Andrei V Nikitin

Michaël Rey

Jean-Paul P Champion

Vladimir G Tyuterev

Extension of the MIRS computer package for the modeling of molecular spectra : from effective to full ab initio ro-vibrational hamiltonians in irreducible tensor form

Keywords: computational spectroscopy, ab initio calculations, vibration-rotation spectroscopy, effective hamiltonians, high-resolution infrared spectroscopy, polyads, irreducible tensors, molecular symmetry

teaching and research institutions in France or abroad, or from public or private research centers.   Extension of the MIRS computer package for the modeling of molecular spectra : from effective to full ab initio ro-vibrational hamiltonians in irreducible tensor form

Introduction

Spectroscopic investigations of polyatomic molecules play a role of primordial importance for atmospheric applications. Effective hamiltonians in irreducible tensor operator (ITO) form have been successfully used to model the crucial effects of ro-vibrational perturbations on line positions and strengths through the analysis of complex interacting band systems (so-called polyads). An overview can be found in [START_REF] Champion | Spherical top spectra[END_REF][START_REF] Hilico | Analysis of the interacting octad system of 12 CH 4[END_REF][START_REF] Rotger | Spectroscopy of XY 5 Z C 4v molecules: Development of the hamiltonian and the transition moment operators using a tensorial formalism[END_REF][START_REF] Nikitin | The MIRS computer package for modeling the rovibrational spectra of polyatomic molecules[END_REF][START_REF] Nikitin | Global analysis of chloromethane: determinability of ground state constants[END_REF][START_REF] Boudon | Symmetry-adapted tensorial formalism to model rovibrational and rovibronic spectra of molecules pertaining to various point groups[END_REF][START_REF] Nikitin | Global analysis of 12 CH 3 35 Cl and 12 CH 3 37 Cl: simultaneous fit of the lower five polyads (0-2600 cm -1 )[END_REF][START_REF] Nikitin | Preliminary analysis of CH 3 D from 3250 to 3700 cm -1[END_REF][START_REF] Rotger | Spectroscopy of XY 2 Z 2 C 2v molecules: A tensorial formalism adapted to the O(3) ⊃ T d ⊃ C 2v chain. application to the ground state of SO 2 F 2[END_REF][START_REF] Nikitin | Analysis of the CH 3 D nonad from 2000 to 3300 cm -1[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] and references therein. This approach proved to be particularly efficient for a full account of the molecular and space symmetry properties. Hereafter we shall refer to these models as ITO spectroscopic models.

Nowadays, modern high-resolution and high-sensitivity techniques as well as direct spectroscopic observations of the planetary atmospheres and stars yield a wealth of spectroscopic data involving highly excited energy levels and relatively weak spectroscopic features. The modeling of such data using empirically adjusted effective hamiltonians becomes more and more complicated for several reasons. Most vibration-rotation bands show irregular perturbations due to accidental resonances. The importance of such effects in high-resolution spectroscopy has been shown in works by J.-M. Flaud, C. Camy-Peyret and co-workers [START_REF] Flaud | Vibration-rotation intensities in H 2 O-type molecules application to 2ν 2 -band, ν 1 -band, and ν 3 -band of H 16 2 O[END_REF][START_REF] Camy-Peyret | Vibration-rotation dipole moment operator for asymmetric top molecules[END_REF] who were among the first researchers developping effective model hamiltonians and dipole transition moment operators for asymmetric top molecules in case of resonance interactions. The number of interaction parameters grows very rapidly as the excitation increases. At high energy ranges which become experimentally accessible new inter-polyad resonances can occur requiring specific terms to be included in theoretical models. A. Barbe and co-workers [START_REF] Barbe | CWcavity ring down spectroscop, of the ozone molecule in the 5980-6220 cm -1 region[END_REF][START_REF] Barbe | Detection and analysis of new bands of 16 O 3 by CRDS between 6500 and 7300 cm -1[END_REF] have shown the crucial role of inter-polyad resonance perturbations and of perturbations due to «dark» (experimentally invisible) states, in particular for ozone specroscopy in the range approaching the dissociation energy. The probability of near-coincidences of line frequencies rapidely increases with the number of atoms. For molecules of high-symmetry the combined effect of degeneracies and resonances makes it nessesary to model a large number of strongly interacting bands simultanously [START_REF] Champion | Spherical top spectra[END_REF][START_REF] Nikitin | Analysis of the CH 3 D nonad from 2000 to 3300 cm -1[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].

The choice of initial values for interaction parameters becomes arduous and the minimization process often reaches a local rather than global minimum. Another reason is a large proportion of dark states in complex and highly exited band systems and the corresponding deficit of experimental information.

To some extent, accounting for unobserved energy levels is equivalent to introducing additional poorly defined parameters in effective models. On the other hand, full nuclear hamiltonians expressed in terms of normal-mode irreducible tensor operators can be obtained from ab initio potential energy surfaces (PES) as reported by Rey et al. [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][START_REF] Rey | The complete nuclear hamiltonian in the irreducible tensor operator formalism for methane[END_REF]. Then, non-empirical effective hamiltonians can be obtained using high-order contact transformations [START_REF] Tyuterev | High-order contact transformations: General algorithm, computer implementation and triatomic tests[END_REF][START_REF] Tyuterev | High-order contact transformations for methane molecule[END_REF][START_REF] Lamouroux | Accurate calculation of transition moment parameters for rovibrational bands from ab initio dipole and potential surfaces: Application to fundamental bands of the water molecule[END_REF][START_REF] Cassam-Chenai | Ab initio effective rotational hamiltonians: A comparative study[END_REF] from a full nuclear hamiltonian. A large proportion of non-diagonal resonance parameters as well as diagonal parameters for dark states can be fixed to theoretical values derived from such transformations. Even though ab initio derived values for non diagonal parameters are approximate, they can describe the resonance couplings in a qualitatively correct way.

Ab initio vibration-rotation hamiltonians can be converted into irreducible tensor expansion [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][START_REF] Rey | The complete nuclear hamiltonian in the irreducible tensor operator formalism for methane[END_REF] which is formally similar to that employed for empirical effective hamiltonians [START_REF] Champion | Spherical top spectra[END_REF][START_REF] Nikitin | Improved algorithms for the modeling of vibrational polyads of polyatomic molecules: Application to T d , O h , and C 3v molecules[END_REF][START_REF] Zhilinskii | Method of irreducible tensorial operators in the theory of molecular spectra[END_REF] but interestingly they can be constructed up to higher order than in pure empirical approaches and are especially suitable for very highly exited band systems. Of course, the handling of ITO hamiltonian expansions built from PES relies on advanced computational efficiency. The performance of programming is important because of the very large number of hamiltonian terms and basis functions to deal with.

For the validation of calculations it is necessary to check the convergence of contact transformations [START_REF] Tyuterev | High-order contact transformations: General algorithm, computer implementation and triatomic tests[END_REF][START_REF] Tyuterev | High-order contact transformations for methane molecule[END_REF][START_REF] Cassam-Chenai | Ab initio effective rotational hamiltonians: A comparative study[END_REF] by achieving variational calculations with full nuclear hamiltonians. Such calculations are very cumbersome and require appropriate computational optimization. The development of efficient computational tools was crucial for the automatic generation and handling of various types of symmetry allowed terms simplifying in turn spectral analyses and spectroscopic data reductions.

The reader is referred to the original MIRS description [START_REF] Nikitin | The MIRS computer package for modeling the rovibrational spectra of polyatomic molecules[END_REF] and to previous review articles for details on the effective hamiltonian tensorial approach and results [START_REF] Champion | Spherical top spectra[END_REF][START_REF] Nikitin | Improved algorithms for the modeling of vibrational polyads of polyatomic molecules: Application to T d , O h , and C 3v molecules[END_REF][START_REF] Boudon | Symmetry-adapted tensorial formalism to model rovibrational and rovibronic spectra of molecules pertaining to various point groups[END_REF]. The present paper is focused on the new computational features, illustrated by the results obtained so far.

Modeling principles

Irreducible tensor method

The irreducible tensor formalism and its computer implementation in MIRS code was already described in details in [START_REF] Nikitin | Improved algorithms for the modeling of vibrational polyads of polyatomic molecules: Application to T d , O h , and C 3v molecules[END_REF][START_REF] Nikitin | The MIRS computer package for modeling the rovibrational spectra of polyatomic molecules[END_REF] and references therein. Let us just remind that within the ITO formalism each elementary vibration-rotation term

h vib-rot = t Ωr(K.kC) n 1 n 2 ...m 1 m 2 ... T Ωr(K.kC) n 1 n 2 ...m 1 m 2 ... = t Ωr(K.kC) n 1 n 2 ...m 1 m 2 ... β( ε V n 1 n 2 ...m 1 m 2 ... ⊗R Ωr(K .kC ) ) A 1 (1)
involved in the hamiltonian expansion is identified by rotational, vibrational and symmetry indices [START_REF] Champion | Complete development of vibration-rotation hamiltonian adapted to study of interactions in spherical top molecules -application to ν 2 and ν 4 bands of 12 CH 4[END_REF] according to the general nomenclature, where V, R and T designate a vibrational, rotational and vibration-rotation tensor operators and t the corresponding parameters. Vibrational operators ε V n 1 n 2 ...m 1 m 2 ... are constructed in tensor form by recursive coupling of creation and annihilation operators associated to the normal modes of the molecule. The lower indices n i and m i being, respectively, the powers of creation and annihilation vibrational operators. The upper indices indicate the rotational characteristics of the considered term: Ω r is the rotational power with respect to the angular momentum components; K is the tensor rank in the full rotation group; C is the rotational symmetry coinciding with the vibrational symmetry to satisfy the invariance condition under the molecular point group operations; β is a normalization constant introduced historically [START_REF] Champion | Complete development of vibration-rotation hamiltonian adapted to study of interactions in spherical top molecules -application to ν 2 and ν 4 bands of 12 CH 4[END_REF] to ensure the coincidence between some of the lower order standard and tensorial spectroscopic constants. Other notations follow references [START_REF] Champion | Spherical top spectra[END_REF][START_REF] Nikitin | Improved algorithms for the modeling of vibrational polyads of polyatomic molecules: Application to T d , O h , and C 3v molecules[END_REF][START_REF] Nikitin | The MIRS computer package for modeling the rovibrational spectra of polyatomic molecules[END_REF].

In MIRS, a specific coupling scheme [START_REF] Nikitin | Improved algorithms for the modeling of vibrational polyads of polyatomic molecules: Application to T d , O h , and C 3v molecules[END_REF] associated to a binary tree is applied directly to an arbitrary number of interacting vibrational modes and to arbitrarily high polyads. The construction of vibrational basis functions is achieved consistently from the action of creation operators on the vacuum function. This method is not only satisfying conceptually but also quite efficient for the computer calculation of matrix elements and commutators. The ro-vibrational terms are easily generated by further tensor couplings. The standard Amat and Nielsen classification scheme [START_REF] Amat | Rotation -vibration of polyatomic molecules[END_REF] is used in which the order of terms of the type H mn is m + n -2. Effective dipole moment (or other molecular property ) expansions are represented in a similar manner as that of the hamiltonian but the full symmetry type has to be appropriately specified according to the molecular symmetry group [START_REF] Champion | Spherical top spectra[END_REF][START_REF] Nikitin | Improved algorithms for the modeling of vibrational polyads of polyatomic molecules: Application to T d , O h , and C 3v molecules[END_REF][START_REF] Zhilinskii | Method of irreducible tensorial operators in the theory of molecular spectra[END_REF]. For transformed dipole transition moments, terms of the type µ mn are of order m + n -1.

Inter-polyad couplings. Effective and full ro-vibrational hamiltonians

The effective model definition and associated quantum numbers are described in detail in Section 2.2 of ref. [START_REF] Nikitin | The MIRS computer package for modeling the rovibrational spectra of polyatomic molecules[END_REF]. Only the basic features are summarized here in order to make the description of the new implementation understandable.

The polyad structure constitutes the key for defining effective hamiltonians. For a given molecule, it is essentially governed by the number of the vibrational modes and their fundamental frequencies.

In the most straightforward way the polyad scheme is build for vibrational normal modes coupled by vibration-rotation resonance interactions. As a preliminary step a reference energy level pattern is determined by resonance conditions defined by the user or by harmonic oscillator energies computed from the fundamental frequencies. Of course, the strength of a resonance perturbation depends not only on the proximity of the zero order energy levels but also on the coupling matrix elements. The MIRS code allows a certain flexibility for the polyad composition by optionally including or excluding some vibrational states according to estimations of coupling matrix elements. It is possible to take into account such physical considerations by playing with the resonance conditions. For instance, a gap between reference levels can be artificially increased in the case of weak vibration-rotation interactions or decreased in the case of strongly coupled but distant vibrational states. The true band centers remain unchanged under this operation. According to this approach the polyad structure of the molecule determines an effective hamiltonian expansion automatically implemented through the formal polyad expansion (see [START_REF] Champion | Spherical top spectra[END_REF] for details)

H polyads = H P 0 + H P 1 + H P 2 + . . . . (2)
Vibrational matrix elements of the hamiltonian are computed in the harmonic oscillator zeroorder approximation. The computation of rotational matrix elements is described in [START_REF] Moret-Bailly | [END_REF][START_REF] Moret-Bailly | Calculation of the frequencies of the lines in a threefold degenerate fundamental band of a spherical top molecule[END_REF][START_REF] Zhilinskii | Reduction of rotational operators to standard form[END_REF].

The irreducible tensor form [START_REF] Zhilinskii | Method of irreducible tensorial operators in the theory of molecular spectra[END_REF][START_REF] Champion | Spherical top spectra[END_REF][START_REF] Nikitin | Improved algorithms for the modeling of vibrational polyads of polyatomic molecules: Application to T d , O h , and C 3v molecules[END_REF][START_REF] Boudon | Symmetry-adapted tensorial formalism to model rovibrational and rovibronic spectra of molecules pertaining to various point groups[END_REF] implemented in the code allows a full account of symmetry properties. The polyad structure also determines the construction of the effective transition moments.

It determines as well the quantum numbers used to label levels and transitions. Strict quantum numbers are related to the usual invariants : the polyad number P , the rotational quantum number J and the ro-vibrational symmetry species C.

It is well known that at a low energy range the polyad structure based on fundamental frequencies usually results in a good approximation for effective hamiltonians of semi-rigid molecules. For higher quantum numbers the modeling of experimental data can be considerably improved by including interpolyad interactions. In the new version, all features concerning intra-polyad terms are conserved but inter-polyad terms can be added. More precisely any inter-polyad operator with rotational tensor powers equal to zero, one or two may be accounted for. This new feature may formally be expressed as

H = H polyads + H inter-polyads (3)
H inter-polyads = H P i /P j . (4)
In this case, the polyad number P is no longer a strict quantum number. Approximate quantum numbers are generated using the eigenvector analysis. These approximate quantum numbers should correspond to quantum numbers of the zero-order harmonic-oscillator + rigid-rotor approximation under the condition that the corresponding physical quantities are kept near-invariant under the effect of perturbations.

In addition to effective ITO spectroscopic models, the new MIRS version offers a possibility of

0-2600 [7]

37 CH3Cl 0-2600 [START_REF] Nikitin | Global analysis of 12 CH 3 35 Cl and 12 CH 3 37 Cl: simultaneous fit of the lower five polyads (0-2600 cm -1 )[END_REF] PH3 700-3500 [START_REF] Nikitin | Modeling of vibrational energy levels of methane from the ab initio constructed potential energy surface[END_REF] CH3F Global variational computation of energy levels and wave functions from a full hamiltonian of nuclear motion in the ITO representation using symmetrized harmonic oscillator / rigid rotor basis set for a singlet electronic state. The tools of the MIRS code allow the generation of all necessary ITO terms up to a requested order of the expansions (1-4). All necessary matrix elements are calculated analytically without loss of precision. The values of t-parameters involved in these expansions have to be provided by the user as the input file for the program. These values can be computed from a given molecular ab initio or empirical PES according to the algorithm described in our recent works [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][START_REF] Rey | The complete nuclear hamiltonian in the irreducible tensor operator formalism for methane[END_REF] up to arbitrary orders. At a given order of expansion a full H vib-rot contains all non-zero symmetry allowed inter-polyad blocks in Eq.( 4) systematically generated from an ab initio PES [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][START_REF] Rey | The complete nuclear hamiltonian in the irreducible tensor operator formalism for methane[END_REF]. Consequently this procedure usually accounts for many more vibrational interactions and the full hamiltonian contains more elementary ITO terms than an effective polyad model. Formally this corresponds to a summation over all inter-polyad couplings in Eq.( 4). But contrary to an effective spectroscopic model all terms with rotational powers Ω r > 2 in the full hamiltonian expansion vanish by definition [START_REF] Watson | Simplification of molecular vibration-rotation hamiltonian[END_REF].

Updated database

The 1).

A quite simple example is the sixth-order model for the ground-state, dyad, pentad and octad of 12 CH 4 . In this case, the vibrational coupling scheme coincides exactly with the one used in the STDS program [START_REF] Wenger | Spherical top data system (STDS) software for the simulation of spherical top spectra[END_REF]. This means that the effective parameter sets are identical in MIRS and STDS providing a good validation of both programs. For the following polyads (from the tetradecad upwards), even though the models are equivalent and include the same number of parameters at a given order of approximation, the two parameter sets are not in one to one correspondence. A similar system is illustrated with the sixth order model for the ground-state, dyad, pentad and octad of PH 3 [START_REF] Nikitin | Modeling of vibrational energy levels of methane from the ab initio constructed potential energy surface[END_REF].

The model for the lower two polyads (triad and nonad) of CH 3 D represents a complex band system of a symmetric top molecule [START_REF] Nikitin | Preliminary analysis of CH 3 D from 3250 to 3700 cm -1[END_REF]. The parameters of the MIRS model have been fitted to some 12589 line positions and to 2400 line intensities with an accuracy close to the experimental precision (about 0.001 cm -1 for positions and 4% for intensities). Altogether a common set of 441 effective hamiltonian parameters has been used to describe 13 vibrational states (ground state, triad, nonad) corresponding to 39 bands (6 fundamental, 3 overtone, 3 combination, and 27 hot bands). The intensities of the 9 bands involved in the nonad-ground state system were modeled at the second order of approximation using 83 effective dipole transition moment parameters.

Finally, results of the global analysis of the lower polyads of 12 CH 3 35 Cl and 12 CH 3 37 Cl are included.

A preliminary analysis of the infrared spectrum in the region from 0 to 2600 cm -1 was performed using 288(303) effective parameters for isotopomers 12 CH 3 35 Cl( 12 CH 3 37 Cl) to model five lower polyads (17 vibrational states) [START_REF] Nikitin | Global analysis of 12 CH 3 35 Cl and 12 CH 3 37 Cl: simultaneous fit of the lower five polyads (0-2600 cm -1 )[END_REF]. The precision on positions is of the order of 0.0007 cm -1 .

Applications

Basic applications

The basic applications already implemented in the original version [START_REF] Nikitin | The MIRS computer package for modeling the rovibrational spectra of polyatomic molecules[END_REF] are maintained in the new one.

Complete files of energy levels can be calculated using the command Calculus > build. Each record contains the energy value, the set of rigorous quantum numbers (total angular momentum J, symmetry point group irreducible representations and ranking number) as well as approximate quantum numbers derived from the coefficients of the eigenvector expansions in the initial basis set. Details on the experimental measurements fitted are also provided. The number and the root-mean-squares (rms) deviations of assigned transitions are given together with the corresponding averaged observed minus calculated residual for every example. The partition function can be derived from the calculated levels involved in the model or set to an appropriate external estimate [START_REF] Wenger | The partition sum of methane at high temperature[END_REF]. Several output formats including those of HITRAN [START_REF] Rothman | The HITRAN 2008 molecular spectroscopic database[END_REF] and GEISA [START_REF] Jacquinet-Husson | The geisa spectroscopic database: Current and future archive for earth and planetary atmosphere studies[END_REF] can be selected. The temperature and intensity threshold can be adjusted to meet specific needs for applications.

Advanced applications

Setting up new models

In order to set up a new model it is necessary to define the polyad structure and to introduce the modeling parameters. The modeling parameters are: (i) the symmetry point group of the molecule, (ii) the harmonic frequencies of the considered normal modes, (iii) the corresponding symmetry species (irreducible representations), (iv) a frequency gap used for gathering the subsequent vibrational sublevels into relevant vibrational polyads. Note that in the modeling procedure the frequencies can be entered in arbitrary units and do not need to match exactly the physical frequencies in order to provide flexibility. For example, the modeling of the CH 3 D molecule was performed by setting the six fundamental frequencies to 2, 2, 1, 2, 1, 1 respectively and the frequency gap to 0.1. This yielded the triad and nonad structure described in [START_REF] Nikitin | Analysis of the CH 3 D nonad from 2000 to 3300 cm -1[END_REF][START_REF] Nikitin | Preliminary analysis of CH 3 D from 3250 to 3700 cm -1[END_REF]. For CH 3 Cl the situation was more complex and the fundamental frequencies were set closer to the actual values: 2968, 1354, 733, 3039, 1452 and 1018, respectively with a frequency gap of 200 (Fig. 1) [START_REF] Nikitin | The MIRS computer package for modeling the rovibrational spectra of polyatomic molecules[END_REF][START_REF] Nikitin | Global analysis of 12 CH 3 35 Cl and 12 CH 3 37 Cl: simultaneous fit of the lower five polyads (0-2600 cm -1 )[END_REF][START_REF] Nikitin | New ground state constants of 12 CH 3 35[END_REF]. This method of the model set up is quite general. For example choosing a large energy gap results automatically in including a maximum of symmetry allowed vibration-rotation interaction terms. Then by removing some of them (via text-editor) one can build a model suitable for describing observed resonance perturbations in molecular spectra. By this way one can easily build an «overlapping-polyad» model in order to account for inter-polyad resonances. In all cases, the resulting polyad structure can be displayed for checking and modifying purposes.

By default, MIRS is set to build effective hamiltonian matrices and transition moments according to the vibrational extrapolation scheme in the normal mode harmonic oscillator basis. This means that for a given molecule a common set of eective hamiltonian parameters is set up to t and predict the subsequent vibrational polyads. The same scheme applies for transition moment parameters. For instance a band system from the ground state to a given polyad and all the corresponding hot band systems are described by a common set of eective parameters. Such features are generated automatically by the program and controlled by a few basic modeling parameters entered through the window menu reproduced in Fig. 1.

The nuclear spin statistical weights involved in intensity calculations are also manually introduced at this stage. Once the vibrational model is set up, the polyads and the transitions to consider are entered through another window menu exemplied in Fig. 2. For each polyad the order of the hamiltonian expansion and the maximum value of has to be entered. Similarly, for each type of transitions the order of the eective dipole moment expansion has to be specied.

At the end of the procedure the build command generates all internal les needed for subsequent calculations. The same command is used to update all les whenever needed. The other commands of the Calculus menu displayed in Fig. 2 are generally self-explanatory. More details are available from the Help command. Thanks to its intrinsic exibility MIRS can also be used to create quantum model hamiltonians not necessarily connected to real molecules. For instance, various systems involving coupled harmonic or anharmonic oscillators can be simulated numerically for further theoretical investigations.

This model includes the ground state and the lower five polyads of CH3Cl. The arrows represent the band systems involved : 5 originating from the ground base, 3 from the ν3 band, 2 from the ν6 band, 1 from polyad 3 and one from the ν3 + ν6 band. The circles represent pure rotational transitions within the ground state and polyad 3 upper states.

All the corresponding observed data (energy levels, line positions and line intensities) can be fitted coherently.

that for a given molecule a common set of effective hamiltonian parameters is set up to fit and predict the subsequent vibrational polyads. The same scheme applies for transition moment parameters.

For instance a band system from the ground state to a given polyad and all the corresponding hot band systems are described by a common set of effective parameters. Such features are generated automatically by the program and controlled by a few basic modeling parameters entered through the window menu reproduced in Fig. 1.

The nuclear spin statistical weights involved in intensity calculations are also manually introduced at this stage. Once the vibrational model is set up, the polyads and the transitions to consider are entered through another window menu exemplified in Fig. 2. For each polyad the order of the hamiltonian expansion and the maximum value of J has to be entered. Similarly, for each type of transitions the order of the effective dipole moment expansion has to be specified.

At the end of the procedure the build command generates all internal files needed for subsequent calculations. The same command is used to update all files whenever needed. The other commands of the Calculus menu displayed in Fig. 2 are generally self-explanatory. More details are available from the Help command. Thanks to its intrinsic flexibility MIRS can also be used to create quantum model hamiltonians not necessarily connected to real molecules. For instance, various systems involving coupled harmonic or anharmonic oscillators can be simulated numerically for further theoretical investigations.

Experimental data fitting (positions and intensities)

MIRS offers extensive set of tools for a non-linear least-squares fit of spectroscopic data (positions and intensities). According to the vibrational extrapolation scheme [START_REF] Champion | Spherical top spectra[END_REF] well suited for global analyses, the program is set to t simultaneously all transitions and/or vibration-rotation levels involved in a given project. The t of intensities is also achieved using a non-linear least-squares procedure with an adequate weighting already described in [START_REF] Wenger | Spherical top data system (STDS) software for the simulation of spherical top spectra[END_REF][START_REF] Wenger | Highly-spherical top data system (HTDS) software for spectrum simulation of octahedral XY 6 molecules[END_REF]. Note that in our approach a rigorous consistency is de facto applied between intensity and position ts. The reader is referred to the original paper [START_REF] Nikitin | The MIRS computer package for modeling the rovibrational spectra of polyatomic molecules[END_REF] for details on the sophisticated algorithms developed to overcome the increasing complexity of the problems to solve.

In the new version, two subprograms have been added to handle the adjustable parameters and their possible constraints. The Tools menu includes the Free_Parameters item to release all parameters that were previously xed automatically by the program. The Copy_Parameters item allows the copy of parameters from one le to another. This is especially useful when one needs to change the model, for example to increase the order of the hamiltonian expansion.

Another improvement has been added to manage situations encountered when approximate quantum numbers are not physically meaningful and well conserved. The assignment of observed to calculated data is based on the rigorous quantum numbers. For an isolated polyad model these are: (polyad number), (total angular momentum), (symmetry species) and the ranking index within each block. In some cases of nearby eigenvalues the ranking indices may undergo permutations from one iteration to another resulting to undesirable assignment jumps. MIRS allows temporarily removing from the t the assigned levels which present such behavior by detecting anomalous variations of the scalar product of the corresponding eigenvectors during the iteration process. For this one needs to calculate at least parts of the eigenvector matrix . If designates the initial eigenfunction within a given block and the current eigenfunction within the same block, the above mentioned scalar product can be expressed as [START_REF] Nikitin | Global analysis of chloromethane: determinability of ground state constants[END_REF] where and denote basis functions. For the great majority of the eigenvectors the scalar product ( 5) is close to unity and equivalently the matrix is almost diagonal. In a minority of cases this is no longer true. For example an matrix of the form 9

Refer to the Web site for explanations about non trivial options the program is set to fit simultaneously all transitions and/or vibration-rotation levels involved in a given project. The fit of intensities is also achieved using a non-linear least-squares procedure with an adequate weighting already described in [START_REF] Wenger | Spherical top data system (STDS) software for the simulation of spherical top spectra[END_REF][START_REF] Wenger | Highly-spherical top data system (HTDS) software for spectrum simulation of octahedral XY 6 molecules[END_REF]. Note that in our approach a rigorous consistency is de facto applied between intensity and position fits. The reader is referred to the original paper [START_REF] Nikitin | The MIRS computer package for modeling the rovibrational spectra of polyatomic molecules[END_REF] for details on the sophisticated algorithms developed to overcome the increasing complexity of the problems to solve.

In the new version, two subprograms have been added to handle the adjustable parameters and their possible constraints. The Tools menu includes the Free_Parameters item to release all parameters that were previously fixed automatically by the program. The Copy_Parameters item allows the copy of parameters from one file to another. This is especially useful when one needs to change the model, for example to increase the order of the hamiltonian expansion.

Another improvement has been added to manage situations encountered when approximate quantum numbers are not physically meaningful and well conserved. The assignment of observed to calculated data is based on the rigorous quantum numbers. For an isolated polyad model these are: P (polyad number), J (total angular momentum), C (symmetry species) and the ranking index n within each P, J, C block. In some cases of nearby eigenvalues the ranking indices may undergo permutations from one iteration to another resulting to undesirable assignment jumps. MIRS allows temporarily removing from the fit the assigned levels which present such behavior by detecting anomalous variations of the scalar product of the corresponding eigenvectors during the iteration process. For this one needs to calculate at least parts of the eigenvector matrix α. If f (0) i designates the i th initial eigenfunction within a given P, J, C block and f (k) j the j th current eigenfunction within the same P, J, C block, the above mentioned scalar product can be expressed as

< f (0) j |f (k) i >= σδ α (0) jσ α (k) iδ < σ|δ >= A j,i , (5)
where σ and δ denote basis functions. For the great majority of the eigenvectors the scalar product ( 5) is close to unity and equivalently the A matrix is almost diagonal. In a minority of cases this is no longer true. For example an A matrix of the form

A =         
           (6)
illustrates a situation where the two vectors corresponding to the middle of the matrix are rearranged while the first vector, associated to the diagonal element equal to 0.98, keeps its initial n index. Sometimes the rearrangement is not obvious, for instance when the diagonal element is not close to unity and simultaneously all non-diagonal values are not very large. This applies for the fourth vector of the matrix 6 with diagonal matrix element equal to 0.65. In this case the corresponding level must be excluded from the fit at least temporarily. The file with the initial eigenvectors is saved before the fit. For every iteration, some of the assigned levels are excluded from the fit and part of the levels previously excluded are re-assigned. If a large proportion of levels are excluded from the fit, then the iteration is unsuccessful and it becomes necessary to decrease the allowed variations for the parameters.

The fit of intensities is also subject to specific problems. Specific features to handle relative signs between transition parameters and interaction hamiltonian parameters are already described in section 4.3 of Ref. [START_REF] Nikitin | The MIRS computer package for modeling the rovibrational spectra of polyatomic molecules[END_REF]. Despite all the above advanced options, it should be emphasized that in any case a purely automatic selection of adjustable terms for positions or intensities in an arbitrarily built model is not recommended. All procedures have to rely on physical considerations and depend on experimental data set and observed perturbations involved.

Group theory re-coupling coefficients and commutator-algebra calculations

The calculations involved in effective hamiltonians for highly exited states can be also very cumbersome and requires program optimization. It was found that one of the bottlenecks is the slow calculation of 9C [START_REF] Champion | Spherical top spectra[END_REF] (other notations: X [START_REF] Griffith | The irreducible tensor method for molecular symmetry groups[END_REF], 9Γ symbols [START_REF] Zhilinskii | Method of irreducible tensorial operators in the theory of molecular spectra[END_REF]). The 12C symbols are required only for intensity calculations while 9C are used for line position calculations. In addition, 9C symbols are used in variational programs designed for the calculation of multidimensional irreducible matrix elements [START_REF] Nikitin | Global modeling of the lower three polyads of PH 3 : Preliminary results[END_REF][START_REF] Nikitin | Vibrational energy levels of methyl chloride calculated from full dimensional ab initio potential energy surface[END_REF][START_REF] Nikitin | Preliminary assignments of 2ν 3ν 4 hot band of 12 CH 4 in the 2µm transparency window from long-path FTS spectra[END_REF]. In the present version of MIRS, the calculation of 6C, 9C and 12C symbols for C 3v , T d , and O h point groups is based on the improved algorithms reported in [START_REF] Nikitin | An efficient code for calculation of the 6C, 9C and 12C symbols for C 3v , T d and O h point groups[END_REF] which is much more efficient than in the program previously used. This new algorithm [START_REF] Nikitin | An efficient code for calculation of the 6C, 9C and 12C symbols for C 3v , T d and O h point groups[END_REF] uses the symmetry relations between even and odd representations. It allows one to speed up the C-symbol calculation and increase the efficiency of spectroscopic programs based on the irreducible tensorial formalism.

MIRS can also be used by expert users to generate various types of coupling coefficients as well as commutators of ITOs in algebraic or numerical formats. Such tools can be useful for the theoretical investigation of the fundamental properties of effective models and for establishing their relations with the full nuclear hamiltonian. It is well-known that, in general, an effective Hamiltonian is not uniquely determined from its eivenvalues [START_REF] Watson | Determination of centrifugal distortion coefficients of asymmetric-top molecules[END_REF][START_REF] Klein | Degenerate perturbation theory[END_REF][START_REF] Tyuterev | Effective hamiltonians, in: Intramolecular interactions and Infrared spectra of Atmospheric gases[END_REF][START_REF] Tyuterev | Generalized contact transformations of a hamiltonian with a quasi-degenerate zero-order approximation -application to accidental vibration-rotation resonances in molecules[END_REF][START_REF] Makushkin | Perturbation Methods and Effective Hamiltonians in Molecular Spectroscopy[END_REF][START_REF] Jorgensen | Effective hamiltonians[END_REF]. The ambiguity of the determination of H ef f from experimental energies is particularly important in the case of resonance interactions due to quasidegeneracies of vibrational states [START_REF] Tyuterev | Generalized contact transformations of a hamiltonian with a quasi-degenerate zero-order approximation -application to accidental vibration-rotation resonances in molecules[END_REF][START_REF] Perevalov | Reduction of the centrifugal-distortion hamiltonian of asymmetric-top molecules in the case of accidental resonances -2 interacting states -lower-order terms[END_REF][START_REF] Perevalov | Ambiguity of spectroscopic parameters in the case of accidental vibration-rotation resonances in tetrahedral molecules -R 2 J and R 2 J 2 terms for E-F 2 interacting states[END_REF][START_REF] Tyuterev | Parameters of reduced hamiltonian and invariant parameters of interacting E and F 2 fundamentals of tetrahedral moleculesν 2 and ν 4 bands of 12 CH 4 and 28 SiH 4[END_REF][START_REF] Lobodenko | Reduced effective hamiltonian for coriolis-interacting ν n and ν t fundamentals of C 3v molecules[END_REF]. In order to improve the convergence of least-square fits to experimental spectra and to obtain a unique set of fitted parameters, the spectroscopic effective

?

hamiltonian models have to be constrained using the so called «reduction theory» [START_REF] Watson | Determination of centrifugal distortion coefficients of asymmetric-top molecules[END_REF][START_REF] Aliev | Higher-order effects in the vibration-rotation spectra of semirigid molecules[END_REF][START_REF] Perevalov | Reduction of the centrifugal-distortion hamiltonian of asymmetric-top molecules in the case of accidental resonances -2 interacting states -lower-order terms[END_REF][START_REF] Perevalov | Reduced effective-hamiltonians for degenerate vibrational-states of methane-type molecules[END_REF][START_REF] Lobodenko | Reduced effective hamiltonian for coriolis-interacting ν n and ν t fundamentals of C 3v molecules[END_REF][START_REF] Tyuterev | Reduced effective-hamiltonians for degenerate excited vibrational-states of tetrahedral molecules -application to 2ν 2 , ν 2 + ν 4 and 2ν 4 of CH 4[END_REF][START_REF] Sarka | Perturbation theory, effective hamiltonians and force constants[END_REF]. The latter relays on extended commutator / anti-commutator algebra calculations. This Lie algebra of ITO plays a key role for a full account of symmetry properties in the contact transformation method [START_REF] Makushkin | Perturbation Methods and Effective Hamiltonians in Molecular Spectroscopy[END_REF][START_REF] Zhilinskii | Method of irreducible tensorial operators in the theory of molecular spectra[END_REF][START_REF] Tyuterev | Effective hamiltonians and perturbation theory for quantum bound states of nuclear motion in molecules[END_REF]. 5 Modified specifications

Vibrational and Ro-vibrational calculation

All MIRS parameter files, dipole moment files and basis function files are text files. In order to calculate highly exited states using full nuclear hamiltonians a new coding of the symmetric powers (l, m, n) [START_REF] Nikitin | Improved algorithms for the modeling of vibrational polyads of polyatomic molecules: Application to T d , O h , and C 3v molecules[END_REF] was necessary. The original version of the program used only digits 0, .., 9 for triplet numbers (l, m, n).

So only symmetric powers up to 9 were available although new applications required numbers with two digits. To ensure the compatibility with the original version and to preserve a good legibility for low symmetric powers, we decided to keep one single symbol according to Table 2. Note that now the number of polyads is limited to 99 which is far beyond the limit of realistic calculations.

Options

For efficient calculations on multi-kernel (multiprocessors) computers, the new MIRS program uses OpenMP version 2.0 (http://openmp.org). Up to 8 kernels can be used simultaneously for matrix element calculations. As mentioned previously the full ro-vibrational hamiltonian contains terms up to the second rotational power only, while effective hamiltonian may contain all possible rotational powers. Different programs may be used for eigenvalue calculation (see Diagonalization option). It is sometimes convenient to store on disk the full matrix for each P, J, C block because the diagonalization of a large number of matrices may be very burdensome.

Limitations

The previous version of the MIRS code was a 32-bit one with considerable restrictions on the number of tensors involved regarding the power of hamiltonian expansions and the size of polyads. The present version extends the number of polyads P from 9 to 99 while the number of terms and the polyad size is limited by the available computer memory. In order to speed-up calculations, sufficient computer resources are required to store all vibrational matrix elements in random access memory. The number of molecular vibration modes is limited to 20 and the maximum total angular momentum quantum number J for the T d point group is limited to 199 [START_REF] Rey | Orientation of O(3) and SU(2) ⊗ C 1 representations in cubic point groups O h , T d for application to molecular spectroscopy[END_REF].

Getting and installing

The

Conclusion

The rapid increase of computer power and the progresses in multiprocessor parallellization have opened the way to considerable improvements of theoretical predictions and data reduction for complex high-energy band systems of polyatomic molecules. This may help in understanding new spectra involving high vibration-rotation excitations which have become experimentally accessible recently or can be available in future. Efficient modeling of spectra requires the development of advanced software specifically optimized for these purposes. Various methods and program implementations have been developed for computation of molecular spectra either from effective spectroscopic models [START_REF] Tashkun | Global fittings of 14 N 15 N 16 O and 15 N 14 N 16 O vibrational-rotational line positions using the effective hamiltonian approach[END_REF][START_REF] Sarka | Perturbation theory, effective hamiltonians and force constants[END_REF][START_REF] Wenger | Spherical top data system (STDS) software for the simulation of spherical top spectra[END_REF][START_REF] Champion | Spherical top spectra[END_REF][START_REF] Flaud | Vibration-rotation intensities in H 2 O-type molecules application to 2ν 2 -band, ν 1 -band, and ν 3 -band of H 16 2 O[END_REF][START_REF] Tashkun | GIP: a program for experimental data reduction in molecular spectroscopy[END_REF][START_REF] Herman | The acetylene ground state saga[END_REF][START_REF] Amyay | Acetylene, 12 C 2 H 2 : Refined analysis of CRDS spectra around 1.52 µm[END_REF] (and references therein) or from potential and dipole moment surfaces using variational approaches [START_REF] Jensen | A variational calculation of the rotation vibration energies for H 2 O from abinitio data[END_REF][START_REF] Jensen | Calculation of rotation vibration linestrengths for triatomic-molecules using a variational approach -application to the fundamental bands of CH 2[END_REF][START_REF] Schwenke | Variational calculations of rovibrational energy levels and transition intensities for tetratomic molecules[END_REF][START_REF] Bowman | Ab-initio-based potential energy surfaces for complex molecules and molecular complexes[END_REF][START_REF] Yurchenko | A variationally computed line list for hot NH 3[END_REF][START_REF] Carter | The variational method for the calculation of ro-vibrational energy-levels[END_REF][START_REF] Nikitin | Rotational and vibrational energy levels of methane calculated from a new potential energy surface[END_REF][START_REF] Zuniga | Optimal internal coordinates, vibrational spectrum, and effective hamiltonian for ozone[END_REF][START_REF] Schwenke | Towards accurate ab initio predictions of the vibrational spectrum of methane[END_REF][START_REF] Matyus | Toward black-box-type full-and reduced-dimensional variational (ro)vibrational computations[END_REF][START_REF] Tyuterev | Global variational calculations of high-resolution rovibrational spectra: isotopic effects, intensity anomalies and experimental confirmations for H 2 S, HDS, D 2 S molecules[END_REF], discrete variable representation (DVR) [START_REF] Light | Discrete-variable representations and their utilization[END_REF][START_REF] Wang | Ab initio potential and dipole moment surfaces for water. ii. localmonomer calculations of the infrared spectra of water clusters[END_REF][START_REF] Wang | Flexible, ab initio potential, and dipole moment surfaces for water. i. tests and applications for clusters up to the 22-mer[END_REF][START_REF] Tennyson | Variational calculations of vibration-rotation spectra[END_REF][START_REF] Tennyson | Methods in Computational Chemistry[END_REF][START_REF] Wang | New ideas for using contracted basis functions with a lanczos eigensolver for computing vibrational spectra of molecules with four or more atoms[END_REF] or filter-digitalization [START_REF] Mandelshtam | The quantum resonance spectrum of the H 3 + molecular ion for j = 0. an accurate calculation using filter diagonalization[END_REF] techniques. A comparative analysis of advantages and limitations of these approaches is beyond the scope of the present paper ( see ref. [START_REF] Tyuterev | Recent advances in global variational and effective calculations of the line positions and intensities for triatomic molecules: some features of a new generation of spectroscopic databanks[END_REF] for some related discussion). Complementary results for non-empirical effective hamiltonians and molecular spectra can be obtained via perturbation theory and Contact Transformation methods [START_REF] Van Vleck | On sigma-type doubling and electron spin in the spectra of diatomic molecules[END_REF][START_REF] Amat | Rotation -vibration of polyatomic molecules[END_REF][START_REF] Aliev | Higher-order effects in the vibration-rotation spectra of semirigid molecules[END_REF][START_REF] Tyuterev | Generalized contact transformations of a hamiltonian with a quasi-degenerate zero-order approximation -application to accidental vibration-rotation resonances in molecules[END_REF][START_REF] Makushkin | Perturbation Methods and Effective Hamiltonians in Molecular Spectroscopy[END_REF][START_REF] Sibert | Theoretical studies of vibrationally excited polyatomic molecules using canonical vanvleck perturbation theory[END_REF][START_REF] Camy-Peyret | Vibration-rotation dipole moment operator for asymmetric top molecules[END_REF][START_REF] Wang | A nine-dimensional perturbative treatment of the vibrations of methane and its isotopomers[END_REF][START_REF] Sarka | Perturbation theory, effective hamiltonians and force constants[END_REF][START_REF] Tyuterev | High-order contact transformations: General algorithm, computer implementation and triatomic tests[END_REF][START_REF] Lamouroux | Accurate calculation of transition moment parameters for rovibrational bands from ab initio dipole and potential surfaces: Application to fundamental bands of the water molecule[END_REF][START_REF] Cassam-Chenai | Ab initio effective rotational hamiltonians: A comparative study[END_REF]]. An efficient computational software for the experimental data reduction in molecular spectroscopy and for the generation of line list parameters allows completing and extending spectroscopic databases which are planned to be linked in frame of the «Virtual Atomic and Molecular Data Center» (VAMDC) European project [START_REF] Dubernet | Virtual atomic and molecular data centre[END_REF].

The reported extension of the MIRS computer package is designed to fully account for symmetry properties by using irreducible tensor representations for all physical properties and operators involved in direct calculations as well as in experimental molecular spectra fitting. Its particular feature is a unified treatment of effective polyad models (with the possibility of including explicitly inter-polyad interactions) and of full ab initio hamiltonian normal mode expansions in the frame of the ITO formalism.

The new MIRS includes more efficient and faster algorithms for the calculation of coefficients related to molecular symmetry properties (6C, 9C and 12C symbols for C 3v , T d , and O h point groups) and for a better convergence of least-square-fit iterations. Several user-friendly tools are included dealing with the ITO algebra and with the generation of symmetry allowed terms in spectroscopic effective hamiltonians, transition moment operators, potential energy function, full ab initio hamiltonian expansions, and unitary transformations involved in the data reduction theory. This approach combined with an appropriate implementation of the high-order Contact Transformation method [START_REF] Tyuterev | High-order contact transformations: General algorithm, computer implementation and triatomic tests[END_REF][START_REF] Tyuterev | High-order contact transformations for methane molecule[END_REF] can be used, in principle, to perform a systematic computation of ab initio values of resonance coupling parameters in spectroscopic hamiltonians which were poorly determined in purely empirical models. The derivation of spectroscopic models for the CH 3 Cl, CH 3 Br, CH 3 I, GeH 4 , AsH 3 molecules constructed in this way is currently in progress. We plan to provide in future a UNIX/LINUX version of MIRS.

Figure 1 :Figure 1 :

 11 Figure 1: MIRS screen copy showing the model definition for CH 3 ClFigure 1: MIRS screen copy showing the model denition for CH Cl

Figure 2 :

 2 Figure 2: MIRS option window

Figure 2 :

 2 Figure 2: MIRS option window

4. 2 . 4

 24 Generation of symmetry allowed terms in the irreducible tensor formWe have added into the new MIRS a possibility for the user to generate a list of symmetry allowed terms involved in the ITO normal mode expansion of either an effective polyad hamiltonian (including inter-polyad interactions), or of an effective dipole moment or of the potential energy function up to an arbitrary order. For example, the list of ITO terms involved in a second order PES expansion for an XY 4 molecule can be obtain from the menu «tools > build parameters» by choosing «type of file = PES», then by choosing a vibrational pattern, and finally by defining the desired model. A similar MIRS tool is also operational for listing all allowed ITO terms involved in generators of unitary transformations which preserve group symmetry, time reversal and hermicity properties of transformed physical properties of a molecule. The present version of the package includes a sample example of symmetry allowed terms in S-generators involved in the problem of the effective hamiltonian reduction for the triad of CH 3 D.

Table 1 :

 1 Data available from the MIRS package

	Molecules Spectral region References
	CH3D	1500-3700	[8]
	CH4	1700-4600	[11]

35

CH3Cl

Table 2 :

 2 Symbolic coding of vibrational powers

	Numbers	Symbols
	0, 1, 2, . . . , 9	0, 1, 2, . . . , 9
	10, 11, 12, . . . , 35 A, B, C, . . . , Z
	36, 37, 38, . . . , 61	a, b, c, . . . , z
	62, 63	{, |
	> 63

  MIRS package can be downloaded freely from our Web sites http://www.iao.ru/mirs/mirs.htm 1 and as supplementary data in the online version of the article. It consists of a self-extractive archive containing various tutorial examples. MIRS, written in C++, includes all necessary executable files for Windows XP and higher. Two versions (32 bit and 64 bit) of executable files are available. MIRS 64 bit works only on 64 bit windows operating systems. The major part of the examples based on effective hamiltonians (except CH4_polyad7 and CH4_polyad9) can be run using the 32 bit version. Fit operations can be run on the 32 bit version only. As a rule, full hamiltonian examples can be calculated using the 64 bit version only. Memory, CPU and disk space requirements depend on the complexity of the project. The package itself requires a minimum of 15 MB of free disk space. The tables of coupling coefficients occupy up to 150 MB when the highest J value considered so far (80) is involved.All examples and projects included in the package can be executed using a computer with at least 2 GB RAM and 500 MB of free disk space. A typical execution time using a processor at 2 GHz is about 10 min to build the CH 3 D project (all matrix elements and prediction files) and less than 15 min to build each of the CH 3 Cl projects. System requirements for complex examples are described in readme.txt files. For example CH4_polyad9 can be calculated only on computer with at least 24GB of memory and 160GB of free disk space. Standard features and options are directly accessible from the tutorial projects in which all necessary modeling files are included. The Help menu can be accessed at any stage of the project building and subsequent calculations. Refer to our Web sites for complete installation and running instructions.

4.2.2 Experimental data tting (positions and intensities)MIRS oers extensive set of tools for a non-linear least-squares t of spectroscopic data (positions and intensities). According to the vibrational extrapolation scheme[START_REF] Champion | Spherical top spectra[END_REF] well suited for global analyses,

also available from http://xeon.univ-reims.fr/Mirs/ or http://icb.u-bourgogne.fr/OMR/SMA/SHTDS/MIRS.html

Acknowledgments

Part of the work was performed in the framework of LEFE-CHAT National Program CNRS (France). Support from Agence Nationale de la Recherche (France) through the project CH4@Titan (ref: BLAN08-2 321467) , from GDRI SAMIA involving Tomsk (Russia), Hefei (China), CNRS (France) and from the VAMDC EC project is gratefully acknowledged.

The full package including on-line documentation and recent data are freely available at http://www.iao.ru/mirs/mirs.htm or http://xeon.

Installation package

The installation package is freely available from the attached file.


É Massot

J Margueron

G Chanfray

On the maximum mass of hyperonic neutron stars

Keywords: 97, 60, Jd - 26, 60, Kp -

Chiral Lagrangian and quark-meson coupling models of hyperon matter are used to estimate the maximum mass of neutron stars. Our relativistic calculations include, for the first time, both Hartree and Fock contributions in a consistent manner. Being related to the underlying quark structure of baryons, these models are considered to be good candidates for describing the dense core of neutron stars. Taking account of the known experimental constraints at saturation density, the equations of state deduced from these relativistic approaches cannot sustain a neutron star with a mass larger than 1.6-1.66 M⊙.

Neutron stars are the most compact stellar objects, and within them, particles that are unstable on earth could be stabilized at densities above 10 14 g cm -3 . The equation of state (EoS) at these densities is still largely unknown due to the poor experimental knowledge on hadron dynamics. However, the recent measurements of the millisecond pulsars PSR J1614-2230 extend the maximum observed mass from (1.67±0.02) M ⊙ [1] to (1.97±0.04) M ⊙ [2]. This imposes new constrains on the EoS of dense matter [START_REF] Lattimer | To appear in Gerry Brown's Festscrift[END_REF]. A major question is whether the EoS including hyperons can be ruled out by this new observation. Relativistic meanfield [START_REF] Schafner-Bielich | [END_REF]5] and Brueckner-Hartree-Fock [6][7][8] models have predicted that the effect of hyperons in dense matter is to soften the EoS and, therefore, to lower the maximum mass. A consistent treatment of the Fock term in a relativistic approach is however missing. In this Letter we analyze the impact of the Fock term in two effective, relativistic meson-exchange models related to the quark nature of hadrons: the chiral Lagrangian model [9] and the quark meson coupling (QMC) model [10].

These models are based on a relativistic meson-exchange theory of the Walecka type, where the mean-field properties of nuclear matter are calculated in the Hartree-Fock approximation [11]. Considering scalar s, ω, δ, ρ and π mesons as well as nucleons and hyperons, the Lagrangian can be written with the usual notations [10]

L = f ψf (iγ µ ∂ µ -M f (s))ψ f -V (s) + 1 2 ∂ µ s∂ µ s - f ψf g f ω ω µ γ µ ψ f + 1 2 m 2 ω ω µ ω µ - 1 4 F µν F µν -g δ f ψf δ • Iψ f - 1 2 m 2 δ δ • δ + 1 2 ∂ µ δ • ∂ µ δ -g ρ ρ µ • f ψf γ µ Iψ f -g ρ κ ρ 2M N ∂ ν ρ µ • f ψf σµν Iψ f + 1 2 m 2 ρ ρ µ • ρ µ - 1 4 F ′ µν • F ′ µν + g A 2f π 3 5 ∂ µ π • f ψf G µ T ψ f + 1 2 m 2 π π • π - 1 2 ∂ µ π • ∂ µ π, (1)
where the sum over the flavor index f means a summation over the baryons: N, Λ, Σ, Ξ. The isospin operator I and the Gamow-Teller operator G µ T are given in Ref. [10]. In the chiral Lagrangian model [9], the meson fields are related to the properties of the QCD condensate. Scalar s and pseudo-scalar π fields are associated with the fluctuations of the chiral quark condensate [12], related to the non-linear "Mexican hat" potential,

V (s) = m 2 s 2 s 2 + m 2 s -m 2 π 2f π s 3 + m 2 s -m 2 π 8f 2 π s 4 , (2)
deduced from chiral spontaneous symmetry breaking. In this model, the chiral partner A 1 of the ρ meson, being too heavy, is ignored and the ω meson is chiral singlet. The first chiral models failed to reproduce the saturation properties of nuclear matter [12]. A better description for the saturation is obtained by introducing the scalar polarizability of the nucleon κ N S in M f (s) [START_REF] Kerman | Berkeley 1974, Proceedings[END_REF][START_REF] Bentz | [END_REF],

M f (s) = M f + g s w f s s + 1 2 κ N S wf s s 2 , (3)
which reflects the effects of confinement as originally proposed in Ref. [15]. The weight factors w f s and wf s will be discussed later.

A further extension of the model consists in replacing κ N S in Eq. ( 3) by κ N S (s) = κ N S (1 + (2s)/(3f π )) where f π is the decay constant of the pion [9]. Another extension is related to the ρ and π tensor interactions, where derivative couplings lead to a contact term that can be partially or totally suppressed by short-range correlations [9,11]. We consider chiral models at the Hartree approximation (such as MC1-H in table 1) as well as in the Hartree-Fock approximation (such as MCi-HF in table 1). The index i in MCi-H(F) models refers to different combinaisons of the extensions discussed: (i=1), constant κ N S in Eq. ( 3) and totally suppressed contact term; (i=2) κ N S (s) in Eq. ( 3) and totally suppressed contact term; (i=3) constant κ N S and partially suppressed contact term; (i=4) κ N S (s) and partially suppressed contact term. More details on these extensions can be found in Ref. [9].

In the quark-meson coupling model QMC700 the scalar potential (2) is quadratic, V (s) = m 2 s s 2 /2, at variance with the chiral Lagrangian model (2); the scalar polarizability of the nucleon κ N S is kept constant as in Eq. ( 3); and only s, ω and ρ vector mesons are considered (κ ρ = 0). The other extensions presented here-before are disregarded. In Ref. [10], self-energies are calculated in the Hartree approximation while a non-relativistic approximation of the Fock contribution to the energy is included. In this paper, we present modified versions of the QMC model in which both Hartree and Fock contributions are treated consistently.

The parameters of chiral Lagrangian and QMC models are given in table 1. In the chiral model, the scalar coupling constant is fixed from the sigma linear model to be g s = M N /f π = 10. Since the dimensionless parameter C = f 2 π κ N S /(2M N ) estimated from lattice simulation is around 1.25 [18], the chiral model fixes C to be consistent with this value but a flexibility is retained for reproducing saturation properties as explained below. In the QMC model, the dimensionless parameter C is smaller and deduced from the nucleon mass in the bag model. The coupling constant g ρ is adjusted to fit the symmetry energy in the chiral Hartree model MC1-H and in QMC models while in chiral Hartree-Fock models MCi-HF g ρ is fixed to satisfy the vector-dominance model giving g ρ = 2.65 and the ρ-tensor coupling constant κ ρ is adjusted to the symmetry energy. The axial coupling constant is fixed to be g A = 1.25 and the δ coupling is fixed to g δ = 1. The other parameters (m s , g ω and C) are chosen to reproduce the saturation properties (n 0 , B 0 and K 0 ) reported on table 1. In QMC model, the couplings g s and g ω are fixed to reproduce n 0 and B 0 ; moreover in the parameterization QMC-HF3 the parameter C is modified to decrease the value of K 0 .

The predictions for the slope of the symmetry energy L 0 and the nucleon Dirac mass M * N given in table 1 for the two models are consistent with experimental values lying in the range 60-90 MeV [16] for the slope and in the range 0.7-0.9M N for the Dirac masses.

Hyperons are included as in Ref. [10]: the coupling constants are written as g f m = w f m g m where w f m is the weight factor for the meson m coupled to the baryon f . For the mesons ω, ρ, π and δ, the meson-hyperon couplings are deduced from the meson-nucleon couplings based on SU (6) symmetry imposing w f ω = (1 + s f /3), where s f is the strangeness of the baryon f . For the isovector ρ or pseudo-vector π mesons, this symmetry is accounted for by the isospin and Gamow-Teller operators. Only the coupling constants of the scalar s meson are allowed to vary around the values imposed by SU (6) symmetry. In QMC700, the weights w f s and wf s are adjusted in the bag model and scale with the bag radius [10]. In the chiral model and in the extensions of QMC in table 1, they are adjusted to the non-relativistic potentials of hyperons at saturation,

V N R (f ) = Σ f S + E f M f (s) Σ f 0 + Σ f 2 S -Σ f 2 0 2M f (s) , (4)
where E f is the single-particle energy and Σ f S and Σ f 0 are the scalar and time-component self-energies [9]. V N R (f ) is fixed to be -30 MeV for f = Λ [17], 30 MeV for f = Σ [19] and -18 MeV for f = Ξ [START_REF] Schafner-Bielich | [END_REF]. While the value for the Λ is quite certain, that for the other hyperons are still under debate. The weights w f s are given in table 2 while we choose wf s = (1 + s f /3) for all baryons f . Being partly related to the underlying quark structure of baryons the chiral and QMC models are good candidates for exploration of the properties of dense core of neutron stars. In table 3 we explore different models for the EoS with and without hyperons and their predictions for the maximum mass M max and associated radius R(M max ). The first three EoS are based only on nucleons. For the following three models in table 3, the EoS is based on non-interacting hyperons. The maximum mass for these models is strongly reduced compared to models without hyperons. Interacting hyperons are taken into account in the next 10 models in table 3. For the five chiral models the maximum mass is increased compared to the models with free hyperons, but it remains smaller than that predicted by the models without hyperons. There is a small variation of about 0.05 M ⊙ between the different HF models depending on the extension of the chiral model under consideration.

In the last four lines of table 3 we present variants of the QMC model whose parameters are given in table 1.

The original QMC700/NY model predicts a maximum mass of 2.00 M ⊙ with hyperons [10]. The model QMC-H/NY is a consistent Hartree model where the s and ω coupling constants are readjusted to reproduce saturation properties (n 0 , B 0 and V N R (f The maximum mass in the model QMC-H/NY is lower than that of the original QMC700/NY since the incompressibility is lower. In the model QMC-HF1/NY the Hartree and Fock terms are consistently treated in the mean-field potential and the same readjustment procedure as in QMC-H/NY is followed. The Fock contribution to the total energy is however treated non-relativistically as in the QMC700/NY model. Including the complete Fock contribution in the energy, we obtain the fully relativistic QMC-HF2/NY model. The maximum mass is not altered by the fully relativistic treatment of the Fock term. Finally, we reduced the incompressibility by increasing the parameter C from 0.51 to 1 (see table 1) and the maximum mass decreases from 1.97 to 1.66 M ⊙ . From this analysis, we conclude that the large mass obtained in the QMC700/NY model [10] is mainly due to an overestimated and incompressibility.

We show in figure 1 the mass-radius relation for the various EoS presented in this Letter. Dotted lines show results for the models without hyperons while the solid lines stand for the models including interacting hyperons. As expected the EoS with hyperons reach lower maximum masses than the models without hyperons, except the model QMC700 due to its large and unrealistic incompressibility. The maximum mass obtained with hypernuclear EoS does not change much for different models. This is due to a self-regulating compensation effect between the softening of the EoS and the onset of the hyperonic degree of freedom [7,8]. The maximum mass of about 1.6 M ⊙ predicted by the relativistic approach is slightly larger than that of 1.4 M ⊙ found in Ref. [8]. The difference of 0.2 M ⊙ is mainly related to the larger incompressibility of the relativistic models. It is also observed that the radii for the Hartree models are smaller than that of the Hartree-Fock ones, and that the predicted radii of the relativistic models and that of Ref. [8] are comparable.

The abundances for nucleons and hyperons are shown for various EoS in figure 2: with free hyperons (MC1-H/NY F G ) and with interacting hyperons (MC1-H/NY and MC1-HF/NY). The order of appearance of hyperons is different for these models and depends strongly on the corresponding interactions. In the model MC1-H/NY F with free hyperons, the Σ -is non interacting and appears first. On the other hand, in the models MC1-H/NY and MC1-HF/NY, the Σ -interaction is repulsive while the Ξ - interaction is attractive, therefore the latter tends to prevent the Σ -from appearing. In the model MC1-H/NY, the Λ appears first followed closely in density by the Ξ -, then at a larger density by the Ξ 0 , while for the model MC1-HF/NY the Ξ -and Ξ 0 appear first and push the other hyperons to larger densities.

We have presented in this Letter different EoS for hyperon matter derived from both chiral and QMC models. These models are thermodynamically consistent and treat the Hartree and Fock terms on equal footing. We have found that the different chiral parameterizations give a maximum neutron star mass that does not exceed 1.6 M ⊙ for HF models with K 0 ≈ 270 MeV. Correcting the QMC700 model [10] by including the complete relativistic contribution to the Fock term, a correct incompressibility and the experimental constraints for the hyperon meanfields, we found a decrease of the predicted maximum mass from 2.00 down to 1.66 M ⊙ . We additionally checked that the effect of rotation with the period of 3.15 ms (as for PSR J1614-2230) does not increase the maximum mass by more than 0.02 M ⊙ . We therefore conclude that it is difficult to reconcile any of the model presented in this work with the observed mass of PSR J1614-2230 and the empirical knowledge of saturation properties of nuclear matter. Unless an unexpected property of the hyperon interaction is missing in the present model, as well as in the BHF models of Refs. [6][7][8]20], our results tends to exclude hypernuclear matter to be present in the core of massive neutron stars. It reinforces alternative models such as for instance the model of deconfined quark matter, for which experimental constraints are almost inexistent. We can therefore conclude that the equation of state of dense matter in the core of neutron star is still not even qualitatively understood.

Fig. 1 :

 1 Fig. 1: Mass of neutron stars versus the radius for selected EoS.

Fig. 2 :

 2 Fig. 2: Particules abundances for the various models of table 3: MC1-H/NYF G (top), MC1-H/NY (middle) and MC1-HF/NY (bottom).

Table 1 :

 1 Parameters of chiral and QMC models and saturation properties:

	density (n0), binding energy (B0), incompressibility

Table 2 :

 2 Weight factor w f s for the hyperons Λ, Σ and Ξ.

	Model	Λ	Σ	Ξ
	MC1-H	0.61	0.18	0.33
	MC1-HF	1.00	0.50	0.55
	MC2-HF	1.00	0.51	0.52
	MC3-HF	0.91	0.38	0.48
	MC4-HF	0.89	0.39	0.47
	QMC-H	0.62	0.35	0.32
	QMC-HF1	0.71	0.51	0.37
	QMC-HF2	0.64	0.41	0.33
	QMC-HF3	0.70	0.33	0.37

Table 3 :

 3 Different models corresponding to different approximations for nucleons (second column) and for hyperons (third column). The last three columns correspond to the maximum mass Mmax, its associated radius Rmax and the corresponding baryonic density n c max .

	EoS	Nucleon	Hyperon Mmax R(Mmax) n c max
		N	Y	(M ⊙ )	(km)	(in n 0 )
	MC1-H/N	MC1-H	No	1.65	10.6	7.8
	MC1-HF/N	MC1-HF	No	1.87	10.8	7.2
	MC3-HF/N	MC3-HF	No	1.85	10.7	7.3
	MC1-H/NY F G	MC1-H Fermi Gas 1.02	11.9	5.6
	MC1-HF/NY F G MC1-HF Fermi Gas 1.12	12.0	5.4
	MC3-HF/NY F G MC3-HF Fermi Gas 1.13	11.9	5.5
	MC1-H/NY	MC1-H	MC1-H	1.45	11.9	6.5
	MC1-HF/NY	MC1-HF MC1-HF 1.58	12.2	5.6
	MC2-HF/NY	MC2-HF MC2-HF 1.60	12.2	5.5
	MC3-HF/NY	MC3-HF MC3-HF 1.55	12.0	5.6
	MC4-HF/NY	MC4-HF MC4-HF 1.55	12.0	5.6
	QMC700/NY	QMC700 QMC700 2.00	12.5	5.1
	QMC-H/NY	QMC-H	QMC-H	1.68	12.2	5.5
	QMC-HF1/NY QMC-HF1 QMC-HF1 1.97	12.3	5.2
	QMC-HF2/NY QMC-HF2 QMC-HF2 1.97	13.3	4.6
	QMC-HF3/NY QMC-HF3 QMC-HF3 1.66	12.3	5.3

We thank I. Vidaña for fruitful discussions during the completion of this work. This work was partially supported by the ANR NExEN and SN2NS project ANR-10-BLAN-0503 contracts and COMPSTAR, an ESF Research Networking Program.