V Boudon

C Richard

L Manceron

High-Resolution spectroscopy and analysis of the fundamental modes of 28 SiF 4 . Accurate experimental determination of the Si-F bond length

Keywords: Silicon tetrafluoride, High-resolution infrared spectroscopy, Tensorial formalism, Line positions, Bond length, Integrated band intensities

Silicon tetrafluoride (SiF 4 ) is a trace component of volcanic gases and is gaining industrial importance. However, a better knowledge of spectroscopic parameters is needed for this molecule in order to derive accurate concentrations. Following a previous reinvestigation of the ν 3 and ν 4 fundamentals [J. Quant. Spectrosc. Radiat. Transfer 260 (2021) 107474], we have undertaken an extensive high-resolution study of its infrared absorption bands. We present here an update of this study. It features the recording of new far-infrared spectra, taking advantage of synchrotron radiation, and a global fit giving a consistent parameter set for the ground and fundamental states of 28 SiF 4 , now including the v 1 = 1 and v 2 = 1 states, by adding to the global analysis the newly measured ν 2 , ν 1 -ν 4 , and ν 4 -ν 2 bands near 264, 412, and 125 cm -1 , respectively. Nearly 20 000 assigned lines pertaining to seven distinct rovibrational bands were fitted with an excellent accuracy (global standard deviation of 0.296). This allows us to obtain an accurate experimental value for the Si-F bond length, r e (SiF 4 ) = 1.5516985(30) Å. Some new band intensity estimates allow us to update our SiF 4 calculated spectroscopic database.

Introduction

Silicon tetrafluoride (SiF 4 ) is the most stable of silicon halides and is thus used in several industrial processes in the electronic and optical industry as a precursor for various Si-based materials [START_REF] Wang | Microcrystalline silicon thin films deposited by matrix-distributed electron cyclotron resonance plasma enhanced chemical vapor deposition using an SiF 4 /H 2 chemistry[END_REF]. A renewed interest in the spectroscopic methods applied to this molecule has been spurred by two different domains. It is produced in volcanic activity [START_REF] Francis | Remote determination of SiF 4 in volcanic plumes: A new tool for volcano monitoring[END_REF][START_REF] Mori | High SiF 4 /HF ratio detected in Satsuma-Iwojima volcano's plume by remote FT-IR observation[END_REF][START_REF] Nicotra | Influx of volatiles into shallow reservoirs at Mt. Etna volcano (Italy) responsible for halogen-rich magmas[END_REF][START_REF] Stremme | Gas composition of Popocatépelt volcano between 2007 and 2008: FTIR spectroscopic measurements of an explosive event during quiescent degassing[END_REF][START_REF] Taquet | Continous measurements of SiF 4 and SO 2 by thermal emission spectroscopy: Insight from a 6-month survey at the Popocatépelt volcano[END_REF][START_REF] Taquet | Variability in the gas composition of the Popocatépelt volcano plume[END_REF] and used as a marker of deep magma ingress in some volcanic chambers [START_REF] Stremme | Volcanic SO 2 and SiF 4 visualization using 2-D thermal emission spectroscopy -Part 1: Slant-columns and their ratios[END_REF][START_REF] Krueger | Volcanic SO 2 and SiF 4 visualization using 2-D thermal emission spectroscopy -Part 2: Wind propagation and emission rates[END_REF], therefore detected by remote infrared (IR) spectroscopic methods. Secondly, the necessity of improving some silicon physical properties for the future development of spintronics imposes drastic purity constraints as well as a need to prepare isotopically pure 28 SiF 4 for material synthesis [START_REF] Sennikov | Towards 0.99999 28 Si[END_REF]. Again, monitoring methods using IR spectroscopy will be useful in these developments. If many low resolution IR and Raman studies have located all SiF 4 fundamentals [START_REF] Königer | Molecular constants of SiF 4 , GeF 4 and RuO 4 . An improvement and the analysis of the IR-band contours of ν 3 (F 2 ) low temperature measurements and by using isotopically pure compounds[END_REF], the knowledge of its rovibrational spectroscopy was however restricted until a recent date to a few IR diode high resolution (HR) studies on selected portions of the strong ν 3 band region [START_REF] Mcdowell | Infrared spectrum and potential constants of silicon tetrafluoride[END_REF][START_REF] Takami | Infrared-microwave double resonance spectroscopy of the SiF 4 ν 3 fundamental using a tunable diode laser[END_REF], followed by double resonance experiment to obtain precious information on the ground state spectroscopic constants [START_REF] Jörissen | The ν 3 fundamental of silicon tetrafluoride. Spectroscopy with laser sidebands[END_REF]. Lately, the other IR-active fundamental (the ν 4 bending mode near 388.4 cm -1 ) was observed at HR using synchrotron radiation (SR) [START_REF] Boudon | High-resolution spectroscopy analysis of the ν 3 , ν 4 and 2ν 4 bands of SiF 4 in natural isotopic abundance[END_REF].

In addition to the industrial and geological applications mentioned above, the study of the structural and spectroscopic properties of SiF 4 is an interesting matter in itself, as the quantum chemical treatment of molecules with second-row atoms stills requires some degree of semi-empirical corrections using experimentally determined properties. Two important milestones in this respect are the equilibrium structure and the rovibrational level positions and IR intensities. Regarding the structure, as the molecule is a spherical top with T d symmetry, the equilibrium structure depends on a single parameter, the equilibrium bond length, r e . Following two electron diffraction studies giving two different estimates: r e = 1.555(2) [START_REF] Beagley | The Si-F bond length in SiF 4 -A new electron diffraction study[END_REF] and 1.552(2) Å [START_REF] Hagen | Interatomic distances and rms amplitudes of vibration of gaseous SiF 4 from electron diffraction[END_REF], several spectroscopic studies improved the situation. Since observation of rotational lines in the vibrational ground state (GS) cannot be made in direct absorption, Jörissen et al. [START_REF] Jörissen | The ν 3 fundamental of silicon tetrafluoride. Spectroscopy with laser sidebands[END_REF] reported a B 0 GS value based on their double resonance experiment on some 182 lines. This resulted in an r 0 value estimated at 1.55982 [START_REF] Hagen | Interatomic distances and rms amplitudes of vibration of gaseous SiF 4 from electron diffraction[END_REF] Å. Yet, vibrational corrections must still be made to obtain an estimate of the equilibrium bond length. J. Breidung et al. devoted next a study to test the different approximate methods for deducing the equilibrium structure [START_REF] Breidung | Equilibrium structure of SiF 4[END_REF]. Depending on the method chosen, Ref. [START_REF] Breidung | Equilibrium structure of SiF 4[END_REF] narrowed substantially the range for possible r e values to 1.5524(8) Å. Nevertheless, there is still room for improvement if one can observe rotationally resolved spectra for all four SiF 4 fundamentals in order to carry out a more exact vibrational correction on B 0 to access B e and thus r e .

The advent of improved instrumentation to couple SR [START_REF] Roy | AILES: La ligne infrarouge et THz sur rayonnement synchrotron SOLEIL[END_REF][START_REF] Brubach | Performance of the AILES THz-infrared beamline on SOLEIL for high resolution spectroscopy[END_REF] to HR interferometry, long path cryogenic cells [START_REF] Kwabia Tchana | A new, low temperature long-pass cell for mid-infrared to terahertz spectroscopy and synchrotron radiation use[END_REF] and improved detectors has opened the way to the observation of rovibrationally resolved very weak bands in the mid to far infrared regions (these being either first-order forbidden fundamentals or weak difference bands involving IR-forbidden states [START_REF] Boudon | Resolving the forbidden band of SF 6[END_REF][START_REF] Faye | High-resolution spectroscopy of difference and combination bands of SF 6 to elucidate the ν 3 + ν 1 -ν 1 and ν 3 + ν 2 -ν 2 hot band structures in the ν 3 region[END_REF][START_REF] Faye | Observation and analysis of the SF 6 ν 2 + ν 4 -ν 5 band: Improved parameters for the ν 5 = 1 state[END_REF]). This study is thus devoted to the observation and analysis of weak bands involving the ν 1 and ν 2 fundamentals or, more generally, the v 1 = 1 or v 2 = 1 states. Secondly, an improved estimate of the equilibrium bond length is proposed.

This paper is organized as follows: Section 2 below presents the spectrum measurements, Section 3 details the theoretical model used, Section 4 gives our results for spectrum analyses, equilibrium bond length and band intensity estimates. Finally, Section 5 presents the TFSiCaSDa (Tetra-Fluoro Silane Calculated Spectroscopic Database [START_REF] Richard | Calculated spectroscopic databases for the VAMDC portal: New molecules and improvements[END_REF]) database update.

Experiment

Because of the weakness of the difference bands or multi quanta transitions, a long optical path length at low temperature was required. In the far and mid-IR, several high resolution spectra were recorded on the AILES Beamline at Synchrotron SOLEIL (Saint-Aubin, France). The Synchrotron light source was coupled to the Bruker 125HR interferometer [START_REF] Roy | AILES: La ligne infrarouge et THz sur rayonnement synchrotron SOLEIL[END_REF][START_REF] Brubach | Performance of the AILES THz-infrared beamline on SOLEIL for high resolution spectroscopy[END_REF] using the AILES cryogenic long path cell regulated at 163 K temperature along the entire optical path, set up to 93 m. This setup has been fully described elsewhere [START_REF] Kwabia Tchana | A new, low temperature long-pass cell for mid-infrared to terahertz spectroscopy and synchrotron radiation use[END_REF]. For these spectra, the SR from the SOLEIL facility (500 mA in 316 bunches mode) was used as a source, without limitation of the interferometer entrance aperture.

For the very weak difference bands analysed in this study, a spectrum with a gas pressure of 14.9 mbar of SiF 4 (Orano, France, better than 99.99 % purity) was recorded at 0.002 cm -1 resolution in the 60-800 cm -1 region, using boxcar apodization, 5.06 cm/s scanner velocity, a 6 µm Si-Mylar beamsplitter and a 4K-cooled Si composite bolometer with a 1.5 ms rise time and a cold, 700 cm -1 low-pass filter. 576 interferograms were averaged in a total recording time of about 38 hours and processed against a background of the empty cell taken at the same 163 K temperature and 0.01 cm -1 resolution. Another spectrum was taken at a lower pressure (0.44 mb) to avoid saturation for the ν 2 band region. A third spectrum was recorded at 0.002 cm -1 resolution using a Ge-KBr beamsplitter and a homemade high sensitivity 4 K-cooled HgCdTe detector [START_REF] Faye | Improved mid infrared detector for high spectral or spatial resolution and synchrotron radiation use[END_REF] using the same 93 m length (see Table 1). Spectra were zero-filled, corrected for channelling effects and calibrated using well-known water and carbon dioxide or carbonyl sulphide rovibrational lines [START_REF] Rothman | The HITRAN 2004 molecular spectroscopic database[END_REF] with a standard deviation of about 3 × 10 -4 cm -1 .

Theoretical model

Vibrational polyads

As already mentioned in our previous work [START_REF] Boudon | High-resolution spectroscopy analysis of the ν 3 , ν 4 and 2ν 4 bands of SiF 4 in natural isotopic abundance[END_REF], silicon tetrafluoride is a tetrahedral spherical top molecule with T d point group symmetry at equilibrium and possesses four normal modes of vibration: ν 1 (non-degenerate, A 1 symmetry 800.7 cm -1 ), ν 2 (doubly degenerate, E symmetry, 264.2 cm -1 ), ν 3 and ν 4 (triply degenerate, F 2 symmetry, 1031.5 and 388.4 cm -1 , respectively). The wavenumbers of these modes are such that the molecule exhibits a nicely defined vibrational structure, with polyads every ca. 130 cm -1 . More precisely, a vibrational level (v 1 , v 2 , v 3 , v 4 ) pertains to polyad

P k if k = 6v 1 + 2v 2 + 8v 3 + 3v 4 . (1)
In this scheme, there is no polyad number 1. The ground state (GS) is the vibrational level (0, 0, 0, 0), and constitutes is polyad P 0 , then level v 2 = 1 (0, 1, 0, 0) constitutes polyad P 2 , etc. Figure 2 schematically displays these polyads. It should be noticed that, along with methane (CH 4 ) [START_REF] Boudon | The vibrational levels of methane obtained from analyses of highresolution spectra[END_REF][START_REF] Albert | Global analysis of the high resolution infrared spectrum of methane 12 CH 4 in the region from 0 to 4800 cm -1[END_REF][START_REF] Amyay | Vibration-rotation energy levels and corresponding eigenfunctions of 12 CH 4 up to the tetradecad[END_REF], silicon tetrafluoride is one of the very few spherical-top molecules with such a well-defined polyad structure.

Effective Hamiltonian operator

Due to the high symmetry of the molecule, we use the tensorial formalism and group theory methods developed in the Dijon group [START_REF] Champion | Spherical top spectra[END_REF][START_REF] Boudon | Spherical top theory and molecular spectra[END_REF][START_REF] Wenger | XTDS and SPVIEW: Graphical tools for the analysis and simulation of high-resolution molecular spectra[END_REF]. Let us just recall briefly the principles of this model.

We are here considering an XY 4 molecule for which the vibrational levels can be grouped into a series of polyads named P k with k = 0, . . . , n, P k=0 being the ground state (GS). The Hamiltonian operator is written as follows (assuming that some perturbative treatment such as a contact transformation [START_REF] Vleck | On σ-type doubling and electron spin in the spectra of diatomic molecules[END_REF][START_REF] Sadovskii | Contact transformations in tensor formalism. Effective Hamiltonian and dipole moment for the ν 2 , ν 4 dyad of tetrahedral molecules[END_REF][START_REF] Champion | Spherical top spectra[END_REF] has been performed to eliminate inter-polyad interactions):

H = H {P 0 ≡GS} + H {P 1 } + . . . + H {P k } + . . . + H {P n-1 } + H {Pn} , (2)
where the different H {P k } terms are expressed in the following form:

H {P k } = all indexes t Ω(K,nΓ)ΓvΓ v {s}{s } β ε V Ωv(ΓvΓ v )Γ {s}{s } ⊗ R Ω(K,nΓ) (A 1 ) . (3)
In this equation, the t

Ω(K,nΓ)ΓvΓ v {s}{s }
are the parameters to be determined, while ε V

Ωv(ΓvΓ v )Γ {s}{s }
and R Ω(K,nΓ) are vibrational and rotational operators, respectively. For each term, Ω v and Ω represent the degree in elementary vibrational operators (creation a + and annihilation a operators), and rotational operators (components J x , J y and J z of the angular momentum), respectively. β is a factor that allows the scalar terms (terms with Γ = A 1 , the totally symmetric irreducible representation of T d ) to match the "usual" contributions like B 0 J 2 , etc. The order of each individual term is defined as Ω + Ω v -2. We then deal with effective Hamiltonians which are obtained, for a given polyad P k , by the projection of H on the P n Hilbert subspace:

H <Pn> = P <Pn> HP <Pn> (4) = H <Pn> {GS} + H <Pn> {P 1 } + . . . + H <Pn> {P k } + . . . + H <Pn> {P n-1 } + H <Pn> {Pn} .
In the present study, we will use data from the transitions indicated in red on Figure 2. We thus use the following effective Hamiltonians:

• The ground state effective Hamiltonian (polyad P 0 ),

H <P 0 > = H <P 0 > {P 0 } . (5)
• The ν 2 bending fundamental effective Hamiltonian (polyad P 2 ),

H <P 2 > = H <P 2 > {P 0 } + H <P 2 > {P 2 } . (6)
• The ν 4 bending fundamental effective Hamiltonian (polyad P 3 ),

H <P 3 > = H <P 3 > {P 0 } + H <P 3 > {P 2 } + H <P 3 > {P 3 } . (7)
• The ν 1 stretching fundamental effective Hamiltonian (in polyad P 6 ),

H <P 6 (ν 1 )> = H <P 6 (ν 1 )> {P 0 } +H <P 6 (ν 1 )> {P 2 } +H <P 6 (ν 1 )> {P 3 } +H <P 6 (ν 1 )> {P 4 } +H <P 6 (ν 1 )> {P 5 } +H <P 6 (ν 1 )> {P 6 (ν 1 )} . ( 8
)
• The ν 3 stretching fundamental effective Hamiltonian (in polyad P 8 ),

H <P 8 (ν 3 )> = H <P 8 (ν 3 )> {P 0 } + H <P 8 (ν 3 )> {P 2 } + H <P 8 (ν 3 )> {P 3 } + H <P 8 (ν 3 )> {P 4 } + H <P 8 (ν 3 )> {P 5 } + H <P 8 (ν 3 )> {P 6 (ν 1 )} +H <P 8 (ν 3 )> {P 7 } + H <P 8 (ν 3 )> {P 8 (ν 3 )} . (9)
For the last two cases, the ν 1 and ν 3 fundamental bands are part of larger polyads (P 6 and P 8 , respectively), owing to the polyad scheme definition in Equation ( 1); however, the present effective Hamiltonien fit (see below) shows no clear perturbation of v 1 = 1 and v 3 = 1 from the other polyad levels. As a consequence, we use here restricted versions of polyads P 6 and P 8 by including only the fundamental levels. Intra-polyad interactions may be included in a future study, when transitions to the other levels in these polyads (which are likely to be weak transitions) will be assigned, which is not the case at present. Polyads P 4 , P 5 and P 6 are included as defined in Equation ( 1) and on Figure 2 in the calculation and fit jobs, but their parameters are not used in the present fits so that their values can be kept fixed to zero.

Effective dipole moment operator

In order to calculate line intensities, we use the dipole moment effective operator, which is expanded in a similar way as for the effective Hamiltonian (see previous section). This development is explained in Refs. [START_REF] Champion | Spherical top spectra[END_REF][START_REF] Boudon | Spherical top theory and molecular spectra[END_REF][START_REF] Boudon | Spherical top theory and molecular spectra[END_REF].

Since we do not consider here absolute intensities for the ν 4 -ν 2 and ν 1 -ν 4 transitions and only approximate intensities for ν 2 -GS (and ν 1 -GS, see below), the dipole moment operator was expanded at the minimum order for these three transitions, that is 1. The ν 2 fundamental band of E symmetry is forbidden in first approximation but gains intensity through the coupling with other bands (presumably mainly ν 4 in this case, as for many other tetrahedral molecules like CH 4 [START_REF] Champion | Analysis of the ν 2 /ν 4 dyad of 12 CH 4 and 13 CH 4[END_REF], CF 4 [START_REF] Carlos | High-resolution spectroscopy and global analysis of CF 4 rovibrational bands to model its atmospheric absorption[END_REF], GeH 4 [START_REF] Ulenikov | High resolution analysis of GeH 4 in the dyad region: Ro-vibration energy structure of 70 GeH 4 and line strengths of M GeH 4[END_REF][START_REF] Richard | Line positions and intensities for the ν 2 /ν 4 bands of 5 isotopologues of germane near 11.5 µm[END_REF], RuO 4 [START_REF] Reymond-Laruinaz | High-resolution infrared spectroscopy and analysis of the ν 2 /ν 4 bending dyad of ruthenium tetroxide[END_REF], OsO 4 [START_REF] Louviot | High-resolution spectroscopy and structure of osmium tetroxide. A benchmark study on 192 OsO 4[END_REF], etc.).

In the present study, we do not analyze it in an interacting dyad, but rather as an isolated band since it appears to constitute the single vibrational level polyad P 2 in the scheme define above. As a consequence, interactions with other bands are small perturbations that can be considered as negligible for line position calculations; isolating the polyads by treating inter-polyad interactions through contact transformations would lead to a first order effective dipole moment operator:

µ ν 2 -GS = µ 1 ν 2 R 1(1,0F 1 ) ⊗ -V 1(A 1 E)E {GS}{ν 2 } (F 2 ) . (10)
We provide below an estimate of the associated parameter µ 1 ν 2 , see Section 4.5

As for the ν 4 -ν 2 difference band, we have an order 1 purely vibrational effective dipole moment operator:

µ ν 4 -ν 2 = µ 1 ν 4 -ν 2 + V 1(EF 2 )F 2 {ν 2 }{ν 4 } , (11)
while for the ν 1 -ν 4 difference band, we have in the same way also at an order 1:

µ ν 1 -ν 4 = µ 1 ν 1 -ν 4 + V 1(F 2 A 1 )F 2 {ν 4 }{ν 1 } . ( 12
)
For these two bands, this thus amounts to only one parameter (µ 1 ν 4 -ν 2 and µ 1 ν 1 -ν 4 , respectively) whose value is simply set to 1 in order to calculate relative intensities.

We show below in Section 4.4 that the totally symmetric ν 1 stretching fundamental with A 1 symmetry is itself slightly observable. Again, this is a "forbidden band", but rovibrational couplings with other bands, as for GeH 4 [START_REF] Boudon | Line positions and intensities for the ν 3 band of 5 isotopologues of germane for planetary applications[END_REF] or OsO 4 [START_REF] Louviot | High-resolution spectroscopy and analysis of the ν 1 /ν 3 stretching dyad of osmium tetroxide[END_REF], etc., can give it some small intensity. Contact transformations would lead to an order 2 effective dipole moment operator:

µ ν 1 -GS = µ 2 ν 1 R 2(2,0F 2 ) ⊗ + V 1(A 1 A 1 )A 1 {GS}{ν 1 } (F 2 ) , (13)
with one parameter, µ 2 ν 1 . In the case of the ν 3 -GS and ν 4 -GS transitions, their effective dipole moment operator is expanded exactly as described in Section 3.2 of our previous SiF 4 paper [START_REF] Boudon | High-resolution spectroscopy analysis of the ν 3 , ν 4 and 2ν 4 bands of SiF 4 in natural isotopic abundance[END_REF]. This allows to calculate absolute intensities of the ν 3 and ν 4 bands in order to update the TFSiCaSDa database, see Section 5 below.

Basis sets and line intensities

The calculation of the effective Hamiltonian and effective dipole moment matrix elements are performed in the coupled rovibrational basis

Ψ (Cv) v ⊗ Ψ (J,nCr) r (C) σ , (14)
where Ψ (J,nCr) r is a rotational wavefunction with angular momentum J, rotational symmetry species C r and multiplicity index n; Ψ (Cv) v is a coupled vibrational basis set with symmetry species C v ; C is the overall symmetry species (C = C v ⊗ C r ), with component σ. In the present case, Ψ (Cv) v contains the relevant functions for the ν 1 , ν 2 , ν 3 or ν 4 normal modes of vibration ,

Ψ (Cv) v σv = Ψ (C 1 ) v 1 ⊗ Ψ (l 2 ,C 2 ) v 2 (C 2 ) ⊗ Ψ (l 3 ,n 3 C 3 ) v 3 (C 23 ) ⊗ Ψ (l 4 ,n 4 C 4 ) v 4 (Cv) σv , (15)
with:

• For the non-degenerate mode ν 1 , we use a symmetrized one-dimensional harmonic oscillator basis set denoted

ψ (C 1 ) v 1 = |v 1 , C 1 , (16)
with v 1 = 1 for the fundamental band, v 1 being the usual vibrational quantum numbers and C 1 = A 1 .

• For the doubly degenerate mode ν 2 , we use a symmetrized doubly degenerate harmonic oscillator basis set denoted

ψ (l 2 ,C 2 ) v 2 σ 2 = |v 2 , l 2 , C 2 , σ 2 , (17)
with v 2 and l 2 being the usual vibrational and vibrational angular momentum quantum numbers, respectively; in the case of the fundamental band, v 2 = l 2 = 1 and C 2 = E, σ 2 = 1 or 2.

• For the triply degenerate modes ν 3 and ν 4 , we use a symmetrized triply degenerate harmonic oscillator basis set denoted

ψ (l i ,n i C i ) v i σ i = |v i , l i , n i , C i , σ i , (18)
with i = 3 or 4, v i and l i being the usual vibrational and vibrational angular momentum quantum numbers, respectively, while n i is a multiplicity index; v i = l i = 1 and

C i = F 2 , σ i =
x, y or z in the case of fundamental bands.

In Eq. ( 15), C 23 is an intermediate symmetry species:

C 23 = C 2 ⊗ C 3 . (19)
The effective Hamiltonian matrix expressed in basis ( 14) is diagonalized numerically, and this leads to eigenfunctions obtained from

H Ψ (J,C,α) σ = E Ψ (J,C,α) σ , (20)
where α = 1, 2, . . . numbers functions with the same symmetry C within a given J block. This eigenbasis set can be expanded in terms of the initial rovibrational basis set [START_REF] Jörissen | The ν 3 fundamental of silicon tetrafluoride. Spectroscopy with laser sidebands[END_REF] and is used to calculate the matrix elements (in Debye) of the Z component µ Z of the effective dipole moment operator µ, in the laboratory-fixed frame. This µ Z component is related to µ through Stone coefficients [START_REF] Stone | Transformation between Cartesian and spherical tensors[END_REF] and the direction cosines tensor as explained in Section 6.1. of Ref. [START_REF] Boudon | Spherical top theory and molecular spectra[END_REF].

The absolute line intensity (especially in the case of the ν 3 or ν 4 band for the TFSiCaSDa database [START_REF] Boudon | High-resolution spectroscopy analysis of the ν 3 , ν 4 and 2ν 4 bands of SiF 4 in natural isotopic abundance[END_REF], see Section 5 below) at temperature T for a transition at wavenumber ν if (in cm -1 ) between an initial state i (with energy E i , in Joules) and a final state f is then obtained through:

S if (cm -1 /(molecule.cm -2 )) = 1 4πε 0 ν if 8π 3 3hcQ e - E i k B T 1 -e - hc ν if k B T R if , (21)
with

R if = 3 α i ,α f Ψ (J i ,C i ,α i ) σ f µ Z Ψ (Jf,Cf,αf) σ f 2 . ( 22
)
Q is the total partition function at temperature T [START_REF] Mcdowell | Rotational partition function for spherical-top molecules[END_REF], c the speed of light in vacuum, h Planck's constant and k B Boltzmann's constant. The line strength S if is expressed here in the so-called "HITRAN unit" [START_REF] Gordon | The HITRAN2016 Molecular Spectroscopic Database[END_REF].

Analysis and discussion

Line positions

We have expanded the effective Hamiltonian up to order 6 for all polyads from P 0 to P 8 (but polyads P 4 , P 5 and P 7 are not used is the present study, see Section 3.1).

Since the aim of the present work is to get accurate effective Hamiltonian parameters for the ground and all four fundamental vibrational states (v i = 1 with i = 1 to 4), we chose to assign and fit together three of the far infrared bands that can be seen in Figure 1, that are ν 2 -GS, ν 1 -ν 4 and ν 4 -ν 2 , together with the ν 3 -GS, ν 4 -GS fundamental bands from our previous paper [START_REF] Boudon | High-resolution spectroscopy analysis of the ν 3 , ν 4 and 2ν 4 bands of SiF 4 in natural isotopic abundance[END_REF]; as in this one, we also included microwave rotational transitions from GS-GS and ν 3 -ν 3 taken from the paper of Jörissen et al. [START_REF] Jörissen | The ν 3 fundamental of silicon tetrafluoride. Spectroscopy with laser sidebands[END_REF].

Some other bands (for instance ν 3 -ν 1 , ν 2 -ν 1 or ν 3 -ν 4 ) shown in Figure 1 were not considered here, being either too weak or overlapping with nearby overtone or combination bands. These ones will be considered at a later stage, when performing an even more global study of full SiF 4 polyads up to the near-infrared region. Isotopologues 29 SiF 4 and 30 SiF 4 were not considered here either, the far-infrared region being somewhat noisy and their lines being too weak to be identified.

The well isolated ν 2 spectrum was quite easy to assign, starting with approximate loworder parameters set up by hand and then assigning and fitting more and more lines. Assignments were performed using the SPVIEW program [START_REF] Wenger | XTDS and SPVIEW: Graphical tools for the analysis and simulation of high-resolution molecular spectra[END_REF] and spectrum fits, calculations and simulations were obtained through the XTDS program suite [START_REF] Wenger | XTDS and SPVIEW: Graphical tools for the analysis and simulation of high-resolution molecular spectra[END_REF] dedicated to the modeling of highly-symmetrical molecules and implementing the tensorial formalism described above.

Since the v 4 = 1 level is well-known from Ref. [START_REF] Boudon | High-resolution spectroscopy analysis of the ν 3 , ν 4 and 2ν 4 bands of SiF 4 in natural isotopic abundance[END_REF], it was easy to perform an initial simulation of the ν 1 -ν 4 difference band; the structure of the totally symmetric v 1 = 1 level is very simple and one can just start by setting by hand its band center. The band was then straightforward to assign.

In the same way, once a first fit gave reliable parameters for v 2 = 1, it was easy to simulate and assign the ν 4 -ν 2 difference band. This is the lowest band we could observe in our spectra (125 cm -1 region).

We could thus perform a global fit of almost 20 000 lines pertaining to seven transitions of 28 SiF 4 : GS-GS (P 0 -P 0 ), ν 2 -GS (P 2 -P 0 ), ν 4 -GS (P 3 -P 0 ), ν 4 -ν 2 (P 3 -P 2 ), ν 1 -ν 4 (P 6 -P 3 ), ν 3 -GS (P 8 -P 0 ) and ν 3 -ν 3 (P 8 -P 8 ). The resulting 59 effective Hamiltonian parameters are given in Table 2. Figure 3 details the fit residuals for line positions for these seven types of transitions, along with some statistics. We can see that we obtain a very satisfying result. In the case of the ν 2 , ν 3 and ν 4 bands, some systematic deviations are observed at high rotational quantum number J values (similarly to what was already observed for ν 3 and ν 4 in Ref. [START_REF] Boudon | High-resolution spectroscopy analysis of the ν 3 , ν 4 and 2ν 4 bands of SiF 4 in natural isotopic abundance[END_REF]), but these stay very small ; this may indicate some small perturbations that will be treated at a later stage, when we perform a more global study.

Spectrum simulations and energy levels

In this section, we illustrate our results by showing some spectrum simulations, compared to the experiment, for the new bands recorded in the far-infrared region. The resulting simulations for ν 3 and ν 4 are extremely similar to those obtained in our previous paper dedicated to these the strongly absorbing fundamentals [START_REF] Boudon | High-resolution spectroscopy analysis of the ν 3 , ν 4 and 2ν 4 bands of SiF 4 in natural isotopic abundance[END_REF].

Figure 4 shows the comparison of the experimental and simulated spectrum for ν 2 , with a detail in its P branch being shown in Figure 5. The fine structure of the band is nicely reproduced.

Figure 6 shows the reduced rovibrational energy levels for the v 2 = 1 vibrational level of 28 SiF 4 , defined by

ν red = ν - Ω t Ω(0,0A 1 )A 1 A 1 {GS}{GS} (J(J + 1)) Ω/2
= ν -B 0 J(J + 1) + D 0 J 2 (J + 1) 2 -H 0 J 2 (J + 1) 4 + L 0 J 6 (J + 1) 6 . . . , [START_REF] Faye | High-resolution spectroscopy of difference and combination bands of SF 6 to elucidate the ν 3 + ν 1 -ν 1 and ν 3 + ν 2 -ν 2 hot band structures in the ν 3 region[END_REF] i.e. we subtract the dominant scalar polynomial terms in order to enhance levels splittings due to molecular symmetry. We give both the calculated and observed reduced energy levels. Observed levels are simply levels reached by assigned transitions which are included in the fit. This gives a good idea of the sampling of the energy spectrum. This one looks quite good here, up to high J values.

Figures 7 and8 compare experiment and simulation for the ν 4 -ν 2 and ν 1 -ν 4 difference bands, respectively. Some inserts show again that the fine structure is very well modeled.

Equilibrium bond length

The determination of the equilibrium bond length r e of a spherical-top molecule (which is the unique geometrical parameter that defines its equilibrium structure) is possible if one knows the value of the rotational constant B 0 in the ground state and rotational constant differences in all the vibrational fundamental levels, say ∆B i = B i -B 0 (i = 1 to 4), where B i is the rotational constant in an excited state with v i = 1 [START_REF] Boudon | High-resolution Raman spectroscopy of the ν 1 region and Raman-Raman double resonance spectroscopy of the 2ν 1 -ν 1 band of 32 SF 6 and 34 SF 6 . Determination of the equilibrium bond length of sulfur hexafluoride[END_REF]. The formula giving B e (equilibrium value) is [START_REF] Louviot | High-resolution spectroscopy and structure of osmium tetroxide. A benchmark study on 192 OsO 4[END_REF]:

B e = B 0 - 1 2 4 i=1 d i ∆B i , (24)
where d i is the normal mode degeneracy (d 1 = 1, d 2 = 2, d 3 = d 4 = 3). The equilibrium bond length is then [START_REF] Louviot | High-resolution spectroscopy and structure of osmium tetroxide. A benchmark study on 192 OsO 4[END_REF]:

r e = 3h 64π 2 cm F B e , (25)
h being Planck's constant, c the speed of light in vacuum and m F the fluorine atom mass, whose values are taken from Ref. [START_REF] Cohen | Quantities, Units and Symbols in physical chemistry[END_REF]. The fitted parameters

B 0 = t 2(0,0A 1 )A 1 A 1 {0}{0}
and ∆B i = t 2(0,0A 1 )ΓvΓv {i}{i} (i = 1 to 4) result from the line position fits in Table 2. Using these fitted values, we get:

B e = 0.13819608(52) cm -1 , (26)
and thus:

r e (SiF 4 ) = 1.5516985(30) Å. ( 27
)
Although this is determined from 28 SiF 4 only, it should be recalled that, within the Born-Oppenheimer approximation, structural parameters like r e do not depend on the isotopologue. It was not possible, using the present data, to assign lines in the ν 2 and ν 1 -ν 4 bands of the 29 SiF 4 and 30 SiF 4 isotopologues since they appeared too weak in our spectra (contrary to the case of the ν 3 and ν 4 very strong fundamental band for which lines of these low-abundance isotopologues were assigned in our previous work [START_REF] Boudon | High-resolution spectroscopy analysis of the ν 3 , ν 4 and 2ν 4 bands of SiF 4 in natural isotopic abundance[END_REF]). However, we could get a very high (1σ) accuracy, due to the very precise B 0 and ∆B i values obtained in the present fit of 19 432 high-resolution infrared transitions (see Table 2). Somewhat higher values can be found in rather old references, as mentioned in the Introduction. The CRC Handbook of Chemistry and Physics [START_REF]CRC handbook of chemistry and physics[END_REF] reports an experimental value r e = 1.553 Å of unknown origin, while Hagen and Hedberg [START_REF] Hagen | Interatomic distances and rms amplitudes of vibration of gaseous SiF 4 from electron diffraction[END_REF] found r e = 1.552(2) Å using electron diffraction. Beagley et al. [START_REF] Beagley | The Si-F bond length in SiF 4 -A new electron diffraction study[END_REF] report r e = 1.555(2) Å using the same technique. The most accurate study was up to now the theoretical one of Breidung et al., using an ab initio calculation and giving r e = 1.5524(8) Å [START_REF] Breidung | Equilibrium structure of SiF 4[END_REF]. This last value is compatible with ours, although just at the limit of the uncertainty interval.

Direct observation of the ν 1 fundamental band

In the case of SiF 4 , the ν 1 fundamental band itself appears to be very weak. As a "forbidden" band (in first approximation) it gets some intensity from coupling with other modes, as in the case of ν 2 . But ν 1 is difficult to observe ; it is also very close to the strong 2ν 4 overtone and partially blended by it. We could, however, observe it in our mid-infrared spectrum that will be exploited in a future paper.

As shown in Figure 9, the accurate ν 1 parameters obtained in the work allowed us to simulate ν 1 (using a single order 2 effective dipole moment parameter, see Section 3.3 and below) and this compares well with the experiment. The region is, however very noisy and the ν 1 lines are too weak to be correctly exploited for the present purpose of an accurate determination of fundamental level parameters; these ones are well fitted thanks to the ν 1 -ν 4 difference band lines. It is however very satisfying that ν 1 is correctly predicted and observed.

Estimates of fundamental band intensities

It is interesting for future theoretical studies to estimate, even roughly, the order of magnitude of the band intensities for all fundamental bands. In the case of the A 1 and E symmetry fundamental regions (IR-forbidden in a first approximation but weakly activated by rotational resonances, see Section 3.3), we did not performed sufficient measurements to attempt a true, line-by-line intensity analysis, but can draw a rough estimate from the total integrated intensities at low temperatures.

So, the situation for all four fundamental bands summarizes as follows, in decreasing order of accuracy:

• In our previous study [START_REF] Boudon | High-resolution spectroscopy analysis of the ν 3 , ν 4 and 2ν 4 bands of SiF 4 in natural isotopic abundance[END_REF], we could make a rather accurate estimate of the ν 3 strong stretching fundamental integrated intensity of 691 km/mol, leading to a dipole moment derivative µ 0 ν 3 = 0.5444(38) D (see Table 3 of [START_REF] Boudon | High-resolution spectroscopy analysis of the ν 3 , ν 4 and 2ν 4 bands of SiF 4 in natural isotopic abundance[END_REF]).

• This same reference [START_REF] Boudon | High-resolution spectroscopy analysis of the ν 3 , ν 4 and 2ν 4 bands of SiF 4 in natural isotopic abundance[END_REF] did not include the case of the ν 4 bending fundamental intensity. But we can make here an estimate of 125 [START_REF] Brubach | Performance of the AILES THz-infrared beamline on SOLEIL for high resolution spectroscopy[END_REF] km/mol for this band from our previous measurements. This last value compares well with the value of 133 km/mol in Ref. [START_REF] Burtsev | Integral intensities of absorption bands of silicon tetrafluoride in the gas phase and cryogenic solutions: Experiment and Calculation[END_REF]. It allows to estimate the dipole moment derivative to be µ 0 ν 4 0.3706 D. As this is just a quick estimate that does not use individual line strengths, we do not provide any uncertainty. The simulation for 28 SiF 4 shows a good overall agreement as illustrated by Figure 10. This result has been used to update our TFSiCaSDa database [START_REF] Richard | Calculated spectroscopic databases for the VAMDC portal: New molecules and improvements[END_REF], see Section 5.

• For the E-symmetry ν 2 band, we can derive a band intensity of about 0.8(1) km/mol.

We can thus estimate the effective dipole moment parameter for the ν 2 band, as described by Eq. ( 10) in Section 3.3, to be roughly µ 1

ν 2 6.2 × 10 -4 D.
• For the estimation of the very weak ν 1 band, which is severely overlapped by the IRallowed 2ν 4 band (0.08 km/mol [START_REF] Boudon | High-resolution spectroscopy analysis of the ν 3 , ν 4 and 2ν 4 bands of SiF 4 in natural isotopic abundance[END_REF]), the situation is more difficult. We could just tune the dipole moment parameter (see Eq. ( 13) in Section 3.3) by hand and got a value of roughly µ 2 ν 1 4.7 × 10 -7 D, leading to an estimate of the ν 1 integrated intensity of 0.02 km/mol. This should be more viewed as an order of magnitude than an accurate determination.

TFSiCaSDa database update

Dijon spectroscopic databases [START_REF] Richard | Calculated spectroscopic databases for the VAMDC portal: New molecules and improvements[END_REF] contain calculated spectroscopic line lists for line positions, line intensities, full rovibrational transition labels and much more. These calculations are derived from experimental spectroscopic analyses where lines are assigned using the SPVIEW software [START_REF] Wenger | XTDS and SPVIEW: Graphical tools for the analysis and simulation of high-resolution molecular spectra[END_REF] while Hamiltonians and dipole moments parameters are fitted with XTDS [START_REF] Wenger | XTDS and SPVIEW: Graphical tools for the analysis and simulation of high-resolution molecular spectra[END_REF] software. By September 2021, 8 molecules are described in 8 databases. These databases are developed in the framework of the international consortium VAMDC (Virtual Atomic and Molecular Data Centre, http://vamdc.org) [START_REF] Dubernet | Virtual Atomic and Molecular Data Centre[END_REF][START_REF] Dubernet | the VAMDC Consortium, The Virtual Atomic and Molecular Data Centre (VAMDC) consortium for astrophysics[END_REF][START_REF] Moreau | The VAMDC portal as a major enabler of atomic and molecular data citation[END_REF][START_REF] Albert | A decade with VAMDC: Results and ambitions[END_REF] and are also part of the Dat@OSU project (http://dataosu.obs-besancon.fr).

The present work allowed us to extend our SiF 4 database (TFSiCaSDa) [START_REF] Richard | Calculated spectroscopic databases for the VAMDC portal: New molecules and improvements[END_REF] with the addition of 147 333 new lines that belong to the ν 4 band, whose absolute intensity parameter has been estimated in this work (see Section 4.5). So far, and for the sake of simplicity, we do not use the polyad scheme as described in Section 3.1 to construct the database and both bands are treated separately. Table 3 presents the result of the database contents. The calculated lines have thus been introduced using two different schemes for the three isotopologues. Scheme 1 stands for ν 3 (i.e. P 1 is just the v 3 = 1 level) and scheme 2 for the newly included ν 4 band (i.e. P 1 is just the v 4 = 1 level). Our webpage (http: //vamdc.icb.cnrs.fr/PHP/SiF4.php) allows to plot the data and download two sorts of file formats: the line by line list is given following the HITRAN 2004 format [START_REF] Rothman | The hitran 2004 molecular spectroscopic database[END_REF], while the cross section list is a simple 2-column flat file. An overview of the TFSiCaSDa database is displayed in Figure 11, as downloadable from our website.

Conclusion

In a continued effort to analyze and to provide reliable spectroscopic simulations of the infrared spectrum of SiF 4 , we have recorded new far-and mid-infrared high-resolution spectra for this molecule. The new data were used to obtain accurate effective Hamiltonian parameters for all the fundamental bands of the 28 SiF 4 isotopologue. These results allowed to perform an accurate determination of the Si-F bond length. We also provided several band intensity estimates and updated the VAMDC/TFSiCaSDa database.

It is noticeable that far-infrared high-resolution spectroscopy provides invaluable spectroscopic information for spherical-top molecules. Traditionally, the "forbidden" fundamental bands ν 1 and ν 2 of such species were studied with high-resolution stimulated spectroscopy, as in our previous works on CF 4 [START_REF] Boudon | High-resolution stimulated Raman spectroscopy and analysis of the ν 1 , 2ν 1 -ν 1 , ν 2 , 2ν 2 , and 3ν 2 -ν 2 bands of CF 4[END_REF], GeD 4 [START_REF] Pierre | Study of the fundamental bands of 70 GeD 4 by high-resolution Raman and infrared spectroscopy: First experimental determination of the equilibrium bond length of germane[END_REF], SF 6 [START_REF] Bermejo | Simultaneous analysis of the ν 2 Raman and ν 2 + ν 6 infrared spectra of the SF 6 molecule[END_REF][START_REF] Boudon | High-resolution Raman spectroscopy of the ν 1 region and Raman-Raman double resonance spectroscopy of the 2ν 1 -ν 1 band of 32 SF 6 and 34 SF 6 . Determination of the equilibrium bond length of sulfur hexafluoride[END_REF][START_REF] Boudon | High-resolution stimulated Raman spectroscopy and analysis of the ν 2 , ν 5 and 2ν 6 bands of 34 SF 6[END_REF] or OsO 4 [START_REF] Louviot | High-resolution stimulated Raman spectroscopy and analysis of the ν 1 band of osmium tetroxide[END_REF]. This technique, however, is not very sensitive and has a quite limited resolution (around 3 × 10 -3 cm -1 ). In the case of the totally symmetric ν 1 fundamental band, only a Q branch can be observed, which limits the amount of lines that can be assigned. Far-infrared absorption spectroscopy can provide higher resolution and larger spectral coverage. In the case of the v 1 fundamental level, if this one can be accessed through a difference band whose other level is already known (like v 3 = 1 for the ν 3 -ν 1 band for SF 6 [START_REF] Faye | High-resolution spectroscopy of difference and combination bands of SF 6 to elucidate the ν 3 + ν 1 -ν 1 and ν 3 + ν 2 -ν 2 hot band structures in the ν 3 region[END_REF] or v 4 = 1 for the ν 1 -ν 4 band of SiF 4 in the present study), this gives access to P , Q and R lines, that is, to many more lines than in the Raman case.

Such results could stimulate future theoretical studies, by challenging ab initio calculations. From our side, we intend to continue the analysis of SiF 4 further, by including as many combination and overtone bands as possible, using the polyad scheme defined in Section 3.1. The aim is to perform a consistent global fit like in the case of CH 4 [START_REF] Amyay | Global analysis of the high temperature infrared emission spectrum of 12 CH 4 in the dyad (ν 2 /ν 4 ) region[END_REF][START_REF] Amyay | New investigation of the ν 3 C-H stretching region of 12 CH 4 through the analysis of high temperature infrared emission spectra[END_REF], GeD 4 [START_REF] Boudon | High-resolution stimulated Raman spectroscopy and analysis of the ν 3 stretching band of GeD 4[END_REF], CF 4 [START_REF] Carlos | High-resolution spectroscopy and global analysis of CF 4 rovibrational bands to model its atmospheric absorption[END_REF] or SiF 4 [START_REF] Faye | The high overtone and combination levels of SF 6 revisited at doppler-limited resolution: A global effective rovibrational model for highly excited vibrational states[END_REF]. Such a global fit approach is especially adapted to a molecule with a well-defined polyad structure, which is the case of SiF 4 (see Section 3.1 and Figure 2). We also intend to try to study more bands of the 29 SiF 4 and 30 SiF 4 less-abundant isotopologues. When complete, this will allow to provide improved synthetic line lists in order to help remote-sensing in situ measurements for this species. 28) R( 29) R( 30) R( 31) P( 28) P( 27) P( 26) P( 25) 35) R( 36) R( 37) R [START_REF] Carlos | High-resolution spectroscopy and global analysis of CF 4 rovibrational bands to model its atmospheric absorption[END_REF] Experiment Simulation, ν 4 Simulation, ν 1 -ν 4

Figures

  Figures

Figure 1 : 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 10 P 11 Figure 2 :

 112345678910112 Figure1: Overview of the far-infrared spectra of SiF 4 used in this paper. We can notice the presence of some impurities (CO 2 , Si 2 F 6 O and another unknown one, marked by "?"). Here, we focus on ν 2 , ν 4 -ν 2 and ν 1 -ν 4 (on the upper edge of ν 4 ); the other combination and overtone bands will be treated in a future paper.

3 Figure 3 :

 33 Figure 3: Fit residuals for line positions for the 7 different types of rovibrational transitions used in the present work.

Figure 4 :

 4 Figure 4: Overview of the ν 2 fundamental band, compared to the simulation. At least 3 Q branches from hot bands (not considered here) are visible.

Figure 7 :

 7 Figure 7: The ν 4 -ν 2 difference band, compared to the simulation. The inserts display a few line clusters in the P and R branches.

Figure 8 :

 8 Figure 8: The R branch of the ν 1 -ν 4 difference band, compared to the simulation including also high-J lines of the ν 4 fundamental band. The insert displays a few line clusters, from J = 35 to 38, showing the excellent agreement.

  ) to R[START_REF] Boudon | High-resolution stimulated Raman spectroscopy and analysis of the ν 2 , ν 5 and 2ν 6 bands of 34 SF 6[END_REF]

Figure 9 :

 9 Figure 9: Mid-infrared transmission spectrum showing the weak ν 1 band, compared to the simulation. The very intense lines on the left correspond to the much stronger 2ν 4 overtone band. The insert shows part of the ν 1 R branch.

Figure 10 :

 10 Figure 10: Experimental spectrum of the ν 4 band from Ref. [15] (at 160 K), compared to the simulation using the dipole moment derivative estimated in the present study.

Figure 11 :

 11 Figure 11: Extracted line by line list plotted on graphs as obtained from the TFSiCaSDa web page (https:// vamdc.icb.cnrs.fr/PHP/SiF4.php). Both bands ν 3 and ν 4 , corresponding to the entirety of the database, are represented on two separate panels, for the bending (top) and stretching (bottom) regions, for the three isotopologues. It should be noticed that the database does not include the isotopic abundance.

  Figure 5: Detail in the P branch of the ν 2 fundamental band, compared to the simulation. Lines not present in the simulation are hot band lines, not considered here.Figure 6: Calculated and observed rovibrational energy levels for the v 2 = 1 fundamental level. Observed levels are those reached by assigned transitions.

		1.6										R(		Simulation
		1.4				28 SiF 4 / calculated						28 SiF 4 / observed
				264.2		P(36)			P(35)		264.2			P(34)	P(33)
		1.2		264							264
			263.8 117.0	117.2	117.4	117.6	117.8	118.0		263.8	132.0	132.2	132.4	132.6	132.8	133.0
			-1						-1
		1.0	cm	263.6					cm	263.6
			/						/
	Absorbance	0.8	wavenumber	263.2 263.4 Simulation			wavenumber	263.2 263.4			Experiment
		0.6	Reduced	262.8 263					Reduced	262.8 263
		0.4		262.6							262.6
				262.4							262.4
		0.2
				262.2							262.2
					0 10 20 30 40 50 60 70 80			0 10 20 30 40 50 60 70 80
		0.0		Experiment	J							J
			254.0 110		254.2 115		120	254.4	125	254.6	130	.8 135	140
									Wavenumber / cm Wavenumber / cm -1 -1

Table 1 :

 1 Experimental conditions for the SiF 4 bands analyzed in this study.

	Tables

Table 3 :

 3 Rovibrational transitions in TFSiCaSDa.

	Isotopologue Transitions	# lines Wavenumber range	Dipolar intensity range
				/cm -1	/(cm -1 /(molecule cm -2 ))
		Scheme 1 (ν 3 )
	28 SiF 4	P 1 -P 0	17 139	1010 -1049	8×10 -24 -2×10 -20
	29 SiF 4	P 1 -P 0	23 660	1000 -1041	8×10 -24 -2×10 -20
	30 SiF 4	P 1 -P 0	22 269	993 -1032	8×10 -24 -2×10 -20
		Scheme 2 (ν 4 )
	28 SiF 4	P 1 -P 0	44 564	356 -426	8×10 -25 -2×10 -21
	29 SiF 4	P 1 -P 0	45 113	354 -424	8×10 -25 -2×10 -21
	30 SiF 4	P 1 -P 0	57 656	353 -421	8×10 -25 -2×10 -21
		Total	210 401

Acknowledgments

The authors wish to warmly thank Mr. Alex Jourdan and Mrs. Julie Muller from the Orano Group, France, for providing the SiF 4 sample that was used to record all the spectra presented in this work, as well as Synchrotron SOLEIL for supporting this work (Project 20200349).

Table 2: Fitted effective Hamiltonian parameters for the fundamental levels fo 28 SiF 4 . Standard deviation is given in parentheses in the unit of the last two digits.

Level Ω(K, nC) {Γv } {Γ v } Value / cm -1 "Usual notation" [START_REF] Herranz | The rotational structure of the fundamental infrared bands of methane-type molecules[END_REF][START_REF] Robiette | The effective vibration-rotation Hamiltonian for triply-degenerate fundamentals of tetrahedral XY 4 molecules[END_REF][START_REF] Champion | Spherical top spectra[END_REF] (when available) GS 2(0, 0A 1 ) 0000A 1 0000A 1 1.3778055( 14)