Hélène Staquet

Pierre-Marc Francois

Baptiste Sandoz

Sébastien Laporte

Philippe Decq

Stéphane Goutagny

Surface reconstruction from routine CT-scan shows large anatomical variations of falx cerebri and tentorium cerebelli

Keywords: Modeling, Brain herniation, Brainstem, Corpus callosum injury, Reference frame

Background Finite element modeling of the human head offers an alternative to experimental methods in understanding the biomechanical response of the head in trauma brain injuries. Falx, tentorium, and their notches are important structures sur-rounding the brain, and data about their anatomical variations are sparse. Objective To describe and quantify anatomical variations of falx cerebri, tentorium cerebelli, and their notches. Methods 3D reconstruction of falx and tentorium was performed by points identification on 40 brain CT-scans in a tailored Matlab program. A scatter plot was obtained for each subject, and 8 anatomical landmarks were selected. A reference frame was defined to determine the coordinates of landmarks. Segments and areas were computed. A reproducibility study was done. Results The height of falx was 34.9 ± 3.9 mm and its surface area 56.5 ± 7.7 cm 2 . The width of tentorium was 99.64 ± 4.79 mm and its surface area 57.6 ± 5.8 cm 2 . The mean length, height, and surface area of falx notch were respectively 96.9 ± 8 mm, 41.8 ± 5.9 mm, and 28.8 ± 5.8 cm 2 (range 15.8-40.5 cm 2 ). The anterior and maximal widths of tentorial notch were 25.5 ± 3.5 mm and 30.9 ± 2.5 mm; its length 54.9 ± 5.2 mm and its surface area 13.26 ± 1.6 cm 2 . The length of falx notch correlated with the length of tentorial notch (r =0.62, P <0.05). Conclusion We observe large anatomical variations of falx, tentorium, and notches, crucial to better understand the biomechanics of brain injury, in personalized finite element models.

Introduction

Brain hemispheres and cerebellum are separated by two rigid structures: the falx cerebri and the tentorium cerebelli. Falx and tentorium are stiffer than brain and pia. They are composed of collagen-rich fibrous tissues which are responsible for the high resistance of these structures to tensile strength [START_REF] Adeeb | The cranial dura mater: a review of its history, embryology, and anatomy[END_REF][START_REF] Van Noort | A study of the uniaxial mechanical properties of human dura mater preserved in glycerol[END_REF].

Falx is a midline reflection of dura mater, sickle-shaped, that extends vertically between the two cerebral hemispheres. The free margin of the falx shapes its inferior limit and borders the corpus callosum. The superior sagittal sinus, which is attached to the skull, limits the outer margin. The tentorium, crescent-shaped, extends between the cerebellum and the cerebral hemispheres. The tentorium forms the upper limit of the posterior fossa. Its aperture, called the tentorial notch, is the communication between supratentorial and infratentorial spaces. The tentorial notch contains one important structure, the brainstem, surrounded by cerebrospinal fluid. The shape of the tentorial is triangular, with two free edges, joining at the tentorial apex. The dorsum sellae forms its anterior edge. The outer margin of tentorium is outlined by the straight sinus on the top, the transverse sinus posteriorly, and the superior petrosal sinus anteriorly [START_REF] Rhoton | Tentorial incisura[END_REF].

Because of their shape, stiffness, and location within the cranium, these structures affect the transfer of mechanical loads within the brain, constraining brain motion, and dampening deformation across the midline [START_REF] Bradshaw | Simulation of acute subdural hematoma and diffuse axonal injury in coronal head impact[END_REF]. Herniation occurs when e.g. a hematoma or a tumor induces brain shift across these notches. Subfalcine and trantentorial herniations can lead to severe disability or death. Hardy et al. examined on using human cadavers the brain displacement and deformation during impact. Dynamic impact events in the vicinity of the falx cerebri influence the brain motion patterns more drastically for impacts in the coronal and horizontal planes rather than in sagittal plan [START_REF] Hardy | A study of the response of the human cadaver head to impact[END_REF]. Using finite element model of the head, Ho et al. studied the strain distribution in the brain and the biomechanical influence of the falx and tentorium during impacts. They demonstrated that falx and tentorium could have an effect on the biomechanical behavior of surrounding brain tissues, especially corpus callosum and brainstem [START_REF] Ho | The peculiar properties of the falx and tentorium in brain injury biomechanics[END_REF]. Thus, the variations of the size of falx and tentorium could modify strain distribution within the cranium.

Little attention has been given to anatomical variations of falx and tentorium. Only one study reporting the variations in the height of the falx had been done so far on living subjects, and was based on carotid angiography in 1969 [START_REF] Galligioni | Anatomic variation of the height of the falx cerebri. Its relationship to displacement of the anterior cerebral artery in frontal space-occupying lesions[END_REF]. Few studies address the anatomy of the tentorial notch. These studies were performed on cadaveric cases [START_REF] Adler | The tentorial notch: anatomical variation, morphometric analysis, and classification in 100 human autopsy cases[END_REF][START_REF] Corsellis | Individual variation in the size of the tentorial opening[END_REF][START_REF] Ono | Microsurgical anatomy of the region of the tentorial incisura[END_REF][START_REF] Sunderland | The tentorial notch and complications produced by herniations of the brain through that aperture[END_REF]. There is a lack of data on anatomical variations and correlations on both tentorium and falx in living subjects.

Anatomical variations of falx, tentorium, and their notches could influence outcomes of brain herniations. Thus, the aim of this study is to characterize anatomical variations of the falx and the tentorium, focusing on their notches. We propose a new method to measure falx and tentorium using routine CT imaging on adult subjects.

Patients and method

Forty anonymized brain CT scans were extracted from institutional database (AP-HP, Approval no. 2017-013 from the CNIL, authorization no. 1980120). CT scans were randomly selected excluding cases with midline deviation. Anonymous DICOM images were implemented in a homemade Matlab program (MathWorks® version 7.0, available on request). One senior surgeon (HS) thoroughly pointed each anatomical structure (falx, tentorium, and venous sinuses) on axial CT slices. Points were identified every c.a 5 mm, one slice out of two. Per subject, on average 1386 points were identified (650 for the falx, 440 for the tentorium and 296 for sinuses). The mean acquisition time was about 20 min per case. Workflow is described on Fig. 1. Briefly, a scatter plot was obtained in Matlab. Height anatomical landmarks were identified on this 3D scatter plot: anterior (A) and posterior (B) crista galli, left (C) and right (D) anterior clinoid process, left (E) and right (F) parieto-mastoid point, the tentorial apex (G), and the torcular (H) (Fig. 1c).

Reference frame of the head

In order to determine the coordinates of the anatomical landmarks for further computing, we defined a reference frame (Fig. 1c). The origin of the frame was defined as the midpoint of the segment between tentorial apex and torcular [GH]. The normal of the least-square falx cerebri plan defined the Y-axis. The cross-product of the Y-axis by the vector (posterior crista galli-tentorial apex) defined the Z-axis. The X-axis is the result of the cross-product of the Y-axis by the Z-axis. The three axis were normalized in order to get an orthonormal frame where each axis is a unit vector (a vector of length one), orthogonal to each other. The following measures have been obtained from the coordinates of the anatomical landmarks and the projection of the set of the points of the tentorium and the falx. The computation time was about 10 min per case.

Falx cerebri

To characterize falx cerebri, we used the 2 parameters depicted Fig. 2. The height of the falx [START_REF] Adeeb | The cranial dura mater: a review of its history, embryology, and anatomy[END_REF] is the distance between the tentorial apex and the superior edge of the falx cerebri along a straight line passing through the midpoint of the clinoid process and the tentorial apex [START_REF] Galligioni | Anatomic variation of the height of the falx cerebri. Its relationship to displacement of the anterior cerebral artery in frontal space-occupying lesions[END_REF]. The surface of the falx (sF) was calculated by a triangulation method, from all the points of the falx projected on the least square falx plan. The notch of the falx was the area limited by the free edge of the falx and the segment between posterior crista galli and tentorial apex [BG]. In order to characterize the notch, we used the 3 parameters depicted Fig. 2. The height of the falx notch [START_REF] Adler | The tentorial notch: anatomical variation, morphometric analysis, and classification in 100 human autopsy cases[END_REF] is the maximal distance between the free edge of the falx and the segment [BG]. The length of the falx notch (3) is the maximal distance between two points of the free edge of the falx. The surface of the falx notch (sFN) was assessed by a triangulation method. Ratio between surfaces was calculated.

Tentorium cerebelli

In order to characterize tentorium cerebelli, we used the 2 parameters depicted Fig. 3. The width of the tentorium (4) is the distance between the 2 parietomastoid points [EF]. The surface of the tentorium (sT) is assessed from all the points projected in the frame, and linked by a triangulation method to obtain a surface. To characterize the tentorial notch, we used 4 parameters. The anterior width [START_REF] Corsellis | Individual variation in the size of the tentorial opening[END_REF] is the distance between the 2 anterior clinoid processes [CD]. The maximal width [START_REF] Dagtekin | Microsurgical anatomy and variations of the anterior clinoid process[END_REF] is the maximal distance between two points of the tentorial notch's free edges. The length of the tentorial notch [START_REF] Fisher | Brain herniation: a revision of classical concepts[END_REF] is the distance between tentorial apex (G) and the middle point of the 2 clinoid processes segment [CD]. The surface of the tentorial notch (sTN) is calculated by a triangulation method and is delineated by the 2 free edges, the segment [CD] and the tentorial apex (G). The tentorial angle [START_REF] Galligioni | Anatomic variation of the height of the falx cerebri. Its relationship to displacement of the anterior cerebral artery in frontal space-occupying lesions[END_REF] is calculated using the Twining's line, drawn from the midpoint of the segment [CD] to the torcular (H), and the straight sinus pictured by the segment [GH] [START_REF] Isherwood | Edward wing twining, 1887-1939[END_REF]. Similarly to the falx, ratio between surfaces was calculated. 4), the anterior width (5), the maximal width [START_REF] Dagtekin | Microsurgical anatomy and variations of the anterior clinoid process[END_REF], and the length (7) of the tentorial notch. Surfaces were calculated by a triangulation method from projected points for tentorium (sT) and tentorial notch (sTN). The tentorial angle (8) was also recorded

Reproducibility study

To evaluate measurement errors, we carried out a reproducibility study on 6 randomly selected CT-scans. Pointing acquisition and determination of the coordinates of anatomical landmarks had been repeated 3 times for each scan, by the same neurosurgeon, with a minimum 24 h-interval between measures. A non-parametric Spearman test was applied. A measure was considered robust enough when variations were 2DS < 3 mm for coordinates, < 5% for distances and surfaces, and < 5°for angles.

Statistical analysis

Results were analyzed using the statistical software XLSTAT 2018.3 (Addinsoft SARL, Bordeaux, FR). All data are presented as mean and standard deviation. The correlation study used a Pearson test, r > or = 0.6 was considered suitable, with a P value < 0.05.

Results

Reproducibility

The reproducibility error on the coordinates of the anatomical landmarks was less than 1 mm. Two parameters had a reproducibility error superior to 5%: the anterior width (5.22%) and the surface of the tentorial notch (5.78%). For the angle, the reproducibility error was less than 2°. These results suggest that the method is reliable and allows further analyses, using our reference frame for the head.

Falx cerebri

Both distances and surfaces presented large inter-subject variations; e.g., the surface of the falx notch ranged from 15 to 40 cm 2 (ratio 2.7) and the height of the falx ranged from 29 to 48 mm (ratio1.6).

Results are given in Table 1. The falx notch surface represented 33% ± 5.2 of the whole surface (falx plus falx notch).

As expected, the height of the falx correlates with the surface of the falx (r = 0.6, P < 0.05). The surface of the falx notch is statistically correlated with its height (r = 0.82, P < 0.05) but not with its length (r = 0.44, P < 0.05).

Tentorium cerebelli

Similarly to falx measurements, distances, angle and surfaces presented large inter-subject variations. Results are given in Table 2. The tentorial notch surface represented 18.79% ± 2.3 of the whole surface (tentorium plus tentorial notch). The width of the tentorium was not correlated with its surface (r = 0.46, P < 0.05). The surface of the tentorial notch was correlated with its length (r = 0.655, P < 0.5), but not with its maximal width (r = 0.420, P < 0.05). We found no correlation between the surfaces of tentorium cerebelli and tentorial notch (r = 0.0, P = 0.99).

Finally, the length of the falx notch was correlated with the length of the tentorial notch (r = 0.62, P < 0.05). Surfaces of both notches were not correlated (r = 0.234, P = 0.055), neither their heights (r = 0.34, P = 0.03).

Discussion

Anatomical variations

The present study presents a consistent description (surfaces and linear measurements) for both falx, tentorium, and their notches, using clinical CT-Scan, on living subjects. Our main results are as follows: the mean surface of the falx was 56.5 ± 7.7 cm 2 , and its height 34.9 ± 3.9 mm. The falx notch has been measured for the first time (28.8 ± 5.9 cm 2 ). The surface of the tentorium (57.6 ± 5.8 cm 2 ) and the ratio of its free area (18%) have been measured for the first time. Important inter-individual variations have been demonstrated in this study for these 2 important structures of the brain. So far the falx and the falx notch have not attracted much attention in the literature. In this study, the height and the surface of the falx notch were the most variable parameters among our data, ranging respectively from 27 to 57 mm and from 15 to 40 cm 2 . We found only one study in the literature, in 1969. It Mean ± SD 34.9 ± 3.9 56.5 ± 7.7 96.9 ± 8 41.81 ± 5.9 28.8 ± 5.9 33.7 ± 5.2 was based on carotid arteriography and brought into evidence anatomical variations in the size of the falx, independently of skull dimensions [START_REF] Galligioni | Anatomic variation of the height of the falx cerebri. Its relationship to displacement of the anterior cerebral artery in frontal space-occupying lesions[END_REF]. In this study, the mean height of the falx was 5.27 cm (ranging from 4 to 6.2 cm). The authors studied the interspace between the falx and the corpus callosum (represented by the inferior longitudinal sinus) and its relation to the displacement of the anterior cerebral artery in frontal lesions. It appeared that a narrow falx notch limited the anterior cerebral artery shift.

In the present study, the height of the falx was smaller (3.49 cm). Indeed, we did not include the superior sagittal sinus and straight sinus in the definition of the falx. This could account for the difference. Brockman et al. measured the variations of the superior sagittal sinus (SSS) on CT-scans [START_REF] Brockmann | Variations of the superior sagittal sinus and bridging veins in human dissections and computed tomography venography[END_REF]. The maximal vertical diameter of the SSS, measured 3 cm dorsally from the coronal suture, was 6.4 mm (SD = 1.2).

The tentorial notch has been more thoroughly studied because of the clinical importance regarding transtentorial herniation. Transtentorial herniation occurs in a wide variety of neurological conditions. If left untreated, transtentorial herniation progresses rapidly to severe disability or death [START_REF] Fisher | Brain herniation: a revision of classical concepts[END_REF][START_REF] Meyer | Herniation of the brain[END_REF]. Previous studies on tentorium anatomy were made on cadavers, using photographs taken from the superior and lateral views [START_REF] Adler | The tentorial notch: anatomical variation, morphometric analysis, and classification in 100 human autopsy cases[END_REF][START_REF] Corsellis | Individual variation in the size of the tentorial opening[END_REF][START_REF] Ono | Microsurgical anatomy of the region of the tentorial incisura[END_REF][START_REF] Sunderland | The tentorial notch and complications produced by herniations of the brain through that aperture[END_REF]. In 1958, Corsellis et al. described unexpected variations in the size and shape of the tentorial notch [START_REF] Corsellis | Individual variation in the size of the tentorial opening[END_REF]. The same year, Sunderland described the relations between this free area and the brainstem, nerves, and vessels [START_REF] Sunderland | The tentorial notch and complications produced by herniations of the brain through that aperture[END_REF]. In 1984, using upper view pictures on cadavers, Ono et al. described a maximal tentorium width of 110.8 mm, seemingly different from our results (99.6 ± 4.6 mm) [START_REF] Ono | Microsurgical anatomy of the region of the tentorial incisura[END_REF]. Nevertheless, on the basis that lateral and posterior tentorium margins are firmly attached to the inner surface of the skull, Ono et al. measured the maximum width of the tentorium as the distance between internal bone surfaces. In our study, the lateral sinuses were not included in the definition of tentorium. This could explain the difference in measures. In 2002, Adler and Milhorat analyzed 100 human cadavers, taking upper view pictures during autopsy. He developed a classification system for the tentorial notch, taking into account the length and the width of the notch [START_REF] Adler | The tentorial notch: anatomical variation, morphometric analysis, and classification in 100 human autopsy cases[END_REF].

Tentorial angle is one of several factors that determine whether the infratentorial approach for surgeries of the pineal or cerebellar vermian region is usable or challenging. One considers to use the supracerebellar infratentorial approach whether the tentorial angle is shallow, or the occipital transtentorial approach whether the tentorial angle is seep [START_REF] Lafazanos | Evaluating the importance of the tentorial angle in the paramedian supracerebellar-transtentorial approach for selective amygdalohippocampectomy[END_REF]. Similarly, the tentorial angle is used to plan the paramedian supracerebellartranstentorial approach for selective amygdalohippocampectomy [START_REF] Syed | A novel method to measure the tentorial angle and the implications on surgeries of the pineal region[END_REF]. A recent trend has emerged to use the Twining line as the reference line to measure the slope of the tentorium. At first, Twining's line, joining the tuberculum sellae and the torcular, on pneumoencephalography, was used to be the landmark of the fourth ventricle [START_REF] Isherwood | Edward wing twining, 1887-1939[END_REF]. The most recent studies cited above, using MRI, focused only on the tentorial angle using images acquired in the sagittal plane. In two recent studies using the Twining's line, mean tentorial angle was calculated at 42°and 43.5°, ranging from 25°to 56.5°, in line with our data (54 ± 5°) [START_REF] Lafazanos | Evaluating the importance of the tentorial angle in the paramedian supracerebellar-transtentorial approach for selective amygdalohippocampectomy[END_REF][START_REF] Syed | A novel method to measure the tentorial angle and the implications on surgeries of the pineal region[END_REF]. Part of the wide range of results could be explained by the precise position of anatomical landmarks used to draw the lines and calculate the angle. The tuberculum sellae is lined up with a line passing through the two anterior clinoid processes and is rather easy to identify in a reproducible way. However, the torcular and the apex are venous confluents with a rather large drop/triangular shape. Depending on the point selected to stand for these landmarks, the calculation of the angle may significantly differ. In this present study, the anatomical landmark called "torcular" is defined as the most posterior and inferior point of the tentorium in the torcular region. The landmark called "apex tentorial" is defined as the most upper point of the tentorium. The exact positions of these landmarks have not been otherwise specified in previous studies [START_REF] Lafazanos | Evaluating the importance of the tentorial angle in the paramedian supracerebellar-transtentorial approach for selective amygdalohippocampectomy[END_REF][START_REF] Syed | A novel method to measure the tentorial angle and the implications on surgeries of the pineal region[END_REF].

Reference frame

For 3D modeling of the head, a precise reference frame is mandatory. In this study, we propose a reference frame of the head, as we found no previous publication or recommendations of the International Society of Biomechanics about it. This reference frame allows 3D calculations and modeling that was otherwise not possible with pictures on cadavers. Hernandez et al. used finite element modeling of the brain to investigate the relationships between head motion, falx displacement, and potential damages to corpus callosum [START_REF] Hernandez | Lateral impacts correlate with falx cerebri displacement and corpus callosum trauma in sports-related concussions[END_REF]. They demonstrated that high coronal and/ or horizontal rotations produce deep lateral displacement of the falx cerebri, which may strain and damage the corpus callosum. They modeled heads with and without falx, and showed that strains in the corpus callosum halved beside when the falx is present. Our study shows important variations of the falx and its notch. One can hypothesize that a large falx notch might behave as an absent falx, thus decreasing damages to the corpus callosum. Pathophysiology of traumatic brain injury is often studied using average finite element model of the head [START_REF] Ho | The peculiar properties of the falx and tentorium in brain injury biomechanics[END_REF][START_REF] Kleiven | Correlation of an FE model of the human head with local brain motion-consequences for injury prediction[END_REF][START_REF] Raul | Finite-element models of the human head and their applications in forensic practice[END_REF][START_REF] Tse | A review of head injury and finite element head models[END_REF]. Inter-individual variability of structures, such as the falx and tentorium notches, is seldom taken into account. Our data suggest that including individual anatomical variations of falx and tentorium notches in personalized models could be crucial to better elicit the mechanisms of brain damages.

In our reproducibility study, two parameters had a reproducibility error superior to 5% (anterior width and surface of the tentorial notch). This variation flows from pixel resolution of routine CT scans (mean = 0.471 × 0.471 mm). For example, 2 SD for the anterior width of the tentorial notch represent an error of 5.22 mm. This represents an error or 5 pixels for each anatomical landmark. Anatomical landmarks used to assess this measure (clinoid process and torcular) have a volume greater than on pixel [START_REF] Dagtekin | Microsurgical anatomy and variations of the anterior clinoid process[END_REF]. This could also explain the lack of reproducibility for the surface of tentorial notch computed from 3D structures.

Correlation

We seek after correlations between the 12 anatomical parameters. We found a significant correlation between the length of falx notch and the length of the tentorial notch (r = 0.62, P < 0.05). No data exist in the literature on correlations between these lengths. These 2 structures both originate from the neural crest, but do not appear at the same time during the embryological development. Falx is first discernible in embryos at the age of 41 days, and tentorium is discernible at 51 days [START_REF] Klintworth | The comparative anatomy and phylogeny of the tentorium cerebelli[END_REF][START_REF] O'rahilly | The meninges in human development[END_REF]. In addition, some authors studying embryos and adult subjects demonstrated that falx and tentorium follow a different growth pattern. The falx has an anteroposterior growth pattern from the frontal part to the occipital part of the cranium, and the tentorium has a centrifugal growth pattern, from both side of the brainstem in the cerebrocerebellar fissure to a fusion in the median plane posteriorly, along the straight sinus [START_REF] Klintworth | The ontogeny and growth of the human tentorium cerebelli[END_REF][START_REF] O'rahilly | The meninges in human development[END_REF].

Conclusion

By precisely defining anatomical landmarks, we showed large variations of the size of falx and tentorium. Because important nervous structures are in close contact with falx and tentorium, their anatomic variations are crucial to better understand the mechanisms of brain herniations. We also propose a reference frame for biomechanical studies of the head.

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflict of interest.

Ethical approval All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee (APHP/CNIL) and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards. For this type of study formal consent is not required. Informed consent was obtained from all individual participants included in the study.

This article does not contain any studies with human participants performed by any of the authors.

  3D computed-tomography (CT) scans were routinely acquired o n a R E V O L U T I O N D I S C O V E RY 7 5 0 H D o r DISCOVERY CT750 (General Electric System, Boston, USA) with voxel resolutions ranging from 0.416 × 0.416 × 1.250 to 0.576 × 0.576 × 1.250 mm.

Fig. 3

 3 Fig. 3 Parameters describing the tentorium cerebelli and its notch. Six anatomical landmarks were used: left anterior clinoid process (C), right anterior clinoid process (D), left parietomastoid point (E), right parietomastoid point (F), tentorial apex (G), and torcular (H). Four distances were calculated: the width of the tentorium (4), the anterior width (5), the maximal width[START_REF] Dagtekin | Microsurgical anatomy and variations of the anterior clinoid process[END_REF], and the length (7) of the tentorial notch. Surfaces were calculated by a triangulation method from projected points for tentorium (sT) and tentorial notch (sTN). The tentorial angle (8) was also recorded

Fig. 1

 1 Fig. 1 Pointing acquisition from CT scan. (a) Example of an axial CT scan used. (b) The falx (squares), the tentorium (points), and venous sinuses (triangles) were specified on axial CT. (c) Computing the points allowed to draw a scatter plot. The anatomical landmarks were identified. Anterior crista galli (A): most anterior and inferior point of the falx. Posterior crista galli (B): most posterior and inferior point of the falx. Left anterior clinoid process (C): most superior and lateral point of the

Fig. 2

 2 Fig. 2 Parameters describing the falx cerebri. Two anatomical landmarks were first determined: posterior crista galli (B) and tentorial apex (G). Three distances were calculated: the height of the falx (1), the height (2), and the length (3) of the falx notch. Two surfaces were calculated by a triangulation method from projected points: the surface of the falx (sF) and the falx notch (sFN)

Table 1

 1 Falx cerebri parameters

		Falx		Falx notch			Surface ratio
		Height (mm) Surface (cm 2 ) Width (mm) Height (mm) Surface (cm 2 )
	Minimum	29	43.2	81.4	27.73	15.8	24.0
	Maximum	48	74.2	113.5	57.72	40.5	45.6
	1st quartile	32.1	50.7	91.5	38.42	25.1	29.9
	Median	33.8	56.8	97.1	40.45	27.3	33.1
	3rd quartile 36.7	59.6	102	144.41	32.1	37.6

Table 2

 2 Tentorium cerebelli parameters

		Tentorium		Tentorial notch				Tentorial angle (°)	Surface ratio
		Width	Surface	Anterior width	Width	Length	Surface
		(mm)	(cm 2 )	(mm)	(mm)	(mm)	(cm 2 )
	Minimum	87.9	45.0	18.0	27.4	45.3	10.6	44.0	13.6
	Maximum	111.6	70.1	36.6	38.2	66.7	16.5	66.0	24.2
	1st quartile	97.1	54.4	23.2	29.0	50.9	11.9	50.6	17.1
	Median	99.0	56.9	25.5	30.4	53.8	13.0	53.5	18.3
	3rd quartile	103.0	62.0	27.4	32.1	58.2	14.3	57.3	20.6
	Mean ± SD	99.6 ± 4.8	57.6 ± 5.9	25.5 ± 3.5	31.0 ± 2.5	55.0 ± 5.3	13.3 ± 1.7	54.0 ± 5.0	18.7 ± 2.3


V Kuchi

P Jardin

C Michel

L Maunoury

M Dubois

P Delahaye

O Bajeat

S Hormigos

V Métayer

M Maccormick

B Roussière

J Guillot

High efficiency ISOL system to produce neutron deficient short-lived alkali RIBs on GANIL/SPIRAL 1 facility

Keywords: neutron-deficient isotopes, fusion-evaporation reaction, ISOL technique, surface ionization, Target Ion Source System, SPIRAL1 Upgrade

published or not. The documents may come

High efficiency ISOL system to produce neutron deficient short-lived alkali RIBs on GANIL/SPIRAL 1 facility

Introduction

The upgraded SPIRAL1 facility [i,ii,iii] aims to extend the production of new radioactive ion beams at GANIL using the Isotope Separator On-Line (ISOL) method [iv]. It provides the opportunity to produce RIBs using a large variety of stable beams, from carbon to uranium, at energies ranging from 4 to 95 MeV/A on different target masses up to Niobium, and with different types of nuclear reactions, such as fragmentation, transfer and fusion-evaporation.

To extend the choice of RIBs available at the SPIRAL1 facility, a technique using fusion-evaporation reaction has been chosen as it favours neutron deficient isotope regions. This region is hardly accessible at other ISOL facilities, where mainly spallation, fragmentation, and fission techniques are used [v,vi]. The fusion reaction requires a low energy primary ion beam which defines the target thickness. The fusion-evaporation channels used here have relatively large cross-sections, of the order of 100 mb and the residual survival probability is optimised by using materials where the thintarget approximation remains valid. In another hand, low energy limits the power deposited in the target, which allows the design of a compact TISS, as the dimensions are closely related to the target size and thus to the primary beam power. In case of short-lived isotopes, the compactness of the TISS is of first importance to limit the release time of the nuclei out of the production system and thus the losses due to their radioactive decay. The objective of the following TISS design was to estimate how efficient could be the TISS performance if it is optimized to favour the atom-to-ion transformation efficiency rather than the number of nuclei produced in the target.

As done within the SPIRAL1 framework, the TISS had also to be as simple as possible to cope with the constraint of operational reliability. The final design and the main related studies are presented in the following paragraphs.

The general principle is outlined in Figure 1. Ions from primary fusion-evaporation processes in a thin self-supporting target are neutralized and thermalized in a "catcher" matrix and undergo temperature-driven diffusion back up to the catcher's surface and into the TISS vacuum cavity. The atoms are accumulated in the cavity and are ionized by the surface-ionization process. An electric field, parallel to the foils, pushes the ions towards an exit hole, beyond which they are strongly accelerated to form a radioactive ion beam. Then they are mass-separated and magnetically selected for experiments.

As ions must be 1)-surface-ionised, 2)-short-lived, 3)-neutron-deficient and 4)-interesting for the SPIRAL1 users, 74 Rb ( = 65 ms) and 114 Cs ( = 570 ms) isotopes have been chosen. They both can be produced with primary beams of 20 Ne and 58 Ni respectively sent on a 58 Ni target.

This approach was first implemented by R. Kirchner and colleagues in the 70's and 80's [vii,viii] with a TISS exploiting fusion-evaporation reactions in a thin production target, and associated with either a FEBIAD ion source or a reduced volume thermal-ion source. In the original design, energetic nuclei produced in the target passed through a window were stopped in a solid catcher, diffused and effused up to an ionizing zone, then guided on out of the source. The cavity was maintained at a high temperature (up to ) to speed up the release of radioactive atoms from the catcher and to reduce sticking time on the TISS walls. Despite the very small cavity volume (of the order of 1 cm ), the atom-to-ion transformation (AIT) time for Kr and Xe was of the order of 1 ms. "Sticking" (effusion) or diffusion times were not considered initially and, had they been so, one would have predicted longer extraction times for chemically reactive elements and low AIT efficiencies for metallic elements with half-lives shorter than 1 ms. In more recent years, gaseous catchers have been designed and improved [ix] to get rid of the sticking time issue but require voluminous vacuum systems, unsuited to SPIRAL1 infrastructure.

Description of the TISS design

The TISS (see Figure 2) mainly consists of two radial foils of 50 mm in diameter intersected by a 12 mm graphite cylinder (0.8 mm thick, 12 mm height and 50 mm in outer diameter). The ring is made of TOYO TANSO graphite (IG-19 type) [x]. The first foil directly impinged by the primary ion beam acts as a target ( 58 Ni) and the second one acts as a catcher for the recoil products and low energy primary beam. The first foil is the production target and is 3 µm or 6.5 µm thick for the production of 74 Rb or 114 Cs respectively. The second foil is made of a flexible graphite "paper", commercially called Papyex®, its microstructure is strongly anisotropic, leading to anisotropic properties between parallel planes and perpendicularly to the plane of the foil. Ni and Papyex® foils are pressed on the cylinder edges with two flanges assembled by screws. This system is the cavity. A current is injected in the walls of the cavity, i.e. walls of the ring and of the foils, through a flexible tantalum band situated at the opposite side from the exit aperture of the TISS. The current leaves the cavity through a graphite wheel connected to the cavity at the exit aperture. Both extremities of tantalum band and graphite wheel are fastened to cooled parts, on the current feedthrough and vacuum chamber respectively.

Owing to the current flowing through the resistive materials of the graphite ring, graphite foil and Ni foil, the cavity is heated (nominally at 1600 K) and an electric field oriented towards the exit aperture is created within the cavity. According to the supplies in the production cave of SPIRAL 1, where this TISS will be installed in fine, 300 A of current is potentially available to heat the cavity walls.

74 Rb ions will be studied using a primary beam of neon 20 Ne (intensity of 1.8.10 13 pps at an energy of 5.5 MeV/A) impinging on the target. In the case of 114 Cs, the primary beam will be 58 Ni (intensity of 3.10 12 pps at an energy 4 MeV/A). The produced isotopes and the primary beam are stopped in the graphite catcher 12 mm beyond the target. The isotopes diffuse and effuse out of the catcher, are ionized by contact with the hot surfaces of the cavity due to their relatively high work function (5 eV for graphite and 4.7 eV for the nickel). Then they are pushed towards the exit hole by an electric field, which reduces the effusion time before their exit from the cavity. The creation of an electric field directly depends on the material resistivity and on the current injected in the TISS. A relevant choice of materials is thus particularly important to simultaneously cope with the constraints of high temperature (melting point, thermal expansion), transport of the electric and beam power (thermal conductivity and emissivity), ionization ability (work function higher than the first ionization potential if possible), sticking time, porosity and electrical resistivity.

Primary beam of heavy ions at ~5.1 MeV/A

Simulations

The three-dimensional finite element model (FEM) shown in Figure 4 has been developed using the ANSYS code. The main purpose of simulations was to study the thermal configuration, the thermomechanical constraints and the electrical potential of the different parts of the TISS cavity. The The results show that the maximum temperature is 1611 K, near to the exit aperture of the cavity. At this temperature, the lifespan of the graphite part is higher than the one required for the on-line operation (2 weeks). This statement is based on the SPIRAL1 experience. The present SPIRAL1 targets irradiated by 1.5 kW of the 40 Ar primary beam and heated at more than 2100 K for two weeks do not present any visible damage. The maximum temperature on the nickel target is lower than 1667 K. This is under the nickel melting point temperature (1728 K), but too closes if we consider an uncertainty of 100 K on the temperature calculation, which is regularly observed during the comparison between experimental and simulated values of temperature. A progressive increase of the ohmic heating while measuring the Ni temperature was thus respected to avoid damaging the Ni foil.

As demonstrated in a previous work [xv], an electric field close to 1 V/cm applied in an ionizing tube of C or Re 4 mm in diameter and 3 cm long can significantly improve the ionization efficiency of alkali atoms. This is explained by a reduction of the probability the atoms have to stick to the tube once ionized. One goal of the present cavity design was to obtain an inner electric field of the same order of magnitude, and eventually slightly lower as the dimensions of the cavity are larger than those of the tube. Obtaining an electric field higher than 1 V/cm in such a large cavity with a maximum current close to 300 A is technically difficult but seemed to be possible. A too important electric field is not desirable as it can limit the distance of the ion jumps and increase the time spend by the ions on the surface. The electric potential within the TISS cavity has been estimated using the electric potential solver in ANSYS Fluent. In Figure 4, the electric potential gradient is similar along the Papyex® foil, the nickel foil and the graphite ring, which was a condition to efficiently lead the ions towards the exit aperture.

The potential difference between the extremities of the TISS cavity is close to 4 V for a current of 300 A. Thus the electric field is close to 0.8 V/cm, as desired.

Monte-Carlo simulations

The effect of the electric field on the effusion time of the ions was calculated with a Monte-Carlo simulation based on the SIMION code [xvi]. A specific routine has been written and introduced in the code. A molecular regime of effusion is assumed. At each collision, the particle sticks to the wall surface (adsorption following the Frankel's equation [xvii]) and then is re-emitted (desorption) into the TISS volume following the cosine Knudsen law distribution [xviii]. The calculation for one ion is stopped when the particle leaves the TISS cavity. For each event, the number of collisions with the surface and the time of flight (TOF) are saved. The mean velocity considered corresponds to the Maxwell-Boltzmann velocity distribution. The effusion time was deduced from the TOF, the sticking time and the mean number of collisions. The results of the simulation for 74 Rb and 114 Cs are presented in Figure 5.

Ni Papyex

Figure 5: Effect of electric field on effusion time. The red and orange lines represent effusion time with and without an electric field for 74 Rb. The green and purple lines indicate the effusion time with and without an electric field for 114 Cs. The sticking times per contact took in the calculations were equal to 0.5 ms and 1 ms for 74 Rb and 114 Cs respectively.

To evaluate the effect of the electric field on the effusion time, the simulations were done with and without an electric field for 74 Rb and 114 Cs. Radioactive decay was not taken into account. Calculated effusion times include the sticking time of rubidium on the graphite, which is equal to 0.5 ms [xix], and which is mainly responsible for the effusion time. The electric field changes the effusion time from ∼100 ms (for 50% of the population) to 12 ms. If the radioactive decay of 74 Rb ( =65 ms) is taken into account, the effusion efficiency without and with electric field goes from 39% to 84%. For cesium, the total effusion time goes from 190 ms down to 27 ms, leading respectively to effusion efficiencies of 75% and 95% if one takes into account the half-life of 114 Cs (0.57 s). For all these calculations, Rb and Cs have been considered as permanently ionized in the cavity.

Total TISS efficiency

Once produced, radioactive isotopes undergo three processes in the formation of ion: diffusion out the target material, effusion and ionization. If one assumes that these processes are sequential, the total efficiency of atom-to-ion transformation is given by:

The release efficiencies of 74 Rb and of 114 Cs out of the catcher are respectively equal to 90% and 97%, according to their diffusion coefficient of 2.10 -8 and 2.5.10 -8 m 2 .sec -1 extracted from [xx]. These high efficiencies are also due to the low recoil energy and thus low implantation depth of the 74 Rb and are equal to 84% and 96%. According to Langmuir equation [xxi], the ionization probabilities per contact for Rb and Cs atoms on graphite surface are equal to ~98% and ~100% respectively at 1600 K. Thus the AIT efficiencies of 74 Rb and of 114 Cs are estimated to be respectively equal to 74% and 93%.

Expected production rates are given in Table 1.

Ongoing tests

The TISS has been installed on the SPIRAL1 test bench at GANIL. The first step is to characterize the thermo-mechanical behaviour of the TISS at an operational temperature of 1600 K. It is installed in a water-cooled vacuum chamber and is heated by the circulation of current passing through the walls of the cavity, foils and graphite ring. Temperatures of the target and catcher were measured using an optical disappearing filament pyrometer [xxiii]. There were two main sources of uncertainty in the temperature measurement: one is related to the operator, adjusting the colour of the filament on the source of light; it is estimated to 1%. One is related to the knowledge of the emissivity of the material considered. In the case of Papyex®, a slight modification of the roughness obtained with a grade 400 (average particles size is 400µm) abrasive paper can make the emissivity vary from 0.44 to 0.46 [xxiv]. In the case of Ni, the emissivity has been taken equal to 0.2. The overall uncertainty is then estimated at 5.3 %. The temperatures of the target and of the catcher are presented in Figure 6 versus the power applied on the cavity. Due to the measurement method, one could observe the Papyex® temperature only through an aperture of 5 mm in diameter performed in the centre of the Ni target (see Figure 7).

Figure 6: Temperature of the target (Nickel, purple squares) and catcher (Papyex®, red circles) versus total power delivered by the power supply. Maximum TISS power is 1560 W.

The temperature of the catcher is higher than the one of Ni, which can be explained by the resistance of the catcher which is lower than the Ni ones. As shown in Figure 7, the temperature of the Ni target is uniform in its central region (~1430 K) and is higher on its periphery (more than 1580 K). This can again be explained by the lower resistance of the graphite ring, leading it to higher heating and temperature.

The potential difference between the extremities of the TISS cavity is equal to 6.5 V, which includes the potential difference on the graphite wheel (0.7 V) and on the tantalum support (1 V). The potential difference present on the TISS cavity is thus equal to 4.8 V, which leads to an electric field of 4.8 V/50 mm close to the targeted value of 1 V/cm. After two days of continuous operation, the nickel target ruptured, first in the hot spot region indicated in Figure 7 and finally all along the perimeter. The region surrounding the exit aperture was identified as the hottest region during the thermal simulation. Despite the risk, the test was performed as the confidence in the absolute determination of the temperature with ANSYS could be subject to an uncertainty of several tens of degrees. This observation will now be taken into account to make the temperature delivered by the code match with the experimental temperature. The design of the cavity will be improved to limit the risk of melting of the Ni foil and to get a more homogeneous temperature throughout the cavity.

Once the window disappeared, the TISS was maintained continuously under operation at a current of 230 A during three weeks without technical problems or observable modification of the voltage/current couple. As expected, this confirms that the graphite parts are reliable for 2 weeks period, as required for online operation. A second test will be performed by 2019 with an upgraded version of the TISS.

Figure 1 :

 1 Figure 1: Principle of the TISS. Radioactive ion production follows steps 1 to 6.

Figure 2 :

 2 Figure 2: Cross-section view of the TISS for the production of 74 Rb isotopesThe nickel target, the Papyex® catcher and the graphite cavity are connected in parallel. In an electrical point of view, their thicknesses have been chosen to avoid short-circuit between two of these three resistors by the third one. As if done so, it would lead to concentrate the electrical heating power on only one resistor, and thus on only one part of the cavity. Owing to the number of parameters involved in the final result and to their interdependency, the final values have been obtained after several iterations using a thermo-electrical simulation described in the following paragraph.The nickel foil and graphite container must have a low open porosity to limit the probability for isotopes to escape. The catcher must present high open porosity to favour the release of isotopes once they are implanted. Recent characterisation of Papyex® porosity [xi] obtained using helium pycnometry technique (Micromeritics-ACCUPYC 1330)[xii] shows that the Papyex® material consists of approximately 49% pores and 51% material. Among the pores, 90% are open and 10% are closed, revealing Papyex® as a good catcher candidate.

  assuming a 20 Ne @ 5.5 MeV/A primary beam power of 308 W, including 38 W lost in the 3 µm nickel target and 270 W deposited in the catcher. A current of 300 A was considered to heat the TISS cavity. In the simulations, electrical resistivity of 1 m.cm [xiii] and 60 µ.cm[xiv], thermal conductivity of 55 W.m -1 .K -1 and 80 W.m -1 .K -1 , and emissivity of 0.45 and 0.2 were taken for Papyex® and Nickel respectively.

Figure 3 :

 3 Figure 3: Thermal simulations of the temperature (°C) over the cavity, taking into account the power deposited by the 20 Ne primary beam in the Ni target (38 W), in the catcher (270 W) and the electrical power of 1350 W.

Figure 4 :

 4 Figure 4: Electric potential along the TISS cavity at a current of 300 A obtained with ANSYS code.

114

  Cs recoils in the catcher material (<12 µm). The effusion efficiencies deduced from the simulations

Figure 7 :

 7 Figure 7: View of the Ni window and of the Papyex® foil through the hole performed in the centre of the Ni foil. The peripheral region of the Ni foil is hotter than the central region.

Table 1 :

 1 Expected 74 Rb and 114 Cs intensities versus production parameters. The in-target production rates are estimated using cross-sections issued from the PACE (Projected Angular-momentum Coupled Evaporation) fusion-evaporation code[xxii].

	Radioactive ion	74 Rb (65 ms)	114 Cs (570 ms)
	Primary beam power	20 Ne (5.1 MeV/A) 308 W	58 Ni (4 MeV/A) 115 W
	Target	58 Ni
	In target production	3.4 10 4	2.4 10 4
	TISS AIT efficiency	74%	93%
	Expected intensity (pps)	2.5 10 4	2.2 10 4


Nicolas Nieuwjaer

Ali Beydoun

Frédéric Lecomte

Bruno Manil

F Cappelluti

Leonardo Guidoni

Debora Scuderi

Charles Desfrançois
email: charles.desfrancois@univ-paris13.fr

IRMPD spectroscopy

Keywords: metal-ligand complexes, acetylacetonate, IRMPD spectroscopy, DFT calculations

come

I. INTRODUCTION

Many recent studies are devoted to functionalized metallic nanoparticles, for many applications: in Physics, due to their localized surface plasmon resonances [START_REF] Cortie | Synthesis and optical properties of hybrid and alloy plasmonic nanoparticles[END_REF]; in Chemistry, due to their specific catalytic properties [START_REF] Turner | Selective oxidation with dioxygen by gold nanoparticle catalysts derived from 55-atom clusters[END_REF]; in Biology, due to their optical [START_REF] Lin | Synthesis of fluorescent metallic nanoclusters toward biomedical application: recent progress and present challenges[END_REF] or magnetic [START_REF] Jung | Functionalized magnetic nanoparticles as chemosensors and adsorbents for toxic metal ions in environmental and biological fields[END_REF] properties coupled to their potential for targeting and vectorization of bioactive molecules within living cells [START_REF] Cobley | Gold nanostructures: a class of multifunctional materials for biomedical applications[END_REF]. In solution, the characterization of the nanoparticle-ligand interface is often performed with NMR or IR methods, but data interpretations are somewhat difficult because of the relative lack of control of the stoichiometry of nanoparticles and of their ligand coverage [START_REF] Zhang | Analytical strategies for characterizing the surface chemistry of nanoparticles[END_REF].

Following a bottom-up approach, we here report on a work to study isolated metal-ligand clusters (few metal atoms, few ligands) that allow for a size selection with mass spectrometry techniques and for a structural characterization in the gas phase by IR spectroscopy. Their specific chemical/physical properties are very dependent on their size and structure and they are intermediate between those of the isolated metal and ligand and those of the liganded nanoparticles in solution [START_REF] Lu | Sub-nanometer sized metal clusters: from synthetic challenges to the unique property discoveries[END_REF][START_REF] Shang | Ultra-small fluorescent metal nanoclusters: Synthesis and biological applications[END_REF]. Although many previous studies have been devoted to organometallic complexes in the gas phase [START_REF] Macaleese | Infrared Spectroscopy of Organometallic Ions in the Gas Phase: from Model to Real World Complexes[END_REF][START_REF] Oomens | Gas-phase metal ion chelation investigated with IRMPD spectroscopy: A brief review of Robert Dunbar's contributions[END_REF][START_REF] Duncan | Infrared spectroscopy to probe structure and dynamics in metal ion-molecule complexes[END_REF][START_REF] Sharpe | Metal-ligand bond energies and solvation energies for gas-phase transition-metal tris (acetylacetonate) complexes and their negative ions[END_REF][START_REF] Hoyau | A Quantitative Basis for a Scale of Na+ Affinities of Organic and Small Biological Molecules in the Gas Phase[END_REF], most of them have concerned simple neutral [START_REF] Caer | Structural characterization of selectively prepared cationic iron complexes bearing monodentate and bidentate ether ligands using infrared photodissociation spectroscopy[END_REF][START_REF] Miller | Entropic Effects on Hydrated Alkali-Metal Cations: Infrared Spectroscopy and ab Initio Calculations of M+(H2O)x=2-5 Cluster Ions for M = Li, Na, K, and Cs[END_REF][START_REF] Fielicke | Vibrational Spectroscopy of CO in Gas-Phase Rhodium Cluster-CO Complexes[END_REF][START_REF] Van Stipdonk | Infrared Multiple Photon Dissociation Spectroscopy of a Gas-Phase Oxo-Molybdenum Complex with 1,2-Dithiolene Ligands[END_REF] or biomolecular ligands [START_REF] Polfer | Infrared Spectroscopy of Phenylalanine Ag(I) and Zn(II) Complexes in the Gas Phase[END_REF][START_REF] Otsuka | Alkali and alkaline earth metal ions complexes with a partial peptide of the selectivity filter in K+ channels studied by a cold ion trap infrared spectroscopy[END_REF][START_REF] Kapota | Vibrational Signature of Charge Solvation vs Salt Bridge Isomers of Sodiated Amino Acids in the Gas Phase[END_REF][START_REF] Boles | Experimental and theoretical investigations of infrared multiple photon dissociation spectra of glutamic acid complexes with Zn2+ and Cd2+[END_REF][START_REF] Günther | IRMPD spectroscopy of metalated flavins: structure and bonding of Mq+-lumichrome complexes (Mq+ = Li+-Cs+, Ag+, Mg2+)[END_REF]. Up to our knowledge, apart from a recent study on holmium complexes [START_REF] Houthuijs | Characterization of holmium(III)-acetylacetonate complexes derived from therapeutic microspheres by infrared ion spectroscopy[END_REF], there is no other previous IR spectroscopy study of isolated complexes between metal ions and anionic acetylacetonate, especially when several metal ions are involved, although they are well known and characterized in their crystalline phase for a long time [START_REF] Diaz-Acosta | Calculated and Experimental Geometries and Infrared Spectra of Metal Tris-Acetylacetonates: Vibrational Spectroscopy as a Probe of Molecular Structure for Ionic Complexes. Part I[END_REF][START_REF] Stabnikov | On the Symmetry of Iron(III)tris-acetylacetonate Crystals[END_REF][START_REF] Slabzhennikov | Normal Vibration Calculations for Iron Tris(acetylacetonate)[END_REF] and although they are often used as precursors for the synthesis of various organometallic complexes and metal oxide nanoparticles [START_REF] Guardia | Controlled Synthesis of Iron Oxide Nanoparticles over a Wide Size Range[END_REF].

We performed IR Multiple Photon Dissociation (IRMPD) spectroscopy experiments on isolated mass-selected metal-ligand ions whose results are compared to Density Functional Theory (DFT) calculations and vibrational spectra computed either at the scaled static harmonic or the unscaled static anharmonic levels or with ab initio molecular dynamics simulations. IRMPD spectroscopy is recognized as a powerful tool for structural characterization of very diverse molecular ions in the gas phase [START_REF] Jasikova | Infrared Multiphoton Dissociation Spectroscopy with Free-Electron Lasers: from Small Molecules to Biomolecules[END_REF][START_REF] Woodin | Multi-photon dissociation of molecules with lowpower continuous wave infrared-laser radiation[END_REF][START_REF] Maitre | Applications of infrared multiple photon dissociation (IRMPD) to the detection of posttranslational modifications[END_REF] through the comparison of infrared signatures with model spectra that are predicted by DFT calculations. Agreement between experimental IRMPD and calculated DFT spectra is generally satisfying, in particular for band positions within tens of wavenumbers, although some larger differences are sometimes observed that can originate either from the experimental method (isomerization during the IRMPD process, red-shift due to the heating, IR intensities depending not only on the linear absorption) and more often from the calculation approximations (DFT method, empirical scaling of harmonic vibrational frequencies…).

In the present study, ions investigated involve light (Al), transition (Fe) and heavy (Ru) metals and one of the simplest anionic bidentate ligands, acetylacetonate (Acac). Starting with small isolated species, our goal is to build some reliable methodology benchmarks for further spectroscopic experiments and DFT calculations concerning larger model systems for real liganded nanoparticles, as used by biochemists. Because of the relative rigidity of the present Acac anion ligand and its strong bidentate interactions with all metal ions, we will see that the low-energy structures of the cluster ions are straightforward in most cases and sometimes almost unique. The present data then constitute mainly a test of the validity of different approaches for simulating IR vibrational spectra for this class of metal-ligand cluster ions.

II. EXPERIMENTAL AND THEORETICAL METHODS

Solid Al(III) and Fe(III) acetylacetonate were purchased from a commercial source (Sigma-Aldrich, purity > 97%) and used without further purification in 1:1 solutions of water and methanol at concentrations of 10 to 100 µM. Solid Ru(III) acetylacetonate was synthesized in the group of J.

Crassous ((Rennes Institute of Chemical Sciences) and used in the same conditions. IRMPD experiments have been performed at the free electron laser (FEL) of the Centre Laser Infrarouge d'Orsay (CLIO) facility, in the 400-1900 cm -1 spectral range. Ion spectra were recorded using the available 7 T hybrid FT-ICR mass spectrometer (APEX-Qe Bruker) coupled to the CLIO FEL beamline. This experimental setup has been previously described in detail [START_REF] Prazeres | Two-color operation of a freeelectron laser and applications in the mid-infrared[END_REF][START_REF] Bakker | Gas-phase structure of a -Allyl-Palladium complex: efficient infrared spectroscopy in a 7 T Fourier transform mass spectrometer[END_REF]. Briefly, ions in solution were delivered in the gas-phase by an electrospray ionization (ESI) source with typical extraction voltage of 4 kV and desolvation temperature of 150-200 °C. Ions were accumulated, premass-selected and thermalized in a quadrupole-hexapole interface and then pulse extracted towards the FT-ICR cell maintained under high vacuum (<10 -9 mbar) and at room temperature. Massselected ions were stored and irradiated for 0.5 to 2 s with the IR-FEL light in the frequency range of interest. Depending on the electron energy used [START_REF] Becke | A new mixing of Hartree-Fock and local density functional theories[END_REF][START_REF] Lee | Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density[END_REF][START_REF] Vosko | Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis[END_REF][START_REF] Ditchfield | Self-consistent molecular-orbital methods. IX. An extended Gaussian-type basis for molecular orbital studies of organic molecules[END_REF][START_REF] Weigend | Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy[END_REF][START_REF] Hay | Ab initio effective core potentials for molecular calculations. Potentials for the transition metal atoms Sc to Hg[END_REF][START_REF] Zhao | The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals[END_REF][START_REF] Cramer | Density functional theory for transition metals and transition metal chemistry[END_REF][START_REF] Daku | Accurate spin-state energetics of transition metal complexes.1. CCSD(T), CASPT2, and DFT study of [M(NCH)[END_REF][START_REF] Møller | Note on an approximation treatment for many-electron systems[END_REF][START_REF] Hutter | cp2k: atomistic simulations of condensed matter systems[END_REF] for covering a given spectral range, the average CLIO laser power is recorded and lies in between 0.5 and 1.2 W. Its frequency resolution, about or lower than 1%, and the spectral rotational broadening of ions at 300 K, lead to an overall linewidth of about 10 cm -1 . If necessary for revealing weak vibrational bands, and in order to increase fragmentation for strongly bound systems, ions were also submitted to an auxiliary broadband CO2 laser synchronized with CLIO-FEL (Universal Laser System, 10 W, cw operation centered at = 10.6 µm), at the beginning of the FEL irradiation period and for times varying between one and few tens of ms. The CO2 laser pulse length was adjusted for each experiment to avoid photodissociation of the molecule by the CO2 laser alone while promoting the fragmentation in the presence of the CLIO laser.

When the IR-FEL light is in resonance with an active vibrational mode of the mass-selected ions, multiple and sequential events of photon absorption and intramolecular vibrational energy redistribution (IVR) ultimately activate their fragmentation [START_REF] Nesbitt | Vibrational energy flow in highly excited molecules: Role of intramolecular vibrational redistribution[END_REF]. At each spectral wavenumber, typically five mass spectra were recorded and accumulated in order to compare the integrated number of precursor (Ip) and fragment (If) ions as a function of the laser frequency. An IRMPD spectrum is then obtained by recording the photofragmentation yield, R = -ln[Ip/(Ip+ΣIf)], as a function of the wavenumber of the IR-FEL radiation.

All static DFT calculations have been performed using the Gaussian 16 package [34] on the HPC platform MAGI of Université Sorbonne Paris Nord. Full geometry optimizations have been performed first at the UB3LYP/6-31G* basic level [START_REF] Becke | A new mixing of Hartree-Fock and local density functional theories[END_REF][START_REF] Lee | Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density[END_REF][START_REF] Vosko | Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis[END_REF][START_REF] Ditchfield | Self-consistent molecular-orbital methods. IX. An extended Gaussian-type basis for molecular orbital studies of organic molecules[END_REF], for different spin multiplicity and starting from different trial structures inspired by those of the crystalline phase. For the Ru atom, we used the Def2TZVP [START_REF] Weigend | Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy[END_REF] or the LanL2DZ [START_REF] Hay | Ab initio effective core potentials for molecular calculations. Potentials for the transition metal atoms Sc to Hg[END_REF] basis set which includes an effective core potential for heavy transition metals. In order to test the validity of the B3LYP functional and the modest 6-31G* basis set, in particular with respect to the spin ground state, we also performed calculations with the M06 functional [START_REF] Zhao | The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals[END_REF], which is supposed to give better energies for transition metal chemistry [START_REF] Cramer | Density functional theory for transition metals and transition metal chemistry[END_REF][START_REF] Daku | Accurate spin-state energetics of transition metal complexes.1. CCSD(T), CASPT2, and DFT study of [M(NCH)[END_REF], at the M06/6-31+G** level. In order to ascertain which spin state is the lowest one, we always performed calculations with different spin multiplicity values around that of lowest total electronic energy. For Al(III), there is no electron in d orbital so that the spin must be zero but for both Fe(III) and Ru(III), there are five d electrons and the resulting spin can thus be in between 1/2 and 5/2, corresponding to one to five unpaired electrons. In the species studied, it appears that all complexes with Fe(III) always display high spin states (5/2), while Ru(III) complexes always correspond to low spin states (1/2). Note that we also performed some ab initio calculations at the UMP2/6-31+G** [START_REF] Møller | Note on an approximation treatment for many-electron systems[END_REF] level which were fully coherent with the DFT energetics and did not bring any interesting added value for IR spectra, so they are not reported here.

Frequency analysis was performed at the same levels of theory either in the harmonic or the anharmonic approximation as implemented in Gaussian 16 [34]. For direct comparison with the measured IRMPD spectra, the calculated vibrational line spectra were convoluted using a Lorentzian line shape function with a full width at half-maximum value of 10 cm -1 . For anharmonic results, no frequency scaling was applied while for harmonic calculations, a frequency scaling factor was determined to fit as best as possible the band positions of the calculated anharmonic spectra. As discussed below, in most cases, a standard scaling factor of 0.975 was suitable both for B3LYP and M06 calculations.

DFT-MD simulations were also performed through Born-Oppenheimer Molecular Dynamics as implemented in the CP2K package [START_REF] Hutter | cp2k: atomistic simulations of condensed matter systems[END_REF] using Galileo and Marconi HPC platforms from Cineca. The cubic box size was set as twice the largest dimension of the complex, therefore around 20-22 Å, depending on the system. IR spectra were obtained by calculating the power spectra of the dipole autocorrelation functions, evaluated using Wannier centers [START_REF] Marzari | Maximally localized generalized Wannier functions for composite energy bands[END_REF] for electron localization. Molecular dynamics simulations in the NVE ensemble have been performed on single molecular ions at the PBE/TZVP [START_REF] Perdew | Generalized gradient approximation made simple[END_REF][START_REF] Schäfer | Fully optimized contracted Gaussian basis sets for atoms Li to Kr[END_REF] level with GTH-SR-MOLOPT pseudopotentials [START_REF] Goedecker | Separable dual-space Gaussian pseudopotentials[END_REF], using a plane wave cutoff of 460 Rydberg and Grimme's D3 correction [START_REF] Grimme | A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu[END_REF] for dispersion interactions (with a 12.5 Angstrom cutoff). Prior to the production simulation time of 20 ps, a 2 ps long NVT equilibration at 298 K has been performed and spectra have been obtained by applying a Lorentzian smoothing with a width of 5 cm -1 but with no frequency scaling. B3LYP-MD simulations have not been conducted due to the too large computational effort required and, for the same reason, we did not perform DFT-MD calculations involving Ru complexes. For comparison with those dynamic calculations and the other static calculations with B3LYP and M06 functionals, static harmonic or anharmonic frequency calculations have also been performed at the same PBE/TZVP level using CP2K with the same computational setup.

III. RESULTS AND DISCUSSION

A. Mass spectrometry

In the condensed phase, metal-acetylacetonate (M-Acac -) complexes are known to form very stable M 3+ (Acac -)3 octahedral neutral species with well-known crystalline structures [START_REF] Diaz-Acosta | Calculated and Experimental Geometries and Infrared Spectra of Metal Tris-Acetylacetonates: Vibrational Spectroscopy as a Probe of Molecular Structure for Ionic Complexes. Part I[END_REF][START_REF] Hon | The Crystal and Molecular Structure of tris(acetylacetonato)-Aluminum(III) and -Cobalt(III)[END_REF][START_REF] Chao | The Crystal and Molecular Structure of Trisacetylacetonatoruthenium(III)[END_REF]. High resolution mass spectrometry allows us to resolve the isotopic patterns of the observed ions. Upon ESI in the positive mode, the main observed ions are the singly charged [M 3+ (Acac -)2(AcacH)] 1+ species that are the cationic counterparts of those neutral in solution, in which one Acac anion has captured one proton and becomes a neutral acetylacetone (AcacH) molecule that can be easily lost by collision-induced dissociation (CID). For Al and Fe (see Fig. 1 and 2 in the supplementary material), we also observed rather high intensities of [M 3+ (Acac -)2] 1+ ions, which can originate from the loss of a neutral AcacH, but also intense [M 3+ 2(Acac -)5] 1+ precursor ions, whose main CID fragments are [M 3+ (Acac -)2] 1+ by loss of stable neutrals M 3+ (Acac -)3. In the case of Al, we also see intense ions at m/z 467.14, which isotopic mass and profile corresponds very well to [Al 3+ 2(Acac - )4OH -] 1+ ions, where one acetylacetonate is replaced by one hydroxyl anion (see Fig. 3 in the supplementary material). For Ru complexes, we also observe [Ru 3+ (Acac -)2(AcacH)] 1+ ions but surprisingly no [Ru 3+ (Acac -)2] 1+ ions, while the main isotopic pattern corresponds to [Ru 2+ (Acac - )1(AcacH)3] 1+ ions (see Fig. 4 and 5 in the supplementary material). We also note that for Ru(IV) we see weak signals corresponding to [Ru 4+ (Acac -)1(AcacH)3] 3+ ions but no signal which could corresponds to [Ru 4+ (Acac -)2] 2+ . It is thus concluded that Ru hardly forms stable complexes with only two Acac ligands but complexes only with three ligands for Ru(III) and four ligands for Ru(II) or Ru(IV), at least from ESI in our experimental conditions. In this report, we will however restrict our results and discussion to M 3+ complexes.

B. [M 3+ (Acac -)2(AcacH)] 1+ , M=Al, Fe, Ru

These species contain three ligands around the metal ion in the oxidation state (III) and are expected to share similar IR spectra corresponding to similar octahedral structures, like the neutral species in the condensed phase, in which the three bidentate ligands form six bonds with the metal ion. As expected, their fragmentation by CID or IRMPD occurs mainly through the loss of the neutral AcacH molecule. Figure 1 displays the experimental IRMPD spectra for the three metals studied. As expected, they share several common features, although the Ru spectrum is red-shifted by 50-70 cm -1 as compared to Al and Fe spectra that are close together, indicating that the Ru-Acac interactions should be stronger than the similar ones with Al and Fe, as discussed below. Experimental data display the two or three runs that have been performed in different frequency ranges and at different IR-FEL laser powers so that bands and lines appear with various relative intensities. Calculated IR intensities are also very different from one method to another and all IR intensities are thus scaled so that the main peak intensity is normalized to about 1 in this figure. All calculated spectra reproduce quite well the position of the two main bands that correspond to different C=O and C-C=O symmetric and asymmetric stretches of neutral or anionic Acac. The only discrepancy concerns the PBE calculations for iron complexes with the band at 1500-1600 cm -1 that is too red-shifted as compared to experiment. It seems that this functional tends to overestimate the interaction between the negatively charge oxygen in Acac -and the aluminum or iron metal ion.

Moreover, there is also a good and even outstanding agreement for the lines of lower intensities and frequencies that appears at 1302, 1163, 1035, very small 946 and 804, and 717 cm -1 for Al and 1293, 1167, 1036, 930, 792 and 677 cm -1 for Fe. Most of these modes can be assigned to either anion Acac -and neutral AcacH species that are respectively more and less red-shifted in the complex. For instance, the line at 1163 cm -1 for Al (or 1167 cm -1 for Fe) corresponds to a red-shifted asymmetric stretching mode involving the five carbon atoms that is calculated in neutral AcacH at 1200 cm -1 .

On the other hand, the line at high frequency, experimentally found at 1733 cm -1 for both Al and Fe, is not so well reproduced by none of the methods used. It is calculated either at too low frequency with PBE or at too high frequencies with B3LYP and M06. This mode corresponds to the symmetric two C=O stretches of neutral AcacH that is calculated at 1755 cm -1 for the isolated neutral. This again indicates that PBE probably overestimates the interactions between metallic cations and neutral oxygen atoms, while B3LYP and M06 underestimate it.

More quantitatively, we have calculated the root-mean-square deviations (RMSD) between the present types of frequency calculations and the experimental values, for the main nine lines and bands that are outlined in Fig. 2 (see tables I and II in the supplementary material). From these values, it clearly appears that static B3LYP scaled harmonic frequency calculations perform much better (RMSD = 15.3 cm -1 for Al and 11.1 cm -1 for Fe) than other methods (RMSD ~ 22 cm -1 with M06 and around 30 cm -1 with PBE) and that unscaled static anharmonic frequency calculation results are always worse than scaled harmonic ones (with B3LYP, RMSD = 28.8 cm -1 for Al and 13.2 cm -1 for Fe). Dynamical DFT-MD frequency calculations do not perform better than static ones (RMSD = 24.2 cm -1 for Al and 34.7 cm -1 for Fe) but we observed a systematic red shift, of the order of 25-30 wavenumbers, that accounts for the main part of those relatively bad RMSD and, when this systematic error is empirically compensated, the RMSD are improved a lot and become the lowest ones (RMSD* = 8.6 cm -1 for Al and 15.6 cm -1 for Fe). We could not find in the literature other examples of this compensation, that can probably be explained as a consequence of the excessive loosening of bonds (and reduction of vibrational frequencies) produced by GGA functionals [START_REF] Sousa | General performance of density functionals[END_REF].

Figure 3 displays the same experimental and calculated IR spectra for Ru, together with the picture of the calculated equilibrium structure. From the two experimental runs displayed, it is clearly seen that pre-heating helps to enhance low intensity lines, particularly the one at higher frequency, but also leads to a widening of all lines, including the two main bands that appear here around 1320 and 1490 cm -1 . For both M06 and B3LYP calculations, these two bands are not as red-shifted, with respect to the previous Al and Fe complexes, as compared to the experiments results: -20 cm -1 instead of 50-70 cm -1 . In that case, PBE functional is performing better, with a calculated red-shift of about 50 cm -1 .

FIG. 3: same as fig. 2 for [Ru 3+ (Acac -)2(AcacH)] 1+ .

The two experimental curves correspond to two runs in the same frequency range but with a CO2 laser pre-heating of ions (red) or not (magenta).

Calculation results are displayed, using LanL2DZ basis set for Ru and either 6-31G* (B3LYP) or 6-31+G** (M06) basis sets for all other atoms. For this complex, no PBE-DFT-MD calculation has been done but only scaled harmonic calculations.

As for the same complexes with Al or Fe, the resulting equilibrium structure displays a symmetric octahedral shape.

As previously, the symmetric two C=O stretches mode of neutral acetylacetone, here at 1665 cm -1 , is still overestimated by at least 50 cm -1 in M06 and B3LYP calculations and largely underestimated, by at least 80 cm -1 in PBE calculations. The calculated equilibrium structure is displayed only for this [Ru 3+ (Acac -)2(AcacH)] 1+ complex but its octahedral shape is also valid for Al and Fe complexes. The three complexes however differ by their M-O distances but also and more noticeably by their dissociation energies, as shown in Table I From these calculated data, it is clear that the M-O distances and the AcacH dissociation energy depend on the size of the metal ion but are even more sensitive to the spin state of the complex. For iron, the ground electronic state is of high-spin 5/2 (all 5 d-electrons are unpaired) but with lower M-O interactions and a dissociation energy just slightly higher (1.63 eV) than that of aluminum with no d-electron (1.50 eV). The higher energy low-spin state 1/2 (four d-electrons paired and only one delectron unpaired) displays stronger M-O interactions (shorter d(M-O) distances) and a much higher dissociation energy (2.45 eV), similar to that of the Ru complex (2.38 eV) whose ground state is the same low-spin state. We conclude that the large red-shift observed in the ruthenium IR spectrum, on several bands involving the electron donor centers of the ligands, in the 1200-1800 cm -1 range, can be used as a signature of strong metal-ligand interactions corresponding to a low-spin state of the complex.

C. [M 3+ (Acac -)2] 1+ , M=Al, Fe

In those species, there are only two anionic bidentate ligands so that their structures are expected to be tetrahedral and strongly bound. Their fragmentation is indeed rather difficult and we never observed the doubly charged [M 3+ (Acac -)1] 2+ . Upon CID, the main fragments correspond to different neutral losses of the ligands while upon IRMPD we also observed a singly-charged fragment corresponding to the loss of one neutral acetylacetonate radical, while it does not appear upon CID.

Figure 4 displays the experimental IRMPD spectra for Al and Fe. Again, experimental data display the two or three runs that have been performed in different frequency ranges without (red) or with pre-heating (magenta and violet) so that spectral bands appear with various relative intensities that are normalized at their maximum.

Compared to previous species, spectra are simpler, as expected because of the higher symmetry due to the fact that all ligands are identical. The two main experimental bands are located around 1345 and 1520 cm -1 for Al and 1300 and 1490 cm -1 for Fe, i.e. 30-50 cm -1 red-shifted as compared to previous species which indicates stronger metal-ligand interactions in those two bidentate ligands complexes. Calculations reproduce quite well the positions of these two main bands, B3LYP performing again better than M06, but static calculations often display a double-peak shape for each experimental band. For those tetrahedral complexes, DFT-MD calculations much better reproduces the experimental spectra both for the band shapes, with only one large peak at the present frequency resolution, and for the positions, with no need of an empirical blue-shift correction as for previous complexes. Only few lines at lower frequencies are hardly detected, only when using CO2 laser preheating of trapped ions, most probably because the dissociative channels are here more endothermic. Again, whatever the theoretical method, calculations well reproduce the position of those lower frequency lines detected, with a slight blue-shift for B3LYP and M06. As expected, ground-state calculated structures (see Fig. 6 in the supplementary material) correspond to a tetrahedral geometry with the same M-O distances for all four oxygen atoms, namely 1.78 Å for Al and 1.88 Å for Fe, i.e. shorter distances than in the previous octahedral complexes, in agreement with the stronger red-shifts observed.

D. [M 3+ 2(Acac -)5] 1+ , M=Al, Fe

In those species, the two metal ions can recover an octahedral environment if one or two bidentate ligands make a bridge between the two cationic centers so that they can share two anionic centers.

Their fragmentation by CID or IRMPD is rather easy and occurs mainly through the loss of the very stable M 3+ (Acac -)3 neutral fragments. Figure 5 displays the corresponding experimental IRMPD spectra for Al and Fe, with two runs that have been performed in the low-and high-frequency ranges.

The two spectra display several similarities in shape and position with those of the mono-metallic [M 3+ (Acac-)2(AcacH)] 1+ complexes (Fig. 2), as expected if they share the same structure symmetry.

The two main experimental bands are located around 1400 and 1560 cm -1 for Al and 1380 and 1580 cm -1 for Fe, i.e. close to the same bands in the mono-metallic ions, with the same small blue-shift of +7 cm -1 for Al and +24 cm -1 for Fe (see Table III in the supplementary material). This is a further indication that they should share the same structure and spin state but with some additional structure constraints in the case of bi-metallic complexes that lead to slightly weaker metal-ligand interactions and thus slightly higher frequencies for those modes involving vibrations of the ligand backbone that are slightly less affected by metal ion binding. This interpretation also holds for the lower frequency modes, below 1100 cm -1 , which are also similar but with small red-shifts in that case (0 to -20 cm -1 ), since they correspond mainly to metal-ligand vibrations that are weaker in the bi-metallic complexes (see Table III, modes 1 to 4, in the supplementary material).

However, we were able to detect one more specific low-frequency band, at 526 cm -1 for Al (very weak) and 447 cm -1 for Fe (stronger), which also appears in the calculations for the lowest double octahedron structure and corresponds to several modes of similar frequencies but all involving the ligands that are shared by the two metal ion centers. The main difference with the mono-metallic complexes comes from the high-energy line (1730 cm -1 ) that is here absent, obviously because there is no neutral ligand.

FIG. 5: experimental IRMPD spectra for [Al 3+ 2(Acac -)5] 1+ and [Fe 3+ 2(Acac -)5] 1+ (bottom curves) as compared to calculated scaled harmonic IR spectra at the B3LYP/6-31G* level, for the lowest energy structure (double octahedron) and for the higher energy configuration (octahedron+plane). Note that the frequency scaling factor was not adjusted but conserved at its previous value of 0.975 and that ferromagnetic coupling was assumed between metal centers.

Finally, we have tested one higher energy structure which consists of a mono-metallic octahedral structure to which a more or less planar [M 3+ (Acac -)2] 1+ species is bound mainly through an interaction between this second metal ion and the C3 carbon atom of an Acac -ligand doubly bound to the first metal ion.

This structure is displayed in Fig. 6, together with the double octahedral structure, but it lies at much higher energy, +72 kJ/mol for Al and +55 kJ/mol for Fe. Moreover, the corresponding calculated spectra exhibit a line at 1690 cm -1 (Al) or 1670 cm -1 (Fe), corresponding to the symmetric C=O stretches of the ligand which C3 atom interacts with one metal ion. Since these lines are not experimentally observed, we rule out the presence of this second conformer in our experiments.

FIG. 6: two equilibrium structures for [Fe 3+ 2(Acac -)5] 1+ , as calculated at the B3LYP/6-31G* level, for the lowest energy conformer (left: double octahedron) and for the higher energy configuration (right: octahedron+plane); in this conformer, there is a specific interaction between an iron ion and a C3 carbon atom of one anionic ligand (dash line in the center), but the total energy is much higher than that of the double octahedron structure.

Finally for Al (as displayed in Fig. 7 of the supplementary material), we were able to obtain the IRMPD spectra of a species at m/z 467 which corresponds either to [Al 3+ 2(Acac -)4OH -] 1+ or to [Al 3+ 2(Acac - )3(AcacH)O 2-] 1+ , i.e. the same complex in which one Acac -ion has been replaced by an hydroxyl OH - anion or two Acac -ions by one neutral AcacH plus a superoxide O 2-anion. From calculations, both isomers are of similar distorted octahedral geometry but the second isomer is located at much higher energy (+172 kJ/mol). In agreement with these calculations, the experimental spectrum fits well to the calculated one only for the first hydroxyl-containing isomer (presence of a line at 920 cm -1 and absence at 1785 cm -1 ).

IV. CONCLUSION

In this work, we have performed IRMPD spectroscopy experiments on strongly bound M 3+ -Acac - complexes with different octahedral or tetrahedral structures. We have shown that the spectral shift of the vibrational modes relative to the O=C-C-C=O part of the Acac ligand, in the frequency range in between 1200 and 1700 cm -1 , is directly related to the strength of the M 3+ -O interactions and to the spin state. The cases of [M 3+ (Acac -)2(AcacH)] 1+ and [M 3+ 2(Acac -)5] 1+ ions are particularly interesting because they can fragment relatively more easily and lead to rich IR spectra even for lowfrequency (400-1000 cm -1 ) low-intensity modes that are characteristic of the ligand structure and, for the lowest frequency ones, of the M 3+ -O interactions too.

Concerning IR spectra simulations, these species are good benchmark systems to compare with different theoretical approaches. In this study, we used DFT calculations with three different functionals (B3LYP, M06 and PBE) and three different spectra calculation (scaled harmonic, static anharmonic and MD simulations). All methods lead to simulated spectra that fit rather well with experimental data, although none of them seems to be always quantitative for the higher fingerprint frequency range (above 1200 cm -1 ), even for the DFT-MD simulations, while the agreement is much better for the low-frequency range (below 1000 cm -1 ). From this observation, we can conclude that, due to the rigidity of the complexes, the spectra calculated using first principles molecular dynamics does not provide a different picture with respect to the static ones. Even a simple and low-cost method, such as scaled static harmonic calculations at the B3LYP/6-31G* level, is already enough to describe the main IR vibrational features that are experimentally observed with the right structure and the right spin state. This conclusion is in good agreement with previous compared IRMPD/theoretical studies [START_REF] Katari | Improved Infrared Spectra Prediction by DFT from a New Experimental Database[END_REF].

This study on simple species allows us to build some methodology benchmarks for further spectroscopic experiments and DFT calculations focused on larger model systems such as real liganded nanoparticles, as used by biochemists. Indeed, several other experiments are underway on more complex ligands and complexes of larger size, in order to further test the above provisional conclusions and to progress towards systems that are more relevant to characterize the physicochemical properties of small nanoparticles.

FIG. 1 :

 1 FIG. 1: IRMPD spectra of the three metal-Acac cations studied. In all experiments, no pre-heating CO2 laser and reduced IR-FEL irradiation times were used (0.25 s for Al, 0.5 s for Fe and 2 s for Ru). As in the following spectra, experimental IR intensities are photofragmentation yields as defined above and scaled in order to be of comparable heights for the main vibrational band.

Figure 2

 2 Figure 2 displays the comparison between experimental and calculated IR spectra for Al and Fe.

FIG. 4 :

 4 FIG. 4: experimental IRMPD spectra for [Al 3+ (Acac -)2] 1+ and [Fe 3+ (Acac -)2] 1+ (top curves) as compared to calculated scaled harmonic (light colors) and unscaled anharmonic (dark colors) IR spectra at the M06/6-31+G** (blue) and B3LYP/6-31G* (black), together with PBE/TZVP calculations performed at the unscaled anharmonic static level (light green) and at the unscaled dynamic level through DFT-MD simulations (dark green). The two main bands around 1300 and 1500 cm -1 are grayed out and few other lines are marked with dotted lines in order to guide eyes.

TABLE I :

 I lower energy spin state, ranges of distances in Å between the metal ion and the oxygen atoms of either anionic Acac -or neutral AcacH ligands and dissociation energy of this neutral ligand, for the three [M 3+ (Acac -)2(AcacH)] 1+ species, calculated at the M06/6-31+G** level. The second line (italic) for iron corresponds to 1/2 spin state, which is of higher total electronic energy but with the same octahedral equilibrium structure and much stronger M-O interactions and a much larger AcacH dissociation energy.

ACKNOWLEDGMENTS

We are grateful to Cineca and PRACE for computational support through grants IsC71_AIMDAcAc and Pra16_3574 and to the MAGI HPC computing center and its engineer Nicolas Grenèche.

We also thank the CLIO facility and its technical staff, in particular Estelle Loire, Fabrice Gobert and Nicolas Jestin for their efficient assistance. The French FT-ICR network (CNRS, FR 3624) is also acknowledged for the access to the MS setup.

Finally, we are also grateful to Jeanne Crassous (Rennes Institute of Chemical Sciences), for providing us a sample of Ru(Acac)3.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

SUPPLEMENTARY MATERIAL

Supplementary material includes full mass spectra of the samples studied together with few isotopic patterns of interest, several tables of experimental and calculated vibrational modes for octahedral mono-and bi-metallic complexes and experimental and calculated IRMPD spectra for species at m/z 467 corresponding to oxidized bi-metallic ions.