Andrei V Nikitin

Michael Rey

Iana S Chizhmakova

Vladimir G Tyuterev

V E Zuev

FIRST FULL-DIMENSIONAL POTENTIAL ENERGY AND DIPOLE MOMENT SURFACES OF 𝐒𝐅 𝟔

Keywords: SF6, ab initio, potential energy surface, dipole moment surface, low resolution spectra Potential Energy Surface, Dipole moment surface, SF 6, Octahedral Group octahedron

A 15-dimensional analytical form for the potential energy and dipole moment surfaces of the SF6 molecule in the ground electronic state is obtained using ab initio methods. In order to calculate the equilibrium S-F distance, we applied coupled cluster CCSD(T) method and several versions of the correlation consistent basis sets -from valence triple-zeta (VTZ) and augmented valence triple-zeta (AVTZ) to core-valence quadruple-zeta (CVQZ) with Douglas-Kroll (DK) relativistic corrections that provided good agreement with empirical equilibrium value. Ab initio electronic energies on 15D grids of nuclear geometries are computed using CCSD(T) method with VTZ and CVQZ-DK basis sets. The analytical representation of the potential energy surface is determined through an expansion in symmetry adapted products of non-linear coordinates up to the 5-th order. The influence of additional redundant coordinates on the quality of the fit was investigated. Parameters of full dimensional dipole moment surfaces are determined up to the 4th order expansion in normal mode coordinates. For validation of ab initio results, the fundamental vibration frequencies and absorption cross-sections of the principal sulfur hexafluoride isotopolgue are calculated giving good agreement with cold (180 K) and room temperature ( 296 K) experimental spectra. Absorption cross-sections calculated from our preliminary line list agree much better with these observations than the simulations using SF6 line-by-line lists constructed from effective models included in currently available spectroscopic databases.

Introduction

Investigation of electronic structure and radiative properties of high octahedral symmetry sulfur hexafluoride ( SF 6 ) molecule are of interest both for the theory [START_REF] Pedraza | Etching-enhanced Ablation and the Formation of a Microstructure in Silicon by Laser Irradiation in an SF6 Atmosphere[END_REF][START_REF] Johnstone | Absolute elastic differential cross sections for electron scattering from SF6[END_REF][START_REF] Malik | A Review of Electrical Breakdown in Mixtures of SF6 and Other Gases[END_REF] and for applications in many domains of industry and medicine. Because of strong absorption bands in the atmospheric transparency windows, SF 6 is a powerful greenhouse gas with the global warming potential (GWP500) by 29300 times larger than that of CO 2 [START_REF] Hodnebrog | Global Warming Potentials and Radiative Efficiencies of Halocarbons and Related Compounds: a Comprehensive Review[END_REF]. At present, the sulfur hexafluoride atmospheric concentration is yet small, but SF6 is one of the most stable gaseous chemicals with an atmospheric life time of 3200 years [START_REF] Ravishankara | Atmospheric Lifetimes of Long-Lived Halogenated Species[END_REF]. According to the conclusions of Earth System Research Laboratory Global Monitoring Division [6], the SF 6 concentration increased from 3.5ppt (1995) to 9.5ppt (2018). A rapid increase of the SF 6 concentration and its long atmospheric life time might result in a considerable accumulation of SF6 in the troposphere.

Thus, in future, sulfur hexafluoride may play an important role in the climate global changing.

The radiative properties of SF 6 were insufficiently studied up to now. Because of a relatively low-frequency vibrations [START_REF] Rey | Understanding global infrared opacity and hot bands of greenhouse molecules with low vibrational modes from first-principles calculations: the case of CF4[END_REF], the Boltzmann population of the excited vibrational levels appears to be significant. Thus, the 'hot bands" corresponded to transitions between excited vibrational states produce a considerable contribution to IR absorption even at room temperature. The ro-vibrational spectra are quite complex and crowded making it difficult to identify isolated lines in experimentally recorded absorption samples. Only few vibrational bands of SF 6 within limited spectral ranges have been studied [START_REF] Boudon | First High-Resolution Raman Spectrum and Analysis of the ν5[END_REF][START_REF] Boudon | High-Resolution Spectroscopy and Analysis of the v4 Bending Region of SF6 near 615 cm -1[END_REF] [10] [START_REF] Boudon | The high overtone and combination levels of SF6 revisited at Doppler-limited resolution: A global effective rovibrational model for highly excited vibrational states[END_REF] [12] using the effective Hamiltonian approach.

Despite many efforts, a simulation of the room-temperature SF6 absorbance using effective spectroscopic models was not sufficiently complete. For this reason, the most advanced spectroscopic databases (as HITRAN [START_REF] Gordon | The HITRAN2016 molecular spectroscopic database[END_REF] or GEISA [START_REF] Jacquinet | The 2015 edition of the GEISA spectroscopic database[END_REF] ) do not contain line parameters for all necessary SF6 bands in order to describe the absorbance at room temperature with high accuracy.

An approach based on ab initio potential energy surfaces (PES) and dipole moment surfaces (DMS) using variational methods for the solution of the full-dimensional nuclear motion Schrödinger equation, has been successfully applied to many 4-5 atomic [START_REF] Yurchenko | Ab initio dipole moment and theoretical rovibrational intensities in the electronic ground state of PH3[END_REF] [16] [START_REF] Huang | Rovibrational spectra of ammonia. II. Detailed analysis, comparison, and prediction of spectroscopic assignments for 14 NH3, 15 NH3, and 14 ND3[END_REF] [18] [START_REF] Owens | Accurate ab initio vibrational energies of methyl chloride[END_REF] [20] [START_REF] Nikitin | First fully ab initio potential energy surface of methane with a spectroscopic accuracy[END_REF] [22] [START_REF] Rey | Understanding global infrared opacity and hot bands of greenhouse molecules with low vibrational modes from first-principles calculations: the case of CF4[END_REF], and also to six-atomic [START_REF] Delahaye | A new accurate ground-state potential energy surface of ethylene and predictions for rotational and vibrational energy levels[END_REF] [24] [START_REF] Rey | First theoretical global line lists of ethylene ( 12 C2H4) spectra at various temperatures in the far-infrared range for quantification of absorption and emission in planetary atmospheres[END_REF] molecules like C2H4. In [START_REF] Manzhos | Computing the Anharmonic Vibrational Spectrum of UF6 in 15 Dimensions with an Optimized Basis Set and Rectangular Collocation[END_REF] some calculated levels of UF6 molecule having the same octahedral symmetry have been reported. However, high precision full-dimensional PESs and DMSs for seven-atomic molecules are still lacking in the literature.

Several works used alternative methods for the nuclear motion calculations for medium and large molecules using vibrational configuration-interaction approach or the perturbation theory ( [START_REF] Burcl | Vibrational spectra of furan, pyrrole, and thiophene from a density functional theory anharmonic force field[END_REF] [28] [START_REF] Charmet | Disentangling the IR spectra of 2,3,3,3-tetrafluoropropene using an ab initio description of vibrational polyads by means of canonical Van Vleck perturbation theory[END_REF] and references herein). However in most cases the rotation states were computed for low J only and the contribution of hot bands have been neglected.

To compute energy levels and IR spectra beyond the harmonic oscillator approximation using ab initio techniques, it is necessary to obtain either the PES on a sufficiently large grid of nuclear displacements or the force field with high order derivatives. Both techniques are often used, but for high-resolution applications the technique based on an analytical PES determined at an extended geometrical grid has proved to produce more accurate results [START_REF] Nikitin | First fully ab initio potential energy surface of methane with a spectroscopic accuracy[END_REF] [30] [START_REF] Manzhos | Computing the Anharmonic Vibrational Spectrum of UF6 in 15 Dimensions with an Optimized Basis Set and Rectangular Collocation[END_REF]. In the full-grid approach, the computational difficulties are related to a simultaneous account of relatively large number of electrons (70 electrons in case of SF6) and of large number of degrees of freedom including representative nuclear geometries for the PES and DMS parametrization.

As a rule, a number of different geometries is several times bigger than the number of the fitted PES parameters. The force field technique often uses the finite difference method for calculating the force constants, and as a rule, a validity of the obtained force field could not be checked for sufficiently large nuclear displacements. To reduce the number of nuclear geometries in the electronic structure calculations, it is necessary to take into account the molecular symmetry.

This paper is devoted to first calculations of the full-dimensional PES and DMS of SF6 molecule and is structured as follows. Equilibrium structure at various basis sets, the coordinate choice and analytical model for PES and DMS surfaces are considered in Section 2 whereas ab initio calculations and fits are discussed in Sections 3 and 4. Calculation of the vibrational-rotation spectra using the variation techniques [START_REF] Rey | Ab initio ro-vibrational Hamiltonian in irreducible tensor formalism: a method for computing energy levels from potential energy surfaces for symmetric-top molecules[END_REF] [32] [33] [34] shows a good agreement with experimental absorption cross-sections in the range of the strongest bands of 32 SF6 and 34 SF6 isotopologues ( section 5), providing a preliminary validation of ab initio results.

The equilibrium geometry and analytical form of PES and DMS

The SF6 molecule belongs to the octahedral Oh point group: at the equilibrium geometry, the molecule is a cube with the S atom at the cube center, and six F atoms at centers of each edge (Fig. 1). Thus, only one geometrical parameter (S-F distance) determines this equilibrium structure unambiguously. Experimental and theoretical studies of the SF6 equilibrium geometry have been considered in [START_REF] Nagarajan | Statistical Thermodynamics Enthalpy, Free Energy, Entropy, and Heat Capacity of Some Hexafluorides of Octahedral Symmetry[END_REF][START_REF] Spiridonov | A practical method for diffraction analysis of equilibrium geometries molecules without refined force fields[END_REF]. In order to calculate the equilibrium S-F distance, we applied coupled cluster CCSD(T) method and several versions of the correlation consistent basis sets - 1.5647 Empirical value [START_REF] Miller | Mean amplitudes of vibration of SF6 and intramolecular multiple scattering[END_REF] 1.5622

*Preliminary result of empirical optimization of S-F distance using variational calculations of rotation energies [32] [7]

Two calculations with two basis sets CVQZ and CVQZ-DK give values closed to the empirical ones. In any case without additional corrections it is impossible to calculate the equilibrium geometry with a precision better than 0.001 A°. Here we use the symmetric coordinates and the molecule orientation from the work [START_REF] Chizhmakova | Potential Energy Surface of SF6[END_REF]. Initially, we begin with linear combinations of bond lengths and bond angles, which will be further adapted using non-linear elementary morse-type and cosine-type functions as described in Section 3. At the first step, six "radial" (stretching) symmetrized coordinates were used to accommodate three irreducible representations: 1g

( ) / 8 u Fx SA q q q q q q q q         (4) 2 (
( ) / 8 u Fx SA q q q q q q q q         (6)
Some of these qij angles in Fig 1 , which take π/2 values at the equilibrium are called "adjacent" angles. Another ones, which take π values at the equilibrium are called "opposition" angles. The number of possible angles is bigger than the number of symmetrized angular coordinates. So two types of additional symmetrized angular coordinates can be added: three coordinates S of symmetry A1g, Eg composed of "opposition" angles q14, q25, q36 or three coordinates S of the same symmetry A1g, Eg from 12 "adjacent" angles :

25 36 14 " ( 2 ) / 6 a E S q q q    36 25 " ( ) / 6 b E S q q  (7) 1 25 36 14
" ( 3 " ) / 3 g Ae S q q q q     2 ) / 24 a E S q q q q q q q q q q q q             g Ae S q q q q q q q q q q q q q

' ( ) / 8 b E S q q q q q q q q         (8)
            
Here re is the equilibrium values for the S-F distance, " e q and ' e q are equilibrium values for "adjacent" or "opposite" angles. The additional angular coordinates can significantly accommodate the fit of the shape of the surface near the bottom of the potential well that will affect the energy levels and wave functions. In the case of methane, this has been shown in the Refs [START_REF] Nikitin | First fully ab initio potential energy surface of methane with a spectroscopic accuracy[END_REF].

The complete set of coordinates and the Oh group together with the matrices of irreducible representations (irrep) were reported in Ref [START_REF] Chizhmakova | Potential Energy Surface of SF6[END_REF]. As explained in [START_REF] Chizhmakova | Potential Energy Surface of SF6[END_REF] , our set of internal angular coordinated is different from that of refs [START_REF] Schatz | Bond Moments and Derivatives in CF4, SiF4, and SF6 from Infrared Intensities[END_REF], [START_REF] Fernandez-Gomez | Calculation of internal valence force constants for XY6(Oh) Octahedral molecules[END_REF], [START_REF] Pistorius | Potential Field and Force Constants of Octahedral Molecules[END_REF]. The definition of the radial symmetrized coordinates (1-3) coincides with that of [START_REF] Halonen | A local mode model for tetrahedral molecules[END_REF]. Matrix representation of group elements [START_REF] Chizhmakova | Potential Energy Surface of SF6[END_REF] (see Table 1 of Appendix 1 for matrices) coincides with that of ref. [START_REF] Delibas | Point groups in solid state physics I: point group Oh[END_REF] as given in our Table 2.

Inversion

Transformation matrix ( 0 0 1 1 0 0 0 1 0 An account of the symmetry and the use of the irreducible tensor representation permitted obtaining an optimal non-redundant grid of nuclear geometries. One grid adapted for the analytical PES development up to the 6-th order (6412 geometries) has been constructed in the work [START_REF] Chizhmakova | Potential Energy Surface of SF6[END_REF] using the geometry set generation technique already applied for other symmetric molecules [49] [50]. Here we use also extended grid of 78739 geometries that unambiguously define the PES development up to 8-th order constructed by the same technique. These two grids of geometries allowed us to carry out ab initio electronic energy calculations for a wide range of nuclear displacements. Initially, the grids were constructed in symmetrized coordinates (1-6) but for ab initio calculations both grids were recalculated in internal coordinates as in ref [START_REF] Chizhmakova | Potential Energy Surface of SF6[END_REF]: six vectors connecting the S atom with the F atoms; six lengths of the S-F bonds {𝑟 1 , 𝑟 2 , 𝑟 3 , 𝑟 4 , 𝑟 5 , 𝑟 6 }; five angles between the bonds {𝑞 12 , 𝑞 13 , 𝑞 14 , 𝑞 15 , 𝑞 16 }, and four torsion angles {𝑡 23 , 𝑡 24 , 𝑡 25 , 𝑡 26 } (see Fig. 1).

) ( 0 -1 0 1 0 0 0 0 1 ) ( -1 0 0 0 -1 0 0 0 -1 ) Equivalent permutations of F atoms (1 to 6) ( 123456
In the analytical expressions for PES it is recommended to use the symmetry coordinates that are linear combinations of the internal coordinates or of 1D elementary functions of these internal coordinates. A seven-atomic molecule has 15 vibrational degrees of freedom. The standard representation in terms of the irreducible tensor operators (ITO) [START_REF] Champion | Spectroscopy of the Earth's Atmosphere and Interstellar Medium[END_REF] [42] was used to obtain analytical expressions for 15-dimensional (15D) PES and DMS developments. Three stretching vibrational modes 1 ( non-degenerate at 774.5 cm -1 ), 2 (two-fold degenerate at 643.3 cm -1 ) and 3 ( three-fold degenerate at 948.1 cm -1 ) transform according to the 𝐴 1𝑔 , 𝐸 𝑔 and 𝐹 1𝑢 irreducible representations while three triply degenerate bending modes 4 , 5 , and 6 transform under the 𝐹 1𝑢 (615.0 cm -1 ), 𝐹 2𝑔 (524.0 cm -1 ) and 𝐹 2𝑢 (347.7 cm -1 ) representations.

The analytic potential function is written in terms of the irreducible tensor operators: ( ,,,,,,,,,,,,,, )

p ip i ip V R
F R r r r r r r q q q q q t t t t   [START_REF] Boudon | High-Resolution Spectroscopy and Analysis of the v4 Bending Region of SF6 near 615 cm -1[END_REF] where each 𝐑 i p tensor is obtained as an appropriate product of radial SR and angular SA nonlinear symmetrized coordinates:

𝑹 𝑖 𝑝 = (([𝑆𝑅 𝐴1 𝑔 ] 𝑝 1 × ([𝑆𝑅 𝐸 𝑔 ] 𝑝 2 × [𝑆𝑅 𝐹1 𝑢 ] 𝑝 3 ) 𝐶 23 ) 𝐶′ × ([𝑆𝐴 𝐹1 𝑢 ] 𝑝 4 × ([𝑆𝐴 𝐹2 𝑔 ] 𝑝 5 × [𝑆𝐴 𝐹2 𝑢 ] 𝑝 6 ) 𝐶 56 ) 𝐶" ) 𝐶 . ( 10
)
These non-linear radial SR and angular SA non-linear symmetrized coordinates involved in our PES model [START_REF] Boudon | High-Resolution Spectroscopy and Analysis of the v4 Bending Region of SF6 near 615 cm -1[END_REF][START_REF] Boudon | Observation and analysis of the SF6 v2 + v4 + v5 band: Improved parameters for the v5 = 1 state[END_REF] were obtained by replacing the combinations (1-6) by the morse-type and cosine-type elementary functions in a similar way as was described in our works for the methane molecule [START_REF] Nikitin | First fully ab initio potential energy surface of methane with a spectroscopic accuracy[END_REF] [49] [START_REF] Nikitin | An efficient method for energy levels calculation using full symmetry and exact kinetic energy operator: tetrahedral molecules[END_REF]. We have tried several choices for an optimal fit to ab initio points. [START_REF] Nikitin | Improved algorithms for the modeling of vibrational polyads of polyatomic molecules: application to Td, Oh, and C3v molecules[END_REF]. An ITO model for the PES or for the DMS expansion at a given total order p corresponds to the summation over various symmetry allowed trees [START_REF] Nikitin | Improved algorithms for the modeling of vibrational polyads of polyatomic molecules: application to Td, Oh, and C3v molecules[END_REF] with the Fip parameters fitted to ab initio values. If the « redundant » symmetrized coordinates of the type S [START_REF] Boudon | First High-Resolution Raman Spectrum and Analysis of the ν5[END_REF] or S [START_REF] Rey | Understanding global infrared opacity and hot bands of greenhouse molecules with low vibrational modes from first-principles calculations: the case of CF4[END_REF] are included in the PES model [START_REF] Chizhmakova | Potential Energy Surface of SF6[END_REF], then two more corresponding factors are added at the left-hand side of the ITO tree [START_REF] Boudon | Observation and analysis of the SF6 v2 + v4 + v5 band: Improved parameters for the v5 = 1 state[END_REF].

The ITO formalism for the PES and DMS expansions can be also applied in terms of normal mode coordinates, as described in [32] [31]. This results in a similar expression to eq.( 10) where the symmetrized SRj and angular SAk coordinates are replaced by normal coordinates Qt . In this work we used the normal mode ITO representation for the dipole moment Energies for all electronic structure calculations were converged to MOLPRO default value 10 -10 a.u.

In order to obtain accurate calculations of vibrational and rotational energy levels from the theoretical PES useful for spectroscopic analysis, it is desirable to use high level ab initio methods with sufficiently large basis sets in electronic structure calculations. However, in case of SF6, the full-grid calculations with large basis sets (like those recently employed for methane [START_REF] Nikitin | First fully ab initio potential energy surface of methane with a spectroscopic accuracy[END_REF] [30]) are too demanding. In this work, an essential part of electronic energy calculations was performed with the CCSD(T)/VTZ ansatz at 6412 nuclear geometries of the grid optimized for the 6-th order PES. To refine the analytical PES form, the same method and the same basis set was additionally used at 2296 geometries of the 8-th order grid. In order to control the surface shape at larger nuclear displacements we carried out supplementary MP2 calculations with VTZ atomic basis on extended grid of 85136 geometries also optimized for the 8-th order PES.

Empirical PES corrections are often used to compensate the lack of accuracy of ab initio calculations. However, for an efficient application of an empirical optimization these corrections have to be relatively small. Typically, simple empirical corrections are used to reduce calculated minus observed deviation for the fundamental vibration frequencies, but this could produce uncontrolled errors for highly excided states if the shape of the initial ab initio PES is not reliably established. This is an important point for a correct modeling of "hot" bands as they could produce a significant contribution the atmospheric absorbance by heavy molecules with relatively low vibrational modes. In case of SF6, only a limited number of experimental vibrational frequencies is known, that can be used for a validation of ab initio results. Ab initio equilibrium geometry obtained with CVQZ and CVQZ-DK basis sets were close to empirical ones ( Table 1), but calculations using CCSD(T)/ CVQZ-DK ansatz for a large grid of geometries is actually too demanding. For this reason, we have applied the algorithm of corrections similar to that of ref. [START_REF] Nikitin | Rotational and vibrational energy levels of methane calculated from a new potential energy surface[END_REF]. First, we have computed the one-dimensional « radial » PES section using the CVQZ-DK basis set as well as electronic energies at 67 nuclear geometries sufficient to determine the force field up to the third order. Then the difference between electronic energies in 67 geometries using CVQZ-DK and VTZ basis sets was fitted to an analytical function 3 ( ( , ))

O F S r q of 15 symmetrized coordinates expanded up to 3-rd order.

One dimensional radial correction functions () X fr were constructed as a difference between two radial potentials calculated for X-basis and for VTZ basis

( ) ( ) ( ) ( ( , , , , )) X X VTZ O e e e f r V r V r F S r r r q    (11)
where VX(r) stands for electronic energies at the radial cut and X=CVQZ-DK or VQZ. The additional one-dimensional function 3 ( , 0, 0, , 0) O Fr was obtained from the third order correction. All other bond lenghts were fixed to their equilibrium values for X and VTZ basis sets whereas all angles were fixed to their equilibrium values. Functions ) (r f X for X=CVQZ-DK is shown in Fig. 3. Note that the one-dimensional radal correction [START_REF] Boudon | The high overtone and combination levels of SF6 revisited at Doppler-limited resolution: A global effective rovibrational model for highly excited vibrational states[END_REF] near the equilibrium geometries of VTZ and X basis sets quite significantly depends on the analytical representation.

The number N of irreducible tensors of the total 𝐴 1𝑔 representation involved in the PES expansion (9-10) increases quite rapidly with the order. Up to the orders 2, 3, 4, 5, 6, 7, and 8

this number N is equal to 9, 31, 123, 406, 1334, 3924, and 10994, respectively. For PES parameterization, to provide the best results using the least square technique, more complex onedimensional basis functions are used in (1-6). Morse-type functions are used as primitive radial functions ( ) 1 exp( ( ))

r i i e f r r r      ( 12
)
involved in the definitions of the symmetrized SR and angular SA coordinates which are used to build our final analytical PES model. Although the function fr(r) 2 for α = 2.0 reproduces very accurately the one-dimensional potential of SF, but the best choice of α in [START_REF] Manceron | First analysis of the v3 + v5 combination band of SF6 observed at Doppler-limited resolution and effective model for the v3 + v5 -v5 hot band[END_REF] for fulldimensional PES was α = 1.9. Consequently, our PES model tends to the corresponding finite asymptote at large inter-nuclear distances contrary to a simple power series expansion in (rre).

The angular function was expressed in terms of the cosine function:

a ij ij ij f q q q   . ( 13
)
Similar to methane PES [START_REF] Nikitin | Rotational and vibrational energy levels of methane calculated from a new potential energy surface[END_REF], it is preferable to use the value α = 1/3 for angular functions.

The relevance of various analytical representation of the PES was checked by the quality of the PES fit on the 8-th order grid of 85136 geometries using MP2/VTZ calculated energies. Various orders of the PES expansion and various forms of the angular dependence were tested. The results of the least-squares fits using weight function w(E) similar to that previously employed for methane studies [START_REF] Schwenke | Vibrational energy levels for CH4 from an ab initio potential[END_REF] [49]

1 tanh( 0.0005( ) 1.002002002) () 2.002002002

EE wE     (14)
where E1=7000cm -1 , are collected in Table 3. It is seen that parameter value α = 1/3 permitted a notable improvement of the standard deviation of the fit for 4-th and 5-th order of the PES expansion. The use of supplementary (redundant) coordinated, as reported in the case of methane [START_REF] Nikitin | An efficient method for energy levels calculation using full symmetry and exact kinetic energy operator: tetrahedral molecules[END_REF] [21] could significantly change the PES near the equilibrium. In this work we have also evaluated a possible impact of supplementary coordinates [START_REF] Rey | Understanding global infrared opacity and hot bands of greenhouse molecules with low vibrational modes from first-principles calculations: the case of CF4[END_REF], [START_REF] Boudon | First High-Resolution Raman Spectrum and Analysis of the ν5[END_REF] for SF6. An introduction of S' coordinate in the PES expansion has somewhat larger effect on the standard deviation than that of S", but in both cases the improvement of the standard deviation was smaller with respect to the inclusion of α = 1/3 in the quadratic terms of Eq.( 13). Finally, we consider that the fifthorder PES expansion provides an acceptable quality of the fit of ab initio energies up to 7000 cm - 1 . The comparison of the observed fundamental band centers of 32 SF6 with our variational nuclear motion calculations using ab initio CCSD(T)/VTZ and CCSD(T)/CVQZ-DK surfaces is given in Table 4. For the VTZ surface defined on a large grid, vibrational and rotational energies were computed using TENSOR variational code in normal mode ITO representation. Technical detail can be found in the previous works devoted to five-and six-atomic molecules [START_REF] Rey | First theoretical global line lists of ethylene ( 12 C2H4) spectra at various temperatures in the far-infrared range for quantification of absorption and emission in planetary atmospheres[END_REF] [32]

[33] [START_REF] Rey | Convergence of normal mode variational calculations of methane spectra: Theoretical line list in the icosad range computed from potential energy and dipole moment surfaces[END_REF]. For the VQZ surface defined on a small grid, we used an incremental scheme accounting for the relative difference i(VQZ)-i(VTZ) between the harmonic frequencies in these basis sets as described below.

The five of six fundamentals calculated with the CVQZ-DK basis set accounting for relativistic contributions show in average better agreement with observed frequencies than VTZ calculations. Note that the most significant contribution in the increase of the fundamental frequencies of angular modes ν4, ν5, ν6 when using the CVQZ-DK PES in comparison with the VTZ PES is due to the decrease of the equilibrium S-F bond length. A rough estimation of the deviations between VQZ calculations and the observed fundamental band centers (as given in column 4 of Table 4) was obtained using approximate relation νi(VQZ)=vi(VTZ)+(i(VQZ)-i(VTZ)) where i are harmonic frequencies.

Table4: The fundamental band centers (cm -1 ) for The S-F bound length computed in CVQZ-DK basis set approached towards the empirical value. This could simplify in future an empirical optimization of the PES parameters. But in order to improve the RMS deviation of the 15-dimensional PES fit with a rather small number of parameters, the slightly increased value re=1.5613 (instead of 1.557 for CVQZ-DK) was used.

The final ab initio PES of this work was constructed using the least-squares fit of the Fp parameters of the 5-th order expansion [START_REF] Boudon | High-Resolution Spectroscopy and Analysis of the v4 Bending Region of SF6 near 615 cm -1[END_REF] to CCSD(T)/VTZ electronic energies at 8708 geometrical nuclear configurations with the CCSD(T)/CVQZ-DK correction as described above.

In order to improve the fit accuracy near the bottom of the PES the weighting function [START_REF] Gordon | The HITRAN2016 molecular spectroscopic database[END_REF] with the parameter E1= 5000 cm -1 was applied. This permitted obtaining RMS deviation of 0.5 cm -1 for 4283 geometries in the range of 5000 cm -1 above the potential minimum. Since the order of the PES was not very high (in comparison with best methane calculations [START_REF] Nikitin | First fully ab initio potential energy surface of methane with a spectroscopic accuracy[END_REF]), the MP2/VTZ electronic energies were not explicitly included in the fit, but were used to check the PES behavior for larger nuclear displacements.

Ab initio DMS fit

Several studies [START_REF] Boudon | First High-Resolution Raman Spectrum and Analysis of the ν5[END_REF][START_REF] Boudon | High-Resolution Spectroscopy and Analysis of the v4 Bending Region of SF6 near 615 cm -1[END_REF][START_REF] Boudon | Observation and analysis of the SF6 v2 + v4 + v5 band: Improved parameters for the v5 = 1 state[END_REF][START_REF] Boudon | The high overtone and combination levels of SF6 revisited at Doppler-limited resolution: A global effective rovibrational model for highly excited vibrational states[END_REF][START_REF] Manceron | First analysis of the v3 + v5 combination band of SF6 observed at Doppler-limited resolution and effective model for the v3 + v5 -v5 hot band[END_REF] have been devoted to line-by-line analyses of experimental Raman and infrared SF6 spectra using effective Hamiltonian and effective transition moments models.

An accurate determination of empirical line intensities of heavy molecules with low frequency

modes is an extremely difficult task because of crowded spectra and blended lines. One of the most challenging issue is a separation of contributions of overlapping cold and hot bands with a significant impact on the uncertainty in the temperature dependence of the absorption or emission of radiation. [18] [START_REF] Rey | TheoReTS -An information system for theoretical spectra based on variational predictions from molecular potential energy and dipole moment surfaces[END_REF] and references therein) permitted constructing quite complete lines lists for triatomic [START_REF] Partridge | The determination of an accurate isotope dependent potential energy surface for water from extensive ab initio calculations and experimental data[END_REF], [START_REF] Polyansky | Calculation of rotationvibration energy levels of the water molecule with near-experimental accuracy based on an ab initio potential energy surface[END_REF], [START_REF] Polyansky | ExoMol molecular line lists XXX: a complete high-accuracy line list for water[END_REF] [69] [START_REF] Tyuterev | Ab initio predictions and laboratory validation for consistent ozone intensities in the MW, 10 and 5 µm ranges[END_REF] , four-atomic [START_REF] Huang | Rovibrational spectra of ammonia. II. Detailed analysis, comparison, and prediction of spectroscopic assignments for 14 NH3, 15 NH3, and 14 ND3[END_REF] [31] [START_REF] Nikitin | G High order dipole moment surfaces of PH3 and ab initio intensity predictions in the Octad range[END_REF] [72] [START_REF] Egorov | Global modeling of NF3 line positions and intensities from far to mid-infrared up to 2200 cm -1[END_REF], five-atomic [START_REF] Rey | Convergence of normal mode variational calculations of methane spectra: Theoretical line list in the icosad range computed from potential energy and dipole moment surfaces[END_REF] [74] [30] [19] [22] [START_REF] Nikitin | Rotational and vibrational energy levels of methyl fluoride calculated from a new potential energy surface[END_REF] and six-atomic [START_REF] Delahaye | A new accurate ground-state potential energy surface of ethylene and predictions for rotational and vibrational energy levels[END_REF] [24] [25] [76] molecules. In some of these cases the accuracy of ab initio line intensities were competitive [77] [78] [79] [START_REF] Nikitin | Accurate line intensities of methane from first-principles calculations[END_REF] with best empirical determinations being more complete and reliable for the hot bands [START_REF] Polyansky | ExoMol molecular line lists XXX: a complete high-accuracy line list for water[END_REF] [81] [82] [START_REF] Rey | Understanding global infrared opacity and hot bands of greenhouse molecules with low vibrational modes from first-principles calculations: the case of CF4[END_REF] [83] [START_REF] Egorov | Global modeling of NF3 line positions and intensities from far to mid-infrared up to 2200 cm -1[END_REF]. On the other hand, knowledge of the shape of ab initio surfaces [START_REF] Vl | Does the "reef structure" at the ozone transition state towards the dissociation exist? New insight from calculations and ultrasensitive spectroscopy experiments[END_REF] [85] helped understanding issues related to molecular dynamics.

One more atom in a molecule produces three supplementary degrees of freedom in the nuclear motion resulting to a very substantial computational effort for ab initio predictions of high-resolution spectra for polyatomic molecules. In a brute-force approach (without symmetry considerations) there would be roughly a factor ~N6 of dimensionality between global spectra calculations of five-atomic molecule ( like methane) and seven-atomic molecule like SF6 where N is a number of ab initio grid points or of basis set function for each degree of freedom.

Here we present the first results for the full 15-dimensional SF6 DMS components and for intensity calculations in the range of the strongest bands. A full account of symmetry permitted drastically reducing the number of redundant geometries and the number of independent DMS parameters. For the construction of the DMS we used the corresponding irreducible tensor representation similar to the analytical expansions [START_REF] Boudon | High-Resolution Spectroscopy and Analysis of the v4 Bending Region of SF6 near 615 cm -1[END_REF][START_REF] Boudon | Observation and analysis of the SF6 v2 + v4 + v5 band: Improved parameters for the v5 = 1 state[END_REF] where the symmetrized internal coordinates were replaced by normal mode coordinates. The corresponding procedure has been described in our previous works [START_REF] Nikitin | New dipole moment surfaces of methane[END_REF] [71] devoted to four-atomic and five-atomic molecules. In these works, the ab initio DMSs were first built in molecular-fixes frames corresponding to the axes choice options of the MOLPRO code. Then the axes were rotated to the Eckart system. As in case of SF6 the relations between internal and normal coordinated Qi were found via numerical transformation, it was easier to proceed by direct DMS fits in normal coordinates.

A progress in methane ab initio calculations has taken a couple of decades [START_REF] Lee | An accurate ab initio quartic force field and vibrational frequencies for CH4 and isotopomers[END_REF] [60] [START_REF] Cassam-Chenai | Lievin J Ab initio calculation of the rotational spectrum of methane vibrational ground state[END_REF] [81] [START_REF] Nikitin | First fully ab initio potential energy surface of methane with a spectroscopic accuracy[END_REF]. The line intensity accuracy in infra-red of 1% to 5% has been recently reported [START_REF] Nikitin | Accurate line intensities of methane from first-principles calculations[END_REF] using the Coupled-Cluster method with ACV5Z one-particle basis sets. In case of SF6 such calculations are presently too demanding. In this work we used CCSD(T) method with two basis sets for the dipole moment calculations. The calculations using VTZ basis were conducted on the first grid containing 1664 nuclear geometries, which was a sub-set of the 6-th order grid relative to PES computations described above. This was then supplemented by AVTZ calculations at 500 geometries. In both cases the optimal samples of geometries near the bottom of the PES were determined. At each geometry, three projections of the dipole moment were obtained. Dipole moments were computed as the derivative of the energy with respect to the weak external uniform electric field using the finite difference scheme with the field variation of 0.0002 a.u. around at the zero field strength.

For the DMS construction we used here the scheme, which has been employed in our previous works for PES and DMS of four-to-six atomic molecules [START_REF] Nikitin | G High order dipole moment surfaces of PH3 and ab initio intensity predictions in the Octad range[END_REF] [86] [22] [START_REF] Delahaye | Accurate 12D dipole moment surfaces of ethylene[END_REF]. After having computed the dipole moments with AVTZ and VTZ basis sets we fitted the corresponding 15-dimensional function diff2_O3(Qi), which describes the corresponding difference DMS(AVTZ) -DMS(VTZ). This function was modeled by the third-order ITO expansion containing 61 adjustable parameters, of which 50 parameters were statistically welldefined. Then, for supplementary geometries from N=501 to N=1664 were the AVTZ calculations were lacking, the AVTZ dipole moments were evaluated as VTZ dipole moments plus diff2_O3(Qi). Finally, the forth-order parameters of the DMS power series expansion in normal coordinated were fitted to the above defined ab initio dipole moment values using weight function [START_REF] Jacquinet | The 2015 edition of the GEISA spectroscopic database[END_REF] with E1=10000 cm -1 . Of 260 fourth-order DMS parameters 194 ones were statistically well-determined. This procedure conducted to high accuracy of the fit corresponding to the RMS deviation of 1.4E-5 a.u. and the weighted standard deviation of 9.9*10 -6 . The distribution of the fit discrepancies is shown in Fig. 4.

Results and Conclusion

Preliminary evaluation of SF6 band intensities using algorithms of variational method described in [START_REF] Rey | Ab initio ro-vibrational Hamiltonian in irreducible tensor formalism: a method for computing energy levels from potential energy surfaces for symmetric-top molecules[END_REF] [32] [START_REF] Rey | First principles intensity calculations of the methane rovibrational spectra in the infrared up to 9300 cm -1[END_REF] [34] permits to conclude that our PES and DMS provided realistic predictions of absorption spectra, at least in the range of strong bands. The comparison of our calculated absorption cross-sections with experimental SF6 cross section [89] [13] shows a very good agreement at cold temperature of 180 K, see Fig. corresponds to the v3 band of the 34 SF6 isotopologue at natural abundance. Our ab initio crosssections agree much better with these observations than the simulations using SF6 line-by-line lists constructed from effective models as available in HITRAN_2016 [START_REF] Gordon | The HITRAN2016 molecular spectroscopic database[END_REF] , GEISA_2015 [START_REF] Jacquinet | The 2015 edition of the GEISA spectroscopic database[END_REF] and Dijon (SheCaSDa) [92] databases as shown in Fig. 5, upper panel. However, ab initio roomtemperature simulations appear to be slightly less accurate than for 180 K (Fig. 5, lower panel).

Calculations at room temperature require an account of more significant contributions of hot bands involving higher vibrational states. The preliminary data obtained in this work will be made available through the TheoReTS [START_REF] Rey | TheoReTS -An information system for theoretical spectra based on variational predictions from molecular potential energy and dipole moment surfaces[END_REF] (http://theorets.univ-reims.fr, http://theorets.tsu.ru) information system. A further improvement could be possibly obtained using lager vibrational basis set that is computationally very demanding. We plan to extend the computation of full line lists in future studies.

.

Figure 1 .

 1 Figure 1. Equilibrium structure and the axes choice for the SF6 molecule

  modeling and for intensity calculations because the matrix elements of the corresponding operators can be computed analytically. However, a sufficiently complete transformation from nonlinear symmetrized coordinates (1-6) to rectilinear normal coordinated Qt in case of SF6 appears to be quite complicated. In this work, we obtained the corresponding relations {SRj, SAk}  Qt numerically via the solution of the system of non-linear equations. A distribution of the RMS deviations of the fit of 15 normal coordinates to 15 internal symmetrized coordinates is shown in Fig 2.The error of the corresponding numerical solution Qt = Qt( SRj , SAk) was much smaller than the precision of the fit of the analytical form[START_REF] Boudon | Observation and analysis of the SF6 v2 + v4 + v5 band: Improved parameters for the v5 = 1 state[END_REF] to ab initio points and thus did not affect the accuracy of the final calculation of spectra.

Figure 2 .

 2 Figure 2. RMS distribution for non-linear fit of normal coordinates Qt to internal symmetrized coordinates SRj , SAk

3

 3

2 (

 2

Figure 3 -

 3 Figure 3 -One-dimensional radial correction E{CCSD(T)-DK/CVQZ-DK} -E{CCSD(T)/VTZ}.

  values in the fourth column were obtained from full dimensional variational nuclear motion calculation at the VTZ level corrected by the difference i(VQZ)-i(VTZ) between the harmonic frequencies. The values of re are given in Å.

Advanced

  ab initio calculations of molecular PESs and DMS together with the development of variational methods for nuclear motion calculations ( [61] [62] [63] [64] [65]

Figure 4 -

 4 Figure 4 -Deviations between fitted analytical representation of the DMS and ab initio dipole moment values versus energies at the corresponding nuclear geometries.

5 ,

 5 lower panel. Note that band centers of 32 SF6 and 34 SF6 isotopologues in Fig.5were shifted to empirical values (964.196 cm -1 and 932.534 cm -1 correspondingly) available at http://vamdc.icb.cnrs.fr/PHP/SF6.php. An agreement of our calculations with experimental absorption cross-sections of Sharp et al [90] at room temperature is also satisfactory (Fig.5, upper panel). The latter ones have been recorded at T=298 K with N2 pressure of 760 Torr with medium resolution of 0.112 cm -1 and included in HITRAN-on-line cross-section compilation [90][START_REF] Gordon | The HITRAN2016 molecular spectroscopic database[END_REF] [91]. The absorption feature near 930 cm-1

Figure 5 .

 5 Figure 5. Comparison of cross section at a temperature of 296K (Upper panel) and of 180K (Lower panel). Upper panel: our simulation from line list of TheoReTS (green -a), the experimentally observed PNNL spectrum available in HITRAN (black -b ), simulation from line list of GEISA 2015 (blue -c ), the theoretical calculation of Dijon (red -d) and simulation from line list of HITRAN 2016 database (yellowe). Lower panel: our simulation from line list of TheoReTS (green -a), the experimental observed spectrum from HITRAN (black -b )

Table 1 .Table 1 .

 11 Energies

	and the equilibrium lengths of the S -F bound obtained with the optimization
	CCSD(T) technique using various basis sets.
	Method / basis set	Energy, Hartree	Equilibrium length S -F, Å
	CCSD(T) / VTZ	-996.07574167	1.57015425
	CCSD(T) / CVTZ	-996.79947177	1.56003654
	CCSD(T) / AVTZ	-996.14253699	1.57357635
	CCSD(T) / CVTZ -DK	-998.23964874	1.56042314
	CCSD(T) / VQZ	-996.30894816	1.56375885
	CCSD(T) / CVQZ	-997.10132168	1.55654004
	CCSD(T) / CVQZ-DK	-998.70581919	1.55702740
	Empirical variational *		1.556
	Ref. [39]

from valence triple-zeta (VTZ) and augmented valence triple-zeta (AVTZ) to core-valence quadruple-zeta (CVQZ) with Douglas-Kroll (DK) relativistic corrections. All ab initio calculation were carried out using MOLPRO suite of programs as implemented in the 2019.1 version

[START_REF] Werner | Molpro: a general purpose quantum chemistry program package[END_REF]

[START_REF] Werner | version 2019.1, a package of ab initio programs[END_REF]

. More details concerning ab initio methods will be given in Section 3. A comparison of results obtained for various basis sets with some empirical values is collected in

Table 2 .

 2 Oh: point group elements

	Group element	C 31 +	C 4z +

  The final expressions of elementary functions used in the present work are given in the next Section 3. Here pj components of p = {p1, …p6} correspond to powers of these coordinates and 𝑝 = 𝑝 1 + 𝑝 2 + ⋯ + 𝑝 6 is the total power of the expansion term. The lower case index i corresponds to the set of symbols C23, C56, C, C, C in the summation (9) that run over nondegenerate (𝐴 1𝑔 ),

doubly ( 𝐸 𝑔 ) and triply degenerate ( 𝐹 1𝑢 , 𝐹 2𝑢 , and 𝐹 2𝑔 ) irreducible representations (irreps) of 𝑂 ℎ point group. The C23 irrep results from the coupling of the symmetrized powers of the second and the third vibrational modes, whereas C stands for the irrep corresponding to the coupling of the latter product with the symmetrized power of the first vibrational mode. In a similar manner the irrep C56 corresponds to the coupling of the fifth and sixth modes coupled with the fourth mode via C. For the PES expansion C'=C", so that only the final symmetry allowed type C of the 𝑅 𝑖 𝑝 tensors is A1g. Very similar expansion can be used for the DMS, but in this case the final symmetry type C of the 𝑅 𝑖 𝑝 tensors is F1u. Each combination of the symmetrized coordinate powers p1, …, p6 and of the coupling irreps C23, C56, C, C, C form a "tree"

[START_REF] Champion | Spectroscopy of the Earth's Atmosphere and Interstellar Medium[END_REF]

Table 3 .

 3 Result of various PES fits using 85136 electronic MP2/VTZ energies.

	Order	Additional angular	#Parameters Alpha	RMS	STD
		coordinates: order, type			cm -1	cm -1
	4	-	123	0.	8.29	3.71
	4	O3, S'	151	0.	6.77	3.16
	4	O3, S"	151	0.	8.08	3.62
	4	-	123	1/3	4.58	2.13
	4	O2, S'	128	1/3	3.98	1.90
	4	O2, S"	128	1/3	4.27	2.05
	4	O3, S'	151	1/3	3.48	1.70
	4	O3, S"	151	1/3	3.89	1.91
	5	-	406	0.	1.96	0.79
	5	O3, S'	434	0.	1.57	0.68
	5	O3, S"	434	0.	1.66	0.68
	5	-	406	1/3	0.94	0.43
	5	O3, S'	434	1/3	0.58	0.28
	5	O3, S"	434	1/3	0.75	0.34

Acknowledgements

The research in IOA was performed as part of the state assignment of the IOA SB RAS. V.T. acknowledges the support from Academic D..Mendeleev program of Tomsk State University and M.R. form LEFE CHAT CNRS program.

Supplementary Materials

In the Supplementary Materials, we provide PES in the internal coordinates and DMS in normal mode expansion in an easy to use explicit form for 32 SF6