads us to the hypothesis that this observed PLI corresponds to an isonitrosyl configuration Rh-O-N, which is also consistent with a decay temperature of about 130 K. Having said that, we might put forward a second hypothesis. The increase of the ADP parameters when refining a nitrosyl (N-bound) configuration points to an overestimation of the actual electron density present on the NO ligand. In other words, the NO might evolve from (formally) NO -in the GS towards NO + in the PLI. Accordingly, the oxidation state of the rhodium would have changed from Rh(III) in the GS to Rh(I) in the PLI. The observed higher ν(NO) vibration energy would be compatible with the NO + assignment. Note however, that the Rh(I) oxidation state is very rare for five-fold coordinated complexes, it occurs in general in four-fold coordination in square-planar configuration. The irradiation with red light seems to affect the NO 2  ligands only slightly, given the minor changes in the photodifference maps. The observed changes in the infrared spectra suggest that a small population of a nitrite isomer is present, however it is too small to be analysed by photocrystallography. The structural changes induced by blue light irradiation show a clear PLI of the NO 2  ligand. The photodifference maps indicate an (O,O')-chelating configuration, which could be refined satisfactorily. With this chelating nitrite ligand, the complex has evolved from a fivefold to a sixfold coordination, and hence the electron density distribution is significantly altered. These changes are responsible for the observed shifts of the ν(NO) vibrations in this case, again to higher wavenumbers. Further, the band areas are significantly smaller than in the GS, indicating a reduced dipole moment in the PLI state, making it difficult to clearly identify the various new bands, which sometimes overlap with GS bands, e.g. for the ρ*(NO 2 ) and δ*(Rh-N-O) bands. A notable exception is the rather intense band at 1148 cm -1 , which we assign to the ν(N-O) vibration of the chelating (O,O')-configuration. The corresponding ν(N=O) vibration at 1470 cm -1 is again overlapping with a GS band and therefore deduced indirectly by the intensity change of this band. Finally, we note, that long-time irradiation with blue light leads to significant NO release, as shown by photocrystallography. Corresponding infrared measurements as a function of irradiation fluence Q = I•t confirm this, as can be seen by the appearance of a band at 2220 cm -1 , corresponding to free NO in KBr, for irradiations beyond Q = 200 J/cm 2 (see Figure S9

Conclusions

Using a combination of photocrystallography and infrared spectroscopy we have shown that in [Rh(NO)(NO 2 ) 2 (Bu t 2 PH) 2 ] two types of PLI can be induced depending on the selected irradiation wavelength. Using red light, the NO ligand undergoes isomerisation, most probably forming an isonitrosyl configuration. When using blue light, the NO 2  ligand forms the (O,O')-chelating isomer. Upon heating, these PLI relax back to the GS via an intermediate state.

Scheme 1 :

 1 Scheme 1: Five main PLI for NO 2  .

Figure 1 :

 1 Figure 1: Ball and stick representation of the structure of [Rh(NO)(NO 2 ) 2 (Bu t 2 PH) 2 ] in the ground state. The two differently sized balls represent the two positions over which the molecule is disordered. The sample was then irradiated at 10 K with a diode laser of 635 nm (P = 60 mW) for 20 min in the dark, until the photostationary state was reached. Complete diffraction data were collected in the photo-stationary state; no space-group change occurs with respect to the ground state (space group Pbca). The unit-cell parameters change only slightly from a = 12.1515(6) Å, b = 12.5518(5) Å and c = 15.2014(7) Å in the GS to a = 12.1855(7) Å, b = 12.5477(6) Å and c = 15.1668(8) Å in the 635 nm photoinduced state. 20199 reflections were measured up to a maximum resolution of sin(θ)/λ = 0.76 Å -1 , and merged to 4056 unique reflections (R int = 0.0447). Empirical absorption correction was performed. All data are given in Table1.

Figure 3 :

 3 Figure 3: Comparison of the two refined structural models considering a nitrosyl (left) and isonitrosyl (right) coordination mode for the 635 nm photoirradiated state.

Figure 4 .

 4 According to these observations, a structural model was built with a fraction of the nitrito ligand switched to the putative orientation. The switched nitrito O2-N2-O3 ligand was refined as a rigid group with N2-O2 and N2-O3 distances, and O2-N2-O3 angle equal to the ground state values. The refinement converged reasonably, allowing for picturing some trends. The refined fraction of the photoirradiated state is P PI = 0.39(2), which is even higher than the value obtained for the 635 nm photo-irradiated state. In the ground state, the Rh atom is connected to the N2 atom with a Rh-N2 bond distance of 2.065(3)Å, while in the photo-induced state, the Rh-O bond distances are Rh-O2B = 2.18(2)Å, and Rh-O3B = 2.29(3)Å.

Figure 4 :

 4 Figure 4: Section of the 445nm photo-difference map in the RhN1N2 plane with isocontour of ±0.2 eÅ 3 (red-negative and blue positive electron density).

Figure 5 :

 5 Figure 5: Evolution of the refined equivalent atomic displacement parameter Ueq for O1 and N1 as a function of the fixed fraction of NO release (the fraction of NO release is directly related to the occupancy factors of N1 and O1 in the structural refinement).

Figure 6 .

 6 Figure 6. Infrared spectra of GS (black line) and after irradiation with 635 nm (red line) at 10 K. The insert shows a zoom onto the region 1750-1900 cm -1 .

Figure 7 .

 7 Figure 7. Infrared spectra of GS (black line) and after irradiation with 445 nm (blue line) at 10 K.Additionally, a strong new band arises at 1148 cm -1 . This band most probably corresponds to the ν(N-O) stretching vibration of the (O,O')-chelating PLI (O=N-O). The corresponding ν(N=O) mode cannot be unambiguously detected, probably it is found around 1470 cm -1 , since this band increases upon irradiation.interpretation is in accordance with the assignment of the two stretching vibrations in [Re(dmb)(CO) 3 (ONO)] based on15 N isotope substitution[20] as well as studies on a [Cu(CH 3 CN) 4 (NO 2 ) 2 ](PF 6 ) 2 complex[21] and a thermally induced nitro-nitrito(O,O) isomerisation in a nickel complex[22]. Photoexcitation as a function of wavelength reveals that this ν(N-O) band at 1148 cm -1 arises by irradiation with 532 nm, has a maximum upon irradiation with 445 nm, and is less pronounced again upon irradiation with 405 nm. This wavelength dependence is reflected also in the behaviour of the ν(P-H) band. Since the hydrogen atoms are only indirectly influenced by the structural rearrangement of the NO and NO 2  ligands, the modification of the ν(P-H) is due to the overall change of the electron density distribution. Upon irradiation the GS band at 2351 cm -1 is shifted to lower wavenumbers: 2348

Figure 8 .

 8 Figure 8. Infrared spectra upon heating after irradiation with 445 nm at 10 K.

Figure 9 .

 9 Figure 9. Infrared spectra upon heating after irradiation with 635 nm at 10 K.

Figure 10 .

 10 Figure 10. Comparison of infrared spectra obtained after irradiation at 100 K with blue (445 nm) and red (635 nm) light, and corresponding spectrum obtained during heating after irradiation with red light at 10 K.

  in the supplementary material).The particularity of the [Rh(NO)(NO 2 ) 2 (Bu t 2 PH) 2 ] complex is that it offers the possibility to induce both, PLI of NO and of NO 2  , by using different wavelengths. This indicates a certain flexibility of the rhodium metal to accommodate the corresponding electron density changes. This is much less observed in hexacoordinated ruthenium nitro-nitrosyl compounds, where up to now only the [Ru(NO)(NO 2 )(bpy) 2 ](PF 6 ) showed this capacity, while other compounds such as [Ru(NO)(NO 2 ) 2 (OH)L 2 ] (L = pyridine or pyridine-derivatives), K 2 [Ru(OH)(NO)(NO 2 ) 4 ], Na 2 [Ru(OH)(NO)(NO 2 ) 4 ], and cis-[Ru(NO)(NH 3 ) 2 (NO 2 ) 2 OH] exhibit only photoinduced NO isomers [6,23,24,25]. Even in the case when the GS contains already the nitro and nitrito form NO 2 and ONO, only PLI of the ligand NO could be induced in PyH[Ru(NO)(NO 2 ) 2 (ONO)py(OH)] and OC-6-34-[RuNO(NO 2 )(ONO)py 2 (OH)][26]. Further interesting effects occur upon heating after irradiation at low temperature. The two PLI obtained by red and blue light irradiation at 10 K decay through intermediate states, which have the same infrared signature. The same state can be generated when irradiating with blue or red light at 100 K, and might correspond to a further isomeric configuration, possibly a chelating (N,O) configuration of one of the NO 2 -. This behaviour leads us to the following hypothesis, summarized in Figure 11: the change of the electronic structure during the relaxation of the PLI as well as possible interactions between the neighbouring NO and NO 2 -ligands result in the adoption of a third PLI, stable in the region of 100-240K. This peculiar behaviour allows to generate different PLI by choosing temperature and irradiation wavelength.

Figure 11 :

 11 Figure 11: Summary of generation conditions of the various PLI in [Rh(NO)(NO 2 ) 2 (Bu t 2 PH) 2 ] as a function of temperature and irradiation wavelength. The structures of PLI-1 and PLI-2 are confirmed by photocrystallography, while the structure of PLI-3 is a hypothesis based on infrared spectroscopic results.

Fig. S1 :

 S1 Fig. S1: Infrared spectra as a function of irradiation wavelength in the spectral range of the PH vibrational bands.

Fig. S2 :

 S2 Fig. S2: Infrared spectra upon heating after irradiation at 10 K with 445 nm: 10 -90 K.

Fig. S3 :

 S3 Fig. S3: Infrared spectra upon heating after irradiation at 10 K with 445 nm: up to 140 K.

Fig. S4 :

 S4 Fig. S4: Infrared spectra upon heating after irradiation at 10 K with 445 nm: up to 190 K.

Fig. S5 :

 S5 Fig. S5: Infrared spectra upon heating after irradiation at 10 K with 445 nm: up to 240 K.

Fig. S6 :

 S6 Fig. S6: Infrared spectra upon heating after irradiation at 10 K with 445 nm: up to 280 K.

Fig. S7 :

 S7 Fig. S7: Infrared spectra upon heating after irradiation at 10 K with 635 nm: 10 -130 K.

Fig. S8 :

 S8 Fig. S8: Infrared spectra upon heating after irradiation at 10 K with 635 nm: 130 -260 K.

Fig. S9 :

 S9 Fig. S9: Infrared spectra as a function of irradiation fluence at 10 K with 445 nm. Beyond about 200 mJ/cm 2 the signature of free NO in KBr at 2220 cm -1 becomes visible.

Table 1 :

 1 Crystallographic data and refinement details for [Rh(NO)(NO 2 ) 2 (Bu t 2 PH) 2 ] in the ground and photo-irradiated states.

		Ground state	Photo-	Photo-
			irradiated	irradiated
			state 635nm	state 445nm
		Crystal data
	Chemical	RhN3O5P2C16H3	RhN3O5P2C16H3	RhN3O5P2C16H3
	formula	8	8	8
	Mr (g.mol -1 )	517.34	517.34	517.34
	Crystal	orthorhombic,	orthorhombic,	orthorhombic,
	system,	Pbca	Pbca	Pbca
	space group
	Z	4	4	4
	a (Å)	12.1515(6)	12.1855(7)	12.2256(15)
	b (Å)	12.5518(5)	12.5477(6)	12.5161(11)
	c (Å)	15.2014(7)	15.1668(8)	15.2656(15)
	α (°)	90	90	90
	β (°)	90	90	90
	γ (°)	90	90	90
	V (Å 3 )	2318.5(2)	2319.0(2)	2335.9(4)
	ρcalc (g cm -3 )	1.482	1.482	1.470
	μ (mm -1 )	0.904	0.903	0.897
	Crystal size
	(mm)
		10	10	10
	No. of	22314	20199	21397
	measured
	reflections
	θ range (°)	3.161-32.955	3.164-32.943	3.146-32.884
	No. of	4104, 0.0452	4056, 0.0447	4107, 0.1063
	independent
	reflections,
	Rint
	No. of	3216	3142	2024
	observed
	reflections [I
	> 2σ(I)]
		Refinement
	No. of	214	157	221
	parameters
	a R1 [F 2 >	0.0759	0.0811	0.1570
	2σ(F 2 )]	[0.0552]	[0.0606]	[0.0798]
	b wR2 [F 2 >	0.1525	0.1646	0.2311
	2σ(F 2 )]	[0.1357]	[0.1501]	[0.1812]
	c GoF	1.049	1.268	0.983
	Δρmax, Δρmin	1.542, -0.937	1.312, -0.829	1.486, -0.798
	(eÅ -3 )
	a

0.18 x 0.25 x 0.33 0.18 x 0.25 x 0.33 0.18 x 0.25 x 0.33 Data collection Temperatur e (K)

Acknowledgements

Measurement time on the X-ray diffraction platform of the Institut Jean Barriol is gratefully acknowledged. Financial supports from the French PIA project "Lorraine Université d'Excellence", reference ANR-15-IDEX-04-LUE, and the CPER are acknowledged

Conflicts of interest

There are no conflicts to declare.

Supplementary Material


Yacine Benatia
email: yacine.ben-atia@imt-nord-europe.fr

Anne Savard
email: anne.savard@imt-nord-europe.fr

Romain Negrel
email: romain.negrel@esiee.fr

E Veronica Belmega

Unsupervised deep learning to solve power allocation problems in cognitive relay networks

Keywords: Unsupervised deep learning, full-duplex relaying, Decode-and-Forward, cognitive radio

In this paper, an unsupervised deep learning approach is proposed to solve the constrained and non-convex Shannon rate maximization problem in a relay-aided cognitive radio network. This network consists of a primary and a secondary user-destination pair and a secondary full-duplex relay performing Decode-and-Forward. The primary communication is protected by a Quality of Service (QoS) constraint in terms of tolerated Shannon rate degradation. The relaying operation leads to non-convex objective and primary QoS constraint, which makes deep learning approaches relevant and promising. For this, we propose a fully-connected neural network architecture coupled with a custom and communication-tailored loss function to be minimized during training in an unsupervised manner. A major interest of our approach is that the required training data contains only system parameters without the ground truth, i.e., the corresponding solutions to the non-convex optimization problem, as opposed to supervised approaches. Our numerical experiments show that our proposed approach has a high generalization capability on unseen data without overfitting. Also, the predicted solution performs close to the brute force one, highlighting the high potential of our unsupervised approach.

I. INTRODUCTION

The ever increasing number of communicating devices and data-hungry applications are challenging the existing technologies and drive the transition to the next generation of wireless communications. Future communication systems target highly ambitious objectives [START_REF] Huang | A survey on green 6G network: Architecture and technologies[END_REF] and among the promising candidate technologies envisioned to reach them are: cognitive radio, cooperative communications and full-duplexing [START_REF] Mobini | Secure cooperative spectrum sharing in full-duplex multiantenna cognitive radio networks with jamming[END_REF].

On the one hand, cognitive radio and full-duplexing are promising technologies to enhance the network's spectral efficiency. In cognitive radio, a non-licensed network -the secondary or opportunistic network -is allowed to communicate in under-utilized licensed bands, provided that the impact on the licensed network -the primary network -is kept below acceptable levels [START_REF] Zhang | A survey of advanced techniques for spectrum sharing in 5G networks[END_REF]- [START_REF] Thampy | Outage probability analysis and optimization of cognitive full-duplex relay networks[END_REF]. Full-duplex nodes are capable to simultaneously transmit and receive information, having the potential to duplicate the spectral efficiency [START_REF] Mobini | Secure cooperative spectrum sharing in full-duplex multiantenna cognitive radio networks with jamming[END_REF].

On the other hand, cooperative communications aim at improving the network capacity by taking advantage of the This work has been supported IRCICA, CNRS USR 3380, Lille, France and by the ELIOT ANR-18-CE40-0030 and FAPESP 2018/12579-7 project.

wireless medium, which makes signals sent by a source available at all receivers within range [START_REF] Kamal | Device-to-device aided cooperative relaying scheme exploiting spatial modulation: An interference free strategy[END_REF]. In this setting, some nodes called relays can enhance other nodes' communications by exploiting its non-intended received signals and, hence, increase the overall network capacity.

The simultaneous blend of all the above techniques holds the potential to improve both the spectral efficiency and network throughput while providing optimal resource allocation policies. In this paper, we consider a relay-aided cognitive radio network similarly to [START_REF] Savard | Full-duplex relaying for opportunistic spectrum access under an overall power constraint[END_REF], where the opportunistic network is assisted by a full-duplex relay performing Decode-and-Forward (DF). Our objective is to find the optimal power allocation policy maximizing the opportunistic achievable Shannon rate under a primary quality of service (QoS) constraint by exploiting machine (deep) learning. The secondary transmitter and the relay have individual power constraints unlike in [START_REF] Savard | Full-duplex relaying for opportunistic spectrum access under an overall power constraint[END_REF], where an overall power constraint was assumed.

Related works: Because of the non-linear and complex relay operations, the resulting power allocation problems in such relay-aided cognitive networks are non-convex and can be solved in closed-form only in special cases, such as: negligible interference links [START_REF] Savard | Optimal power allocation in a relay-aided cognitive network[END_REF], negligible opportunistic direct links [START_REF]Optimal power allocation policies in multi-hop cognitive radio networks[END_REF], Compress-and-Forward relaying [START_REF] Savard | Full-duplex relaying for opportunistic spectrum access under an overall power constraint[END_REF]. Outside these very specific cases, such power allocation problems become difficult to tackle and even intractable using traditional techniques based on convex optimization and game theory.

More recently, machine learning techniques based on deep learning have been exploited to solve various resource allocation problems [START_REF] Lee | Deep power control: Transmit power control scheme based on convolutional neural network[END_REF]- [START_REF] Guo | An energy-efficiency multirelay selection and power allocation based on deep neural network for Amplify-and-Forward cooperative transmission[END_REF]. In [START_REF] Lee | Deep power control: Transmit power control scheme based on convolutional neural network[END_REF], a convolutional neural network is trained to maximize the spectral and the energy efficiency of a wireless network composed of several interfering single antenna transceiver pairs under individual power constraints. The authors of [START_REF] Liu | Power allocation schemes based on machine learning for distributed antenna systems[END_REF] developed a deep neural network (DNN) approach for distributed antenna systems, which learns the nonlinear mapping between channel realizations and power allocation schemes of the traditional sub-gradient algorithm to maximize both the spectral and energy efficiency. In [START_REF] Sun | Learning to optimize: Training deep neural networks for wireless resource management[END_REF], power control policies based on DNNs in multiuser wireless networks were proposed. In [START_REF] Liang | Towards optimal power control via ensembling deep neural networks[END_REF], the authors maximized via deep learning the sum rate of a fading multiuser network under minimum rate constraints. In [START_REF] Eisen | Learning optimal resource allocations in wireless systems[END_REF], the authors first solved a general convex functional optimization problem with stochastic constraints via DNNs and then applied the general solution to maximize the rate of a point-to-point additive white Gaussian noise (AWGN) channel and of an interference channel. Unfortunately, the problem under study in this paper is not convex and the approach in [START_REF] Eisen | Learning optimal resource allocations in wireless systems[END_REF] cannot be applied. The authors of [START_REF] Ahmed | Deep learning for radio resource allocation in multi-cell networks[END_REF] exploited DNNs based on autoencoders to solve the sum-rate maximization with respect to the sub-band power allocation in a downlink multi-cell network. In [START_REF] Guo | An energy-efficiency multirelay selection and power allocation based on deep neural network for Amplify-and-Forward cooperative transmission[END_REF], the authors studied the power allocation and relay selection problem maximizing the energy efficiency of an Amplify-and-Forward relay channel with the help of a DNN.

In the context of cognitive radio networks, various resource allocation problems have also been tackled via deep learning [START_REF] Zhou | Resource allocation based on deep neural networks for cognitive radio networks[END_REF]- [START_REF] Shah-Mohammadi | Deep reinforcement learning approach to QoE-driven resource allocation for spectrum underlay in cognitive radio networks[END_REF]. In [START_REF] Zhou | Resource allocation based on deep neural networks for cognitive radio networks[END_REF], the authors have proposed a deep learning approach for resource allocation problems in cognitive radio networks maximizing the spectrum and energy efficiency. In [START_REF] Lee | Resource allocation for multi-channel underlay cognitive radio network based on deep neural network[END_REF], a DNN is used to determine the power allocation over multiple channels of the opportunistic users. At last, in [START_REF] Shah-Mohammadi | Deep reinforcement learning approach to QoE-driven resource allocation for spectrum underlay in cognitive radio networks[END_REF], a deep reinforcement learning approach is applied to solve a distributed resource allocation problem.

None of the above works consider jointly cooperative communications (via relaying) and cognitive radio networks, which significantly increases the difficulty of the problem under study. Also, these existing methods cannot be readily applied in our setting.

Our main contributions: In this paper, we propose to exploit the high potential of deep neural networks to solve nonconvex power allocation problems to maximize the opportunistic achievable Shannon rate under a primary quality of service (QoS) constraint and individual device power constraints. To the best of our knowledge, machine learning techniques have not been employed yet to derive optimal power allocation policies for relay-aided cognitive radio networks.

More specifically, we leverage unsupervised learning and design a novel neural network architecture able to solve the non-convex constrained resource allocation problem. Our experimental results show that the proposed DNN architecture is able to minimize a customized loss function that takes into account both the secondary rate and the primary QoS constraint, without any overfitting effect while ensuring good generalization properties over the validation set. Furthermore, the secondary rate predicted by our proposed DNN for unseen system parameters (i.e., in the test data) is close to the optimal one obtained by exhaustive search (or brute force) while meeting the primary QoS constraint.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system under study, depicted in Fig. 1, is composed of a primary user or transmitter U P and its destination D P ; a secondary full-duplex relay; and a secondary user U S and its destination D S , similarly to [START_REF] Savard | Optimal power allocation in a relay-aided cognitive network[END_REF], [START_REF] Savard | Full-duplex relaying for opportunistic spectrum access under an overall power constraint[END_REF], [START_REF]Optimal power allocation policies in multi-hop cognitive radio networks[END_REF]. The received signals at the relay, primary and secondary destinations write as where i ∈ {P, S}, j ∈ {P, S}\i; X P , X S and X R , of average power P P , P S and P R respectively, denote the message send by the primary user, the secondary user and the relay respectively; Z R and Z i denote the AWGN at the relay and at destination D i of variance N R and N i respectively. Without loss of generality, we assume that all noise processes are of unit variance N R = N S = N P = 1 or equivalently consider channel gains normalized by the receiver noise variance, defined as

Y R = h P R X P + h SR X S + Z R (1) Y i = h Ri X R + h ii X i + h ji X j + Z j , (2)
g ij = h 2 ij
Nj . Furthermore, we assume that the channel gains follow a common fading and path-loss model given as

h ij ∼ N (0,σ 2 g ) √ 1+d γ ij
, where d ij denotes the distance between the nodes i and j and γ is the path loss factor [START_REF] Ding | On the performance of non-orthogonal multiple access in 5G systems with randomly deployed users[END_REF]. We let g (g ij , ∀i, ∀j) denote the vector collecting all channel gains in the network. We assume that the relay performs full-duplex decode-and-forward (DF) and that the relay can cancel out any self-interference as in [START_REF] Savard | Optimal power allocation in a relay-aided cognitive network[END_REF], [START_REF] Savard | Full-duplex relaying for opportunistic spectrum access under an overall power constraint[END_REF], [START_REF]Optimal power allocation policies in multi-hop cognitive radio networks[END_REF]. Also, the messages sent by the secondary user and the relay are treated as additional noise at the primary destination; and the primary message is treated as additional noise throughout the secondary network. Let R i , i ∈ {P, S} denote the achievable rate of the primary and secondary user respectively; and R P denote the primary achievable rate in the absence of the secondary transmission:

R P = 1 2 log 2 (1 + g P P P P ).
The primary network allows the opportunistic communication provided that the following minimum QoS constraint is met in terms of achievable primary rate [START_REF] Savard | Optimal power allocation in a relay-aided cognitive network[END_REF], [START_REF] Savard | Full-duplex relaying for opportunistic spectrum access under an overall power constraint[END_REF], [START_REF]Optimal power allocation policies in multi-hop cognitive radio networks[END_REF]:

R P ≥ (1 -τ )R P , τ ∈ [0, 1]. (3)
As shown in [START_REF] Savard | Full-duplex relaying for opportunistic spectrum access under an overall power constraint[END_REF], under DF relaying, the primary achievable rate is expressed as

R P = C g P P P P g RP P R + g SP P S + 2α √ g SP g RP P S P R + 1 ,
where C(x) = 1 2 log 2 (1 + x) denotes the capacity of the point-to-point AWGN channel and α ∈ [0, 1] comes from the used superposition coding technique and represents the tradeoff between sending the message from the previous block and sending a new one. Hence, the QoS constraint in (3) can be rewritten as Q(g, α, P S , P R ) ≤ A, with Q(g, α, P S , P R ) = g SP P S + g RP P R + 2α g SP g RP P S P R , A = g P P P P (1 + g P P P P ) 1-τ -1 -1. Our main objective is to maximize the opportunistic achievable rate R S , when both the relay and secondary user are constrained by a maximum power expressed as P R and P S respectively. The achievable rate region is discussed in details and provided in [START_REF] Savard | Full-duplex relaying for opportunistic spectrum access under an overall power constraint[END_REF]. Contrarily to [START_REF] Savard | Full-duplex relaying for opportunistic spectrum access under an overall power constraint[END_REF], we consider here individual power constraints on each secondary device, i.e. secondary user and relay. To sum up, the resulting optimization problem under study writes as

(OP) max P R ,P S ,α min {f R (g, α, P S , P R ), f S (g, α, P S , P R )} s.t. Q(g, α, P S , P R ) ≤ A, (C1) 0 ≤ P S ≤ P S , 0 ≤ P R ≤ P R , (C2) 0 ≤ α ≤ 1, with (C3)
f R (g, α, P S , P R ) = g SR (1 -α 2 )P S g P R P P + 1 , f S (g, α, P S , P R ) = g SS P S + g RS P R + 2α √ g RS g SS P S P R g P S P P + 1 .

In the above, the objective function follows from the achievable rate of the secondary user: R S (g,α,P S ,P R ) = C(min{f R (g,α,P S ,P R ),f S (g,α,P S ,P R )}) , as shown in [START_REF] Savard | Full-duplex relaying for opportunistic spectrum access under an overall power constraint[END_REF]; constraint (C1) is the primary QoS constraint; constraints (C2) correspond to the individual power constraint of the relay and secondary user; constraint (C3) follows from superposition coding.

III. DEEP LEARNING POWER ALLOCATION POLICY

A close analysis of the problem (OP) reveals a non-convex objective function coupled by a non-convex QoS constraint (C1), this is due to the non-linear operations performed by the relay. Hence, solving the non-convex (OP) is a very challenging task. For this, we propose an unsupervised approach based on deep neural networks (DNN). At the opposite, a supervised approach would require computing a labeled training dataset containing the solutions to the non-convex problem (OP) for a large number of sampled system parameters. This would be too computationally heavy and, henceforth, we opt for an unsupervised approach relying on a training dataset composed of only samples of the inputs (i.e., system parameters) of our DNN and exploits a specifically tailored communication loss to perform the training in an unsupervised manner.

A. Custom loss function

A key component of our proposed approach is the loss function that the network will be trained to minimize. Solving constrained optimization problems with DNNs is highly non-trivial, unless the constraints are of box-type such as (C2), (C3). This is not the case of the QoS constraint (C1) which is a difficult non-convex constraint. Nevertheless, as opposed to the power constraints (C2), the primary QoS constraint is not a physical (hard) constraint but rather a requirement, which can be relaxed and included as a penalty in the objective function below

L = N =1 -R S (g , α, P S , P R ) + λ[Q(g , α, P S , P R ) -A] + ,
with [x] + = max{0, x} and N denoting the total number of channel realizations g , ∈ {1, . . . , N } in the training dataset.

The hyperparameter λ denotes the unit price in bits/Watt of the primary QoS violation. A small value of λ will result in maximizing the achievable opportunistic rate without taking into account the primary QoS constraint; whereas large values of λ will strictly satisfy the primary QoS constraint but at the cost of opportunistic rate. This tradeoff between opportunistic rate and primary QoS will be further investigated via numerical results.

B. Proposed DNN architecture

Our proposed DNN architecture to solve (OP) is composed of four fully connected hidden layers and is depicted in Fig. 2. The input consists of the channel gains vector g which is used to predict the outputs: (α, PR , PS ), i.e., the solution of (OP). The fully connected architecture is justified because of its generality and given that there is no a priori structural or temporal information within the input vector g to be exploited via more specific architectures such as convolutional or recurrent networks.

The four hidden layers are composed of M -2M -2M -2M neurons with M = 128 and are followed by a rectified linear unit (ReLU) activation function due to its low computational complexity. This specific architecture is chosen based on extensive empirical experiments, as we discuss in the numerical section.

The final layer is followed by sigmoid activation functions: a standard one sgm α (x) = 1/(1+e -x ) to map the predicted α into its feasible set [0, 1], and two modified ones sgm Pi (x) = P i /(1 + e -x ), i ∈ {S, R}, to map the predicted powers P R and P S into [0, P R ] and [0, P S ] respectively. This final layer ensures that the hard constraints (C2), (C3) are met.

IV. NUMERICAL SIMULATIONS Below, we discuss our numerical setup and DNN training procedure as well as the performance evaluation of our approach. Complete details and source codes of our experiments are available online at https://github.com/yacine074/ USDNN-to-solve-powerallocation-problems-in-CRAN.git. A. Experimental setup 1We consider a square cell of dimension 10 × 10 m in which the relay is positioned in the center whereas both primary and secondary user positions are uniformly distributed in the cell, unless specified otherwise. The path loss factor is set to γ = 3 and the channel gain standard deviation σ g = 7. We assume that P P = P R = P S = 10 W and set the threshold τ = 0.25 for the maximum primary rate degradation.

Dataset: To the best of our knowledge, the majority of related works exploiting DNNs use simulated data [START_REF] Sun | Learning to optimize: Training deep neural networks for wireless resource management[END_REF], [START_REF] Liang | Towards optimal power control via ensembling deep neural networks[END_REF], [START_REF] Zhou | Resource allocation based on deep neural networks for cognitive radio networks[END_REF], [START_REF] Lee | Resource allocation for multi-channel underlay cognitive radio network based on deep neural network[END_REF], given the lack of real data that is available and open access. Thus, unless specified otherwise to train and test our proposed DNN architecture, we use a dataset composed of three disjoint parts: i) a training set containing 10 6 channel realizations g ; ii) a validation set containing 2 × 10 5 channel realizations g jointly with the ground truth, i.e., the corresponding optimal resource allocation policies (α * , P * R , P * S ) obtained by brute force (or exhaustive search) for evaluating the generalization capability and for tuning the hyperparameters of our approach; iii) a test set containing 2 × 10 5 channel realizations jointly with the ground truth to evaluate the optimality gap of our predicted solution.

DNN training: In our numerical simulations, we used the ADAM optimizer to iteratively update the weights of our DNN. The batch size is set to 4096, the learning rate to 10 -4 ; these values allows the DNN weights optimization to converge within 1000 epochs.

Performance metrics and benchmark: We define here the relevant performance metrics used for the evaluation purpose. First, we define the relative gap between the predicted achievable rate via our DNN and the achievable rate obtained by brute force, our ideal benchmark, as follows:

G = 1 N N =1 RS, -R * S, 1 N N =1 R * S, (4)
where RS, = R s (g , α, PS , PR ) denotes the secondary achievable Shannon rate obtained based on our DNN prediction for the -th sample in the dataset and R * S, denotes the optimal rate for the -th sample in the dataset obtained via brute force or exhaustive search. We chose brute force as benchmark thanks to its implementation simplicity and because it approximates the optimal solution with an adjustable precision.

Second, the degradation of the primary achievable rate caused by the opportunistic interference is defined as:

∆ = 1 -RP, /R P, , (5)
where RP, = R P (g , α, PS , PR ) denotes the achievable primary rate of our predicted allocation. Based on this metric, we can also define the empirical outage as the proportion of samples in the dataset (or channel settings) for which the target primary QoS constraint is not met and the average primary rate degradation when in outage:

Outage = 1 N N =1 I [∆ > τ ] , (6)
∆ out = N =1 I [∆ > τ ] × ∆ N =1 I [∆ > τ ] , (7)
where I [x] equals 1 when x is true and 0 otherwise. To choose the architecture in Fig. 2, we have performed extensive simulations over the validation set. In Fig. 3, we report the most significant results. On the left, we analyze the impact of the number of layers and compare four different architectures composed of one up to four hidden layers as follows: M , M -2M , M -2M -2M , and M -2M -2M -2M , with M = 128. We see that there is a significant gain in secondary rate when increasing the number of layers from 1 to 3; moving to 4 layers helps to decrease the outage. Hence, a 4layer architecture is a good compromise between performance and computational cost. Now, on the right, we compare three different such four-layer DNNs, by varying the number of neurons per layer M ∈ {64, 128, 256}. We see that increasing M from 64 to 128 neurons leads to a significant gain in secondary rate; increasing further the number of neurons does not seem justified given the incurred computational cost. For these reasons, we choose M = 128 coupled with the 4-layer architecture in Fig. 2 henceforth.

C. No overfitting: performance on train and validation sets

In Fig. 4, we plot the evolution of our custom loss function L over the number of epochs within the training and validation sets for λ ∈ {10 0.5 , 10 2 }. First, notice that the DNN training converges within 1000 epochs. The superposed performance over the training and validation sets hints towards a good generalization capability of our proposed DNN. More importantly, since the loss function does not increase within the validation set, our approach does not suffer from overfitting. Finally, the decay rate of the loss function is much faster for λ = 10 0.5 . Indeed, when λ becomes small, the custom loss is mainly rate-driven and not much emphasis is put on the primary QoS constraint; this leads to a much easier optimization problem to solve. At the opposite, for larger values of λ, the custom loss puts an emphasis on satisfying (C1), which leads to a more difficult problem. Of course, this parameter needs to be tuned carefully as discussed below. rate and satisfying the primary QoS constraint. The primary QoS constraint is satisfied on average even though some outlayer points can be observed for secondary rate-driven losses. At last, Decode-and-Forward relaying was found to perform well when the relay is close to the secondary user.

Fig. 1 .

 1 Fig. 1. Cognitive relay-aided network.

Fig. 2 .

 2 Fig. 2. Our proposed DNN architecture.

Fig. 3 .

 3 Fig.3. Impact of the number of layers and number of neurons on the prediction performance over the validation set.

Fig. 4 .

 4 Fig. 4. Evolution of the loss function evolution over the training epochs over the training and validation sets (no overfitting effects).

Fig. 5 .

 5 Fig. 5. Relative average gap G and outage as functions of the hyperparameter λ over the validation set.

  Average and maximum primary rate degradation and average degradation when in outage (∆out) as functions of λ over the validation set.

  Histogram of the primary rate degradation over the validation set.

Fig. 6 .

 6 Fig. 6. Impact of the hyperparameter λ.

  Average primary rate degradation (%).

  Average predicted relay power (W).

  Average secondary rate (bpcu).

Fig. 7 .

 7 Fig. 7. Impact of the relay position (x R , y R ).

Our proposed DNN architecture in Fig.4is robust to changes within this experimental setup. Although the training needs to be relaunched for different setups, our core conclusions carry over in general.

D. Choice of the hyperparameter λ

In Fig. 5, the relative gap in (4) and the outage in [START_REF] Thampy | Outage probability analysis and optimization of cognitive full-duplex relay networks[END_REF] are depicted as functions of λ over the validation set. For small values of λ (rate-driven custom loss), the relative gap G is positive, which means that the secondary rates obtained via the DNN are larger than the optimal ones via brute force. The reason is that the primary QoS constraints are not necessarily met by our DNN solutions, as illustrated by the high outage levels. At the opposite, for large values of λ (primary QoSdriven custom loss), the outage goes to zero as expected at a cost in terms of secondary rates. Indeed, our predicted secondary rates are smaller than the optimal ones (negative relative gain G), but this gap is kept below 10 %.

In Fig. 6, we investigate closer the impact of λ on the primary rate degradation. For this, we plot the average and maximum values of the primary rate degradation as well as the average degradation when in outage ∆ out in [START_REF] Kamal | Device-to-device aided cooperative relaying scheme exploiting spatial modulation: An interference free strategy[END_REF] in Fig. 6(a). Also, in Fig. 6(b), we illustrate the histogram of the primary rate degradation (∆) for λ ∈ {10 0.5 , 10 2 }. The mean primary rate degradation falls quickly below the threshold τ = 25 %. For small values of λ, the worst case primary degradation can reach up to 90 %. Nevertheless, such extreme degradation is obtained only for a small number of out-layer data points. This is indicated by the curve ∆ out hitting the 25 % threshold reasonably fast as well as by the histogram of the degradation in Fig. 6(b).

To sum up, the value λ = 10 0.5 achieves a good tradeoff between the achievable secondary rate and the primary QoS degradation in our setting and will be used in the test phase below.

E. Prediction performance on test data

Henceforth, we evaluate the performance of our predicted solution ( α, PR , PS ) obtained with new samples (i.e., test data) unseen during the training or validation phases.

First, over our initial test set in Sec. IV-A, we obtain a relative average gap G = -1.39 %, Outage = 2.60 % and the average degradation when in outage ∆ out = 25.18 %; this performance is coherent with the validation results demonstrating the high generalization capability of our approach on unseen data.

Impact of the relay position: We will now change the setting of Sec. IV-A and fix the positions of the primary and secondary user/destination nodes: U S (5, 2.5), D S (7.5, 5), U P (2.5, 5), D P (5, 7.5) as in Fig. 7; and consider that the relay position (x R , y R ) varies within the cell. Hence, we generate a second test set composed of 10 4 channel realizations. In Fig. 7(a), Fig. 7(b), and Fig. 7(c), we illustrate the average primary rate degradation, the average predicted relay power PR and the average secondary rate RS , respectively as functions of the relay position (x R , y R ) over this new test set.

In Fig. 7(a), we see that the average degradation in the primary rate always falls below the fixed threshold of τ = 25 % irrespective from the relay position. When the relay is very close to the primary nodes, the degradation drops below 20 %, since very little power is allocated to the relay as shown in Fig. 7(b). The worst case degradation arises when the relay lies between the secondary nodes, since the secondary rate improvement overcomes the damage the relay causes to the primary link. At last, in Fig. 7(b), we also notice that more power is allocated to the relay when it is close to the secondary user. This is to be expected since DF relaying is known to perform well in terms of achievable rate in these cases, which is indeed confirmed in Fig. 7(c).

V. CONCLUSIONS

In this paper, we investigated the power allocation problem in a cognitive radio network where the secondary user is assisted by a full-duplex relay node performing Decode-and-Forward. Because of the complex relaying operations, neither the secondary rate nor the primary QoS constraint are convex, leading to a difficult constrained Shannon rate maximization problem. To this aim, we proposed an unsupervised deep learning method based on fully connected architectures and a communication-tailored loss function modeling the secondary rate and the primary QoS constraint. We evaluated the performance of our approach via extensive numerical simulations. Our results show the high generalization capability of our DNN solution on unseen data and without overfitting. More importantly, we highlighted the tradeoff between the secondary