Andrei V Nikitin

Alexander E Protasevich

Alena A Rodina

Michael Rey

Attila Tajti

Vladimir G Tyuterev

V E Zuev

Ro-Vibrational levels and (e-f) splitting of acetylene molecule calculated from new potential energy surfaces

Keywords: acetylene, potential energy surface, vibrational energy levels, kinetic energy operator

Ro-vibrational energy levels of C2H2 are reported using variational nuclear motion calculations from new ab initio and empirically optimized full six-dimensional ab initio potential energy surfaces in the ground electronic state of the acetylene molecule. Ab initio calculations are based on extended electronic structure coupled-cluster calculations for dynamic electron correlations. The calculations account for the triple, quadruple and pentuple excitations as well as relativistic and diagonal Born-Oppenheimer corrections. Variational nuclear motion calculations were performed using the exact kinetic energy operator in orthogonal coordinates.

The convergence of energy levels calculations versus the size of the vibrational basis set functions was verified. Our best ab initio potential energy surface that includes the above mentioned contributions provides the RMS (obs.-calc.) errors of 0.95 cm -1 for five fundamental energy levels. The largest contribution to the RMS error is caused primarily by a significant deviation of the ν4 fundamental frequency. Experimental values of 120 vibrational band origins were used to empirically adjust few lower-order parameters of the potential energy surface. The average error drops down to 0.45 cm -1 or 0.25 cm -1 for empirically optimized potential energy function with two or seven adjusted parameters corresponding to quadratic force field terms and one third order term. The splitting of (ef) rovibrational doublets and their J dependence were calculated with high accuracy due to the full account of Coriolis interactions. Computed (ef) splittings allow us to check the correctness of the assignment of empirical energy levels. The estimation of the accuracy for the calculated vibrational levels in an extended range up to 9500 cm -1 shows that the set of ab initio vibrational levels can be used for future assignmens of empirically unknown ro-vibrational energy levels. The comparison of the calculated and experimental ro-vibrational energy levels of the C2D2 and 13 C2H2 isotopologues is also reported.

Introduction

Calculation of energy levels for acetylene molecule from the force field parameters or from the potential energy surface (PES) with application to spectroscopy analyses was the subject of many studies during last decades [1] [2] [3] [4] [START_REF] Allen | [END_REF] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]. Acetylene is one of the most studied species among four-atomic linear molecules [18] [19] [20] [21] [22] [23]. A large set of empirically determined energy levels has been obtained in [24] [25] [26] [27] [28] [29] [23] due to the importance of acetylene absorption/emission spectra for various applications [30] [31] [17]. The PES appears to be flat with respect to the torsion angle near the linear equilibrium geometry if one of the CCH angles is close to . For this reason, the vibrational wave functions are delocalized toward the torsion degree of freedom for the corresponding set of geometrical configurations. In this context the calculation of energy levels faces some problems similar to those of nonrigid molecules. For linear A2B2 molecules, there remain 9 vibrational-rotational degrees of freedom after excluding the translational motion. It is well-known that the separation between vibrational and rotational degrees of freedom is different in the case of linear and non-linear molecules [32]. This requires a special care for the basis set construction in variational calculations [33], particularly in case of flat PES cuts in some vibrational motions as we have for the acetylene molecule.

The point group of the equilibrium geometrical configuration of an A2B2 linear molecule is D∞h that possesses non-degenerate / ug   and an infinite number of doubly degenerate irreducible representations / ug  , / ug  , … which are often used to label energy levels. However, when the vibrational-rotational motions are excited, the molecule is no longer linear [34] [35], and it is more appropriate to consider molecular symmetry groups defined as D∞h(M) [START_REF] Bunker | Molecular Symmetry and Spectroscopy[END_REF], [START_REF] Chubb | [END_REF], accounting for permutation-inversion operations of identical nuclei. This latter one is isomorphic to the Klein abelian group [START_REF] Klein | Lectures on the icosahedron and the solution of equations of the fifth degree (Vorlesungen über das ikosaeder und die auflösung der gleichungen vom fünften grade)[END_REF] possessing only 4 elements and 4 one-dimensional irreducible representations. Thus rovibrational wave functions for each J value are partitioned into four isolated blocks distinguished by the parity under inversion p and the parity under permutation  of identical nuclei. The states with even value of Jp  are labeled by e , whereas the states with the odd value of by f [START_REF] Brown | [END_REF]. Even p   values correspond to g states and odd values to u states.

The Hamiltonian eigenvalues in various blocks are, in general, different. There appear e/f doublets in rovibrational energy levels spitted by the Coriolis interactions. In this work we show that the (ef) splitting for these doublets can be very accurately predicted from ab initio PES for various J values. Note that in ref. [40] an artificial, formally infinite molecular symmetry group D∞h(AEM) has been introduced for linear molecules, but wave functions were finally classified according to one-dimensional representations.

The spectroscopic data of acetylene including highly excited states are important due to a wide range of applications: it is used as a fuel and in industrial applications linked to the high temperature of the flame. The acetylene is involved in various reactions as a chemical building block and serves for carbon dating of organic materials. It was found in atmospheres of giant planets and exoplanets. The accurate knowledge of rovibrational transitions is important for the interpretation of near-infrared spectra in the ground-based and satellite observatories [41] [42].

The development of efficient methods for variational nuclear motion calculations based on ab initio potential energy surfaces (PES) and of dipole moment surfaces (DMS) has permitted significant progress in predictions of vibrational-rotational bands. This was, for example, the case with the spectroscopy of water [43] [44] [45] [46], carbon dioxide [47] [48] [49], hydrogen sulfide [50] [51], sulfur dioxide [52], and ozone [53] [54] [55]. In the latter case, ab initio predictions were used for precise quantification [56] of band intensities, for understanding the PES properties in the transition state range towards the dissociation threshold [57] [55], the interactions between potential wells [58], and modeling of the isotopic exchange reactions [59].

Recently, theoretical calculations of rotationally resolved spectra have helped extending assignments and modeling of high-resolution spectra for 4,5, and 6 atomic molecules [60] [61]

[62] [63] [64], including hot bands. Many ab initio PESs have been subsequently refined by a fit to experimental data to achieve better accuracy in line positions. Theoretical line lists for ammonia [65] [49] [66] and phosphine [67] [68] [69] are successful examples of this trend for four-atomic species. New PESs and DMSs have been recently reported for five-atomic [70] [71],

six-atomic [72] [73], [74] and seven-atomic [75] molecules.

In the present work, we report theoretical vibrational levels of acetylene computed from a new accurate ab initio PES using the algorithms similar to the one recently applied for the methane [76] and formaldehyde molecules [77] [78] and compare our results with other recent calculations [77] [79] and with experimental data. A very large set of observations for acetylene makes this molecule an interesting benchmark object for high-level quantum-chemical calculations. For many-electron molecules, the "spectroscopic accuracy" in ab initio vibrational levels of about 0.1 cm -1 ( at least in the low and medium energy ranges ) was achieved in few works as reported in refs. [46], [80] for the water molecule, and in ref. [76] for methane. Both of these molecules contain 10 electrons, whereas the acetylene molecule has 14 electrons. Two recent studies [77] [78] reported quite accurate theoretical calculations for formaldehyde (16 electrons), that gave a hope to achieve similar results for acetylene as well.

In contrast to formaldehyde, acetylene is an experimentally much better studied molecule, but due to symmetry properties the origins (J=0 levels) for many acetylene bands cannot be observed directly. They are usually implicitly deduced from experimental spectra using effective rotational constants. For this reason, a reliable comparison with observation cannot be limited to purely vibrational levels and has to include J > 0 series. The corresponding calculations reported in this paper can be considered as an additional source of information that could be useful for the extended assignment and modeling of high-resolution experimental spectra of acetylene and its isotopologues.

The paper is structured as follows. The choice of the coordinates and the analytical representations for the PES and their comparison in terms of the flexibility of the fit of ab initio electronic energy points on the geometrical grid is discussed in Section 1. Exact kinetic energy operator (KEO), ro-vibrational calculations, and the basis set convergence for the nuclear motion are discussed in Section 2. Section 3 is devoted to the study of the dependence of the theoretical vibrational levels on the electronic basis set. The impact of relativistic corrections, diagonal Born-Oppenheimer corrections (DBOC), and of high-order electron correlation including connected triple and quadruple excitations in the coupled-cluster method is also considered. A detailed comparison with available experimental data in Sections 3 and 5 shows that the resulting full-dimensional ab initio PES provides currently the most accurate theoretical band origins and low-J rotational levels for acetylene. The high-precision calculations of the (e-f) splitting of rovibrational levels due to Coriolis interactions and their J-dependence are discussed in Section 4. Two versions of an empirically optimized PES obtained with a fine tuning of two and seven PES parameters and the corresponding predicted vibrational levels are also reported.

Coordinates and analytical PES representation

For calculations of the vibrational-rotational energy levels of the acetylene molecule we used the kinetic energy operator (KEO) in the orthogonal coordinates [81] [8] [82] [83] written in the form originally presented in ref [84]. The positions of atoms in the molecular fixed coordinates (MFC) are related to their positions in the laboratory system i R as follows: (see Figure 1).

𝑹 = 𝑚 1 𝐑 1 +𝑚 3 𝐑 3 𝑚 1 +𝑚 3 - 𝑚 2 𝐑 2 +𝑚 4 𝐑 4 𝑚 2 +𝑚 4 , 𝒅 1 = 𝐑 2 -𝐑 4 , 𝒅 2 = 𝐑 1 -𝐑 3 , (1)
with 1 m , 4 m being the nuclear mass of carbon and 2 m , 3 m those of hydrogen. The PES is expressed as a power series of elementary functions in the symmetrized coordinates involving Morse-type functions for "radial" and a sine function of angular variables:

1 2 1 2 1 2 3 4 5 6
( , , , , , )

i j k l m n ijklmn ijklmn V d R d f S S S S S S     
, where

    1 1 2 2 5 1 2 ( ) ( ) / 2, ( ), ( ) ( ) / 2 S y d y d S y R S y d y d      , (2)
      3 1 2 6 1 2 4 ( ) ( ) / 2, ( (3)
( ) / 2, S f f S f f S f           . )
A Morse-type function ( ) of identical nuclei and the operation of reflection in the plane that inverts the sign of τ. This group is isomorphic to the Abelian C2v point group, or the Klein group [START_REF] Klein | Lectures on the icosahedron and the solution of equations of the fifth degree (Vorlesungen über das ikosaeder und die auflösung der gleichungen vom fünften grade)[END_REF]. Therefore, we use almost the same analytical representation for the PES shape as for the H2CO molecule [78]. The only exception concerns some terms in Eq.( 2) in which there is no dependence on the angle for k=0 and n=0. These terms also cannot depend on the angle τ. Therefore, the parameters corresponding to k=0 and n=0 and l>0 were not included in the expression (2 In the present study, we applied the same tensor techniques for an optimal sampling of the grid of nuclear configurations as described in our previous works for non-abelian symmetry groups [85] [86]. This permitted accounting for the full symmetry of the molecule in order to reduce the number of geometrical nuclear configurations for the ab initio calculation of electronic energies. To build the corresponding grid of geometries, we used one-dimensional sections of the PES. Though the radial one-dimensional sections are similar to the corresponding sections of formaldehyde, the angular sections differ considerably. Figure 2 shows two angular one-dimensional sections S3, S6 and four torsional sections for 12 10, 20, 40, 60

       
degrees. It is evident that the dependence on the torsion angle is weak at small energies.

To describe the PES up to the 12-th order of the expansion (2) in general (but limited to the 6-th order for CH radial coordinates), a grid of geometries was built. The corresponding initial reference grid (G(R)) contained 4876 points. Additionally, a grid of 7171 geometries (with a step of 30 degrees within the range of 0 -180 degrees) was built to refine the description of the torsion dependence at the bottom of the potential well at energies up to 5000 cm -1 . Thus, the largest grid included 12047 geometries.

The set of the grid points was chosen in a way which ensures that a maximum number of parameters of our analytical PES representation would be well-defined in the least-squares fits to the calculated ab initio electronic energies. The PES parameters responsible for the coupling among various vibrational modes were systematically included. This relatively dense grid provides a possibility for testing various high-order PES expansions in elementary functions of the symmetrized coordinates. The initial grid of points was arranged in a way that their energy cut-off would correspond to the extent of the studied infrared bands. For this grid, the initial calculations were carried out with the coupled-cluster CCSD(T) method using the quintuple-zeta cc-pACV5Z [87] basis set (referred to as ACV5Z sets in abbreviated notation), and the MOLPRO quantum chemistry program package [88] [89].

To validate various representations of the PES using the least-squares technique, an extended grid of geometries was used. As in the case of formaldehyde, the grid was augmented by adding supplementary sets of points corresponding to 180- values (for 180  

) in the initial grid.

        degrees.
To compare the efficiency of the analytical form (2) in various coordinate sets, the benchmark PES calculations were carried out using a least-squares fit to the ACV5Z grid values, without corrections discussed in the following sections. Even though the ACV5Z basis set does not provide the best quantitative accuracy, we believe that it is preferable to use all PESs at the same theoretical level, without corrections for a reliable comparison of the general qualitative shape of the surfaces. Ab initio potential energies were fitted using the analytical symmetry adapted representation (2)-( 4), and the weight function employed by Schwenke and Partridge in ref [83]:

. 2 2.00200200 2) 1.00200200 ) ( 0005 . 0 tanh( ) ( 1     E E E w ( 5
)
The weight function defined in equation ( 5) decreases with the energy E (expressed in cm -1 ) in order to de-emphasize the contribution of large grid displacements for vibrational energies beyond E1=15000 cm -1 to the PES fit. The least-squares fit of the PES in various sets of coordinates leads to similar results. The row III of Table 1 shows the number of parameters which were statistically well determined by fitting of the PES to ab initio electronic energies, versus the total number of parameters of the corresponding order of the expansion. The PES III contains all sixth-order terms augmented by 8 th order angular terms, but the overall fit is not better than for the PES I. Although the fit of the PES II is slightly better than that of PES I, the energy levels up to 7000 cm -1 obtained from these two PESs are close, but the calculation of energy levels with the PES I is considerably faster. Table 2 gives the equilibrium geometry parameters. It can be seen that the calculation using the CV6Z basis set gives smaller equilibrium distances in comparison to CV5Z. Figure 6 shows that ACV6Z gives a shorter distance in comparison to ACV5Z. To expand the PES, we use the CCSD(T) / ACV5Z value of the equilibrium geometry as described in our previous works [76], [78]. For CCSD(T) / ACV5Z, the re(CC) and re(CH), distances in the equilibrium geometry are slightly larger than the values optimized using standard quantum chemical procedures. Note that due to the presence of linear terms in the expansion of the potential, the vibrational-rotational levels in our calculations practically do not depend on the reference geometry point with respect to which the PES is expanded in a Taylor series.

Rovibrational calculations

The rotational-vibrational wave function is constructed in the form

      1 1 2 2 1 1 2 2 1 1 2 2 1 2 0 , , , , , 2 , , , ,
l m l m J M l m l m Y Y J m m M         , (6) where  
, lm Y  are spherical functions [90], and

      2 21 , , exp exp 8 J MK J J K M iM d iK      
is a symmetric top fuction [START_REF] Bramley | [END_REF], depending on the Euler angles  ,  ,  , which determine the orientation of the molecule frame system (MFS ) with respect to the laboratory frame (LF) , and 1

  , 2        (instead of 2   
 in Ref. [84]). Note that in (6), the sum 12 mm  plays the role of K (the projection onto the Z axis in the MFS) in the ,, J K M function. Therefore, in this work, we use the notation :

12 K m m . The kinetic energy operator from Ref [84] was used for the calculation of energy levels.

   1 2 1 2 2 2 2 2 2 2 ortho 2 2 2 1 1 1 2 2 2 2 2 1 2 2 2 2 2 1 2 2 2 1 2 2 1 2 1 2 2 2 2 2 ˆ2 2 2 1 1 1 1 1 1 ˆ2 2 1 ˆˆˆˆˆˆˆ2 2 2 d d R d R d R z z z R T M d d d M d d d M R R R l l M d R M d R J l l J l l l l l MR                                                                                     1 2 1 ˆ, l J l l J       (7)
where Mis the total nuclear mass of the molecule ; vectors connecting the nuclei C4 and H2, the nuclei C1 and H3, and the centers of mass of the CH nuclear groups, respectively (see Figure 1), and the operators l, J in ( 7) are:

ˆjz j li     ,   ˆêxp ctg j j j jz j l i l              , ( 1, 2) j  12 ˆẑ z z J l l ,   1 1 ˆˆê xp ctg sin z J i J J i          
Ref [84] introduced a basis adapted for parity with respect to space inversion:

      12 1 1 2 2 1 1 2 2 1 1 2 2 0 0 , , , , , , , , , , , 1 , , , , , / 2 1
,

Jp mm J M p l m l m J M l m l m J M l m l m         (8)
where

{0,1} p  , 1 1 max m l L  , 2 2 max m l L  , 12 0 m m J   
The basis of Eq.( 8) can be adapted to the symmetry that corresponds to the permutation of identical atoms. This basis is described in Section 4 below. On the other hand, the basis ( 8) can be adapted to the complete symmetry in a simpler way by contracting the set of angular functions. The contraction of the angular basis ( 8) is necessary because of its large dimension.

For example, for Lmax=28, and p=0, the number of functions ( 8) with values of 12 mm  equal to 0, 5, 10, 15, 20 is 8555, 15604, 13864, 104200, and 122639, respectively. Due to the rapid increase of the energy with increasing 12 mm  , the angular basis vectors with 12 10 mm  are far beyond a natural cut-off for the considered range of applications. In addition, spectral transitions to energy levels for which the vectors 12 3 mm  , produce contributions to very weak ines inaccessible for experimental registration. When contracting, the eigenfunctions of the operator

1 2 ortho angular 2 2 1 2 2 2 2 2 1 2 1 1 1 1 1 1 ˆ2 2 d R d R T l l M d R M d R                         (9)
for fixed 12 mm  values are used as an angular basis set. When solving the ro-vibrational problem with the KEO of Eq. ( 9) at certain value of J, the eigenfunctions for 12 K m m J    in the low energy range are chosen in the following form:

1 2 1 2 1 2 1 2 1 2 ,, , , , 1 1 2 2
, , , , ,0, , , , ,

K p K p n l l m m l l m m m m K F c K p l m l m    . ( 10
)
The eigenfunctions , Kp n F defined by Eq.( 10) are either symmetric or antisymmetric versus the permutation of the pairs of indices 1 1

2 2 ( , ) ( , ) l m l m 
with a high degree of accuracy. Below, we denote such a permutation operator as  with eigenvalues , which will be added as upper case indices to the functions ,, Kp n F  to specifiy their parity.

To select an optimal basis set, an analysis of the convergence of vibrational-rotational energy levels was made. The dependence of the quantized angular energies on the parameter Lmax is shown in Figure 3. It is evident that up to 8000 cm -1 the convergence is no worse than 10 - 3 cm -1 and Lmax=28 is quite sufficient for convergence. The convergence of energy levels at J=20

depending on the number of angle functions is shown in Figure 3. Figure 4 shows the convergence of the angular energy levels at J=20 depending on the number of angle functions. It can be seen that for J=20 a sufficiently high number of angular functions is required. At the same time, as a rule, the transition intensities for highly excited angular functions are low.

Therefore, 1200 or 1800 angular functions are quite sufficient for a calculation of spectra at room temperature.

Figure 5 shows the convergence of vibrational energy levels depending on the number of radial functions. Below 4500 cm -1 , the influence of the PES type on the energy levels is not significant:

the differences in the level values do not exceed 0.1 cm -1 .

Improvement of the electronic basis sets and corrections to the PES

First, we conducted calculations using the augmented core-valence basis set ACV5Z in the whole set of 12047 nuclear geometries. As the next step, we accounted the following three corrections to the PES: "mass-velocity-Darwin" (MVD) relativistic corrections, the diagonal Born-Oppenheimer correction (DBOC), and higher-order dynamic electron correlations. The adiabatic DBOC corrections [START_REF] Born | Dynamical theory of crystal lattices[END_REF] [93] [94] [95] [96], which make main first-order contributions to the electronic energy beyond the Born-Oppenheimer approximation have been included in accurate PES calculations for several small molecules ([A1] , [43]

, [A2], [A3], [65], [76], [78],

[97] [98] [A4]) , and references therein), but the acetylene molecule was not yet considered.

It has been argued in many previous works [99] [100] [101] [102] [103] [104] that dynamic electron correlations beyond single and double excitations are important to approach spectroscopic accuracy. Such correlations have been included in recent spectroscopically accurate PESs of methane [76] [105] and a formaldehyde [77] [78]. A thorough study of the impact of triple and quadruple excitations on the equilibrium geometry and the quartic force field of formaldehyde was reported in [77].

In this work, we calculated MVD relativistic corrections with MOLPRO [88] [89] and DBOC correction with the CFOUR [START_REF] Stanton | CFour program[END_REF] program suites on the full reference grid of 12047 nuclear geometries. To check the accuracy of relativistic corrections, DK=4 was also calculated, but the difference between DK and MVD calculations was negligible. DBOC correction is calculated in the cc-pVTZ basis and the accuracy of this calculation does not raise questions [95]. Much more expensive and doubtful are calculations for higher-order dynamic electron correlations. The T(Q) correction was calculated with the CFOUR program [START_REF] Stanton | CFour program[END_REF] using the noniterative quadruple CCSDT(Q) method [101] [102] [103] [104] with the cc-pVTZ oneparticle basis and frozen core. The correction term was calculated at 5216 points of two grids of nuclear geometries and approximated by the 4 th order expansion of the type (2). Using this form in a way similar to ref [76], the energy differences T(Q) were calculated for all other 12047-5216 geometries.

This gave a quite smooth analytical function of all six symmetrized coordinates for the sum of these three corrections:

Δ(3corr) (Si) = ΔRel (Si) + ΔDBOC (Si) + ΔT(Q) (Si). ( 11
)
In this step we have constructed the ab initio corrected surface

PES(ACV5Z+3Corr) = PES(ACV5Z) + Δ(3corr) (Si) . ( 12
)
Table 5 provides the energy levels using the PES(ACV5Z) and PES(ACV5Z+3Corr) defined by Eqs. (11,12) that include the above considered corrections. It is seen that the values of the T(Q) dynamic correction for acetylene are several times higher than those for the methane molecule [76] and are of the same order as for formaldehyde [78]. The differences between the calculated and measured levels are considerably larger for the ν2 and ν4 band origins. Although the PES (ACV5Z + 3Corr) for acetylene is constructed in a similar way to the PES of methane, its accuracy is significantly lower. A similar situation was observed for formaldehyde [78]. It was necessary to improve the potential energy surface, particularly along the CC bond. Since the relativistic corrections and DBOC are relatively small, we explored further contributions related to higher-order dynamic electron correlations including quadruple excitations, as well as to the larger, aug-cc-pCV6Z one-particule basis set. To perform this operation, in the next step, we used the iterative CCSDTQ [103] [104] [106] [START_REF] Matthews | [END_REF] method which provides a more rigorous treatment of dynamic electron correlation corresponding to connected quadruple excitations.

However, the calculations that use this method converge very slowly for low-symmetry geometries or for large rCC bond distances. At this level of theory, full PES calculations are extremely demanding, but it was possible to study the corresponding contributions on the most relevant one-dimensional cuts [76] [78].

In the case of the formaldehyde molecule [78], we calculated one-dimensional corrections ΔQ = CCSDTQ -CCSDT(Q) (cc-pVTZ basis) and ΔACV6Z = ACV6Z -ACV5Z basis (CCSD(T) method) for 20 geometries. At the first stage, similar one-dimensional corrections were calculated for the acetylene molecule: ΔQP6(rCC), ΔQ6(rCH), and ΔQ6(qCH). The correction ΔQP = CCSDTQP -CCSDT(Q) was evaluated for about 33 points along both bond stretching coordinates. The CCSDTQ(P) -CCSDT(Q) term was calculated in the cc-pVTZ basis, while for CCSDTQP-CCSDTQ(P) the cc-pVDZ basis set was used. The one-dimensional ΔACV6Z = ACV6Z -ACV5Z corrections were also computed at the same sets of geometries. Onedimensional radial corrections along the CC bond are shown in Figure 6. These corrections result in an increase of the potential energy at distances that exceed re(CC). At the distances that are shorter than re(CC), the potential energy slightly decreases. For the CC bond, the ΔQ = CCSDTQ-CCSDT(Q) correction in the cc-pVTZ basis considerably exceeds the ΔACV6Z = ACV6Z-ACV5Z correction calculated with the CCSD(T) technique.

For the CH bond, the contributions of these corrections are of a similar magnitude. We summarized two corrections ΔQ+6Z = ΔQ + ΔACV6Z and approximated this sum analytically using a power series expansion in Morse-type functions ΔQ+6Z (S1,S2,S5) of radial coordinates (2b). The choice of analytical representations had a marginal effect on the vibrational energy levels. The one-dimensional corrections made ν2 considerably closer to the empirical values, but for the vibrational frequency ν4, a significant discrepancy remained.

In the case of acetylene, due to the absence of the 2 () f  parameter in the PES and the weak dependence of the PES on  at small angles q, it was impossible to get by with only onedimensional corrections. A minimum requirement is a 3-dimensional angular correction evaluated at a significant number of points. However, even an approximate shape of the correction surface ΔQ + ΔACV6Z is not known a priori. Therefore, the correction was calculated in a randomized set of points near the minimum of the potential well and approximated by the analytical form (2) up to the third order. The calculation of ΔQ6(S1,S2,S3,S4,S5,S6) = ΔQ + ΔACV6Z required considerable computer resources. As quantum-chemical energies were calculated in new geometries, the parameters of series (2) were adjusted. The calculation of 331 points turned out to be sufficient for a reliable determination of the parameters for shape (2)-( 4) of the surface ΔQ6. In this case, none of the parameters of the interaction between angular and radial coordinates was determined. Therefore, in the final version, one-dimensional radial sections were used for the radial part, as in the case of formaldehyde.

One-dimensional potential energy surfaces ΔQ6 (q2) and ΔQ6 (q3) were subtracted by fitting the parameters of the three-dimensional angular surface Δ * Q6(S3, S4, S6) = ΔQ6(S3, S4, S6) -ΔQ6(q2) -ΔQ6(q3). Note that when fitting the parameters Δ * Q6(S3, S4, S6) , in addition to 331 randomized geometries, 39 geometries previously obtained for one-dimensional angular sections Q were taken in account. The analytical form of Δ * Q6(S3, S4, S6) was adjusted with an RMS deviation of 0.088 cm -1 .

This accuracy makes it possible to estimate that the contribution of the determination error of Δ * Q6 to the lower vibrational levels does not exceed 0.3 cm -1 . The contribution of ΔQ6 brings ν4 closer to the empirical values. However, the calculated ν4 levels differ from the empirical values by more than 1.45 cm -1 . Table 3 summarizes the contributions of various corrections to vibrational levels and our final ab initio results. The experimental energy levels are collected from papers [108], [109], [110], and [111]. Previous experiences in the determination of this type of PES corrections for various molecules showed that corrections evaluated along the stretching of chemical bonds provide the maximum contribution. In the case of the CC bond, the onedimensional correction P obtained as the difference CCSDTQP-CCSDTQ in the cc-pVDZ basis, calculated with the MRCC program [112] turned out to be quite large. Our final ab intio PES that includes these four corrections was constructed as follows:

PES(ACV5Z+4Corr) = PES(ACV5Z) + Δ(3corr) (Si) + Δ * Q6(S3, S4, S6) + ΔCC_QP6 (r2) + ΔCH_Q6(r1) + ΔCH_Q6(r3) + ΔqHC_Q6(q2) + ΔqHC_Q6(q3) (13) The fourth correction ΔQ+6Z improves the accuracy of the calculated fundamental frequencies and brings them closer to the measured values, especially for ν2. Also, the correction diminished the calculated value of the equilibrium geometry rCCe . The rCH correction has a noticeably smaller effect on energy levels than the rCC correction. As a rule, the caculated origins of the nν4 bands deviate from observations by values slightly smaller than the product of n and the error in the ν4 band origin (-1.45cm -1 ). At the same time, the deviation of other four fundamental frequencies is considerably smaller, the average value being 0.25 cm -1 . The large deviation of ν4 could be explained by the significant contributions of T(Q) and ΔQ6. The contribution of T(Q) is always negative while the contribution of ΔQ6 is always positive. In this case, the value of the ΔQ6 contribution is greater than half of the T(Q) contribution. A significant part of the ΔQ6 contribution is precisely due to the difference Q -(Q). Thus, we can assume that the (P) contribution could be negative, reducing the calculated value of ν4. In general, it is not clear which factors could significantly increase the value of ν4.

The maximum contribution to (13) is caused by corrections related to dynamic electron correlation, and they are mainly calculated in the cc-pVTZ basis. Due to their computational complexity, it is quite difficult to calculate corrections in a larger basis. To assess the possible influence of the basis, we have compared some one-dimensional rCC corrections in the cc-pVTZ and cc-pVDZ basis sets. The comparison showed that the difference in the contribution of the corrections can vary up to 20% when the basis is changed. Reducing the contribution of the T(Q) correction could bring ν4 closer to the empirical value, but the recalculation of the corrections in the larger cc-pVQZ basis is too expensive, especially for calculations in a full 6-dimensional space. In addition, the QP+6Z correction is also quite significant. This correction can be calculated with a maximum accuracy of 25%. To further improve an accuracy of the C2H2 PES, an account of full multy-dimencional corrections would be necessary.

Energy levels calculated from the purely ab initio PES are compared with the empirical values of ref. [113] in Table 4 up to the 8-th polyad. The last two colums of Table 4 give also the comparison with calculations using the empirically optimised PESs including two or seven fitted parameters. It can be seen that the PES with two adjustable parameters brings the ν4 band origin noticeably closer to the empirical value. At the same time, the PES with seven adjustable parameters slightly improves the RMS (calc.-obs.) deviation in comparison with the twoparameters case. Starting from polyad 8, the RMS deviation begins to increase even for the PESs with adjustable parameters. Note that our vibrational assignment is not always in agreement with the identification from work [113]. For four empirical levels, including two of the polyad 7, and two of the polyad 8, it was not possible to find calculated levels that have consistent identification.

RMS deviation

*) Experimental values in wavenumber units (cm -1 ), from ref [23]. C is the symmetry type.

Splitting for (е -f) doublets in rovibrational energies

It is known that the rotational patterns of Σ, Π, Δ, and Φ vibrational states form (e-f) near degenerate doublets -closely lying levels of different parity with respect to inversion. Despite the limited accuracy of the PES, the use of the exact kinetic energy makes it possible to calculate the splitting in (ef) doublets and determine its J-dependence with high accuracy. The symmetry-reduced basis set can be represented as:

        1 1 2 1 2 1 1 2 2 2 2 1 1 1 1 2 2 ,0 , , , , , , 1 , , , , , , , , , , , , , 1 21 p
m m m l l J M p l m l m J M p l m l m J M p l m l m          (14) Here 1 1 2 2
, , , , , , J M p l m l m is defined by formula (8), p -is the parity with respect to inversion:

  , 1 , p E p p   
, and  -is the parity with respect to permutation of a pair of indices

1 1 2 2 ( , ) ( , ) l m l m  in eigenfunctions (10):   ˆ, 1 , pp     
. This definition was not commonly used in previous works, but it is convenient for the software implementation, since it does not depend on J. In the conventional notations, the blocks with even p   are denoted by g and correspond to even n3+ n5 values. The blocks with odd p   are denoted by u and correspond to odd n3+ n5 values. For each vibrational band, the parity g or u is defined by the number mod(n3+ n5, 2).

The complete space of basis functions is divided into four independent blocks having even or odd values of , p  . In the J = 0 case the basis set is orthogonal, but in a general case with J > 0, the basis set ( 11) is non-orthogonal. Note that for 12 0 mm  , the function

1 1 2 2
, , , , , , J M p l m l m always contain two components Let us first consider the case J = 0. Each of the basis functions , , , , , J M l m l m defined by Eq.( 6). The number of basis functions containing 4 components , , , , , J M l m l m is the same in each block.

The block 0, 0 p   is the only block, for which basis functions contain one component. In the

          1 2 2 1 0 1 1 0 2 2 0 1 1 0 2 2 , , , , / 2 l l l l Y Y Y Y         
, and their amount is equal to

          1 1 1 1 , 1 1 , 2 2 1 1 2 2 , , , , / 2 l
m l m l m l m Y Y Y Y          
, and their amount is also equial to

    2 1 2 1 2 1 ˆ2 corr R H l l J l l J MR              
from the Hamiltonian (7), then all levels of a certain vibrational state in the block 1 Jp  would coincide exactly with the levels in the block 0 Jp  for K > 0 . In this case, it really makes sense to talk about the point group D∞h [START_REF] Chubb | [END_REF], [40], for which doubly degenerate representations K (K=1,… ∞) would occur. But this scenario is only true in an approximation where the Coriolis interaction corr H is neglected. In reality, due to the weak Coriolis interaction between the K = 1 and K = 0 states, the J = 1 levels in doublets are shifted differently in each of the blocks , p  . Since the Coriolis interaction in our approach is described by a simple exact KEO, the magnitude of the corresponding (e-f) splitting and its J-dependence are modeled with high accuracy. Note that the absolute accuracy of the levels is much lower as it is limited by the accuracy of the PES. Similarly, the levels with higher values K=2, K=3 etc are also splitted due to the Coriolis interactions. Though K is not an exact quantum number, the contribution of the basis functions with a certain K value is, as a rule, dominant for the wavefuctions of low-lying states. Eigenfunctions with dominant contribution of large K contain insignificant amplitudes of the basis functions with K = 0. The higher is K, the smaller is the splitting. But even for large values of K the vibrational-rotational energy levels are always non-degenerate. This behavior is caused by weak Coriolis interaction. Figure 8 shows the energy levels values for the bands (1001 1 0 0 ) 1 g and (0000 0 3 1 ) 1 u calculated in this work and the empirical values [23 ] determined using MARVEL procedure [ A4]. It can be seen that our calculation describes the splitting of e and f levels with high accuracy. As a rule, for the most of vibrational bands, the calculated and empirical levels, as well as the levels splitting, look similar and are described by smooth curves. But for some bands, the difference looks like a broken line. For example, Figures 9 and Fifure 10 show the J-dependence of the splittings for the Π states and the Δ states. One of the reasons for the erratic behaviour could be possible misassignments of some levels in experimental works. The irregular splitting often correlates with poor uncertainty of empirical level included in MARVEL database [23] . Thus, variational calculation can be useful for testing empirical values of energy levels. As a rule, the J-dependence of the splittings is well approximated by a third-order polynomial.

To estimate the accuracy of the model, we fitted the difference between the (e-f) splittings from MARVEL and our calculation. Table 5 shows the differences between the calculated and empirical splittings as a power function

23 0 1 2 3 y C C x C x C x    
. For the  bands, the shift is large, and it is calculated inaccurately (large coefficient C0) since the shift depends on the PES.

But at the same time, the J dependence of the calculated shift of  bands is correct (small coefficients C1, C2, C3). For the Π, Δ, and Φ bands, the difference is small, the J dependence is weak and is well described by polynomials up to the third order. A smoothness of the J-dependence can be visualized using the second derivatives of (Ee-Ef) with respect to J computed by the finate difference method. The situation is quite different depending on the vibrational state. Figure 11 shows two examples of the behaviour of these second derivatives versus J for our calculates splittings and for empirical MARVEL values [23].

At the left-hand side of Figure 11 for the (0001 1

Calculations for the C2D2 and 13 C2H2 isotopologues

The vibrational levels of C2D2 given in Table 6 were calculated with nuclear masses using both ab initio and empirically fitted PESs. The calculation Calc(C2D2) corresponds to our best ab initio PES(ACV5Z+4Corr) + ΔDBOC, while Calc++(C2D2) corresponds to the empirically optimized PES (7 adjusted parameters) fitted to 120 vibrational levels of the main acetylene isotopologue as described in the previous section. The contribution of ΔDBOC was considered as the difference between DBOC for 12 C2D2 and 12 C2H2. The C2D2 isotopologue is heavier than 12 C2H2, thus the DBOC correction for C2D2 is expected to be less important than this correction for 12 C2H2. To calculate the contribution of the ΔDBOC, we expressed it in the form of a fourth-order Taylor expansion. Experimental levels were taken from work [113]. Note that work [113] reports only pure vibrational levels. In order to obtain rovibrational levels, we add the value B0*J, where rotational constants B0 are well known from analyses of experimental spectra: B0 = 0.8475 cm -1 for C2D2 and B0 = 1.1195 cm -1 for 13 C2H2. As for the main isotopologue, the energy levels calculated from ab initio PES have large deviations for all ν4 overtones. Note that our calculated levels C2D2 unambiguously correspond to experimental levels in all cases except for 4ν5 and 5ν5.

The vibrational levels of 13 C2H2 given in Table 7 were also calculated with nuclear masses using both ab initio and empirically fitted PESs. The comparison of the DBOC parameters for the two isotopologues shows that the contribution of DBOC for 13 C2H2 is about 90-95% of that for 12 C2H2. The DBOC effect being weak, the accordingly scaled contribution of the main isotopologue (see the respective column of Table 3) could be applied to the calculated 13 C2H2 levels. For the 3ν3 level, it is not possible to find a one-to-one correspondence between the empirical and calculated energy levels, most likely due to differences in the assignments. The last rows of Tables 6 and7 give the RMS deviations between experimental and calculated levels.

In both cases, the main contribution to the error in the energy levels calculated from the ab initio PES is caused by the ν4 overtones. [113] in cm -1 . E" levels are empirical levels + B0*J (B0 = 1.1195 cm -1 ) **) RMS deviation without the level 3ν3.

Conclusion

Comparisons with experimental data show that the ab initio PES of the electronic ground state of acetylene constructed in this work is currently the most accurate one. We note however, that the modeling of the PES shape versus the angular motion was more complicated than in the case of the recent ab initio study of another four-atomic moleculeformaldehyde [78]. This is because the harmonic term does not provide a dominant contribution to the frequency of the torsional motion, contrary to the case of formaldehyde. A quadratic approximation in the torsional variable does not work at all for the corresponding PES cut, which requires several higher-order terms for a satisfactory modeling. This particular shape could be thus more error sensitive in the calculation of rovibrational levels.

A full account of the Coriolis interactions using exact KEO together with the ab initio PES has permitted reliable predictions of splittings in (e-f) rovibrational doublets for Π, Δ, and Φ states. For the first time, the accuracy of ab initio values of the splittings is comparable to the experimental accuracy in high-resolution spectra. Note that the values of the splitting are quite robust with respect to an empirical optimization of the PES. This permitted testing the corresponding assignments in various bands. As a rule, the J-dependence of the splittings is smooth, but in some cases erratic outliers appear. This gives hints for possible errors in experimental assignments, which have to be checked during further analyses of the spectra. Firstprinciples calculations thus provide an independent insight into uncertainties of available experimental data. A similar accuracy is obtained for vibrational levels of the isotopologues С2D2 (Table 9) and 13 С2H2 (Table 10), confirming the reliability of the ab initio PES.

We have also produced an empirically optimized PES by adjusting 2 and 7 parameters to 120 experimental J = 0 and J=1 levels derived from high-resolution spectra with the RMS (obs.calc.) deviation of 0.45 and 0.25 cm -1 . Band origins of C2H2 computed from these PES up to 7000 cm -1 are given in Table 4. Both the ab initio and the empirically optimized PES are provided in the Supplementary Materials as a C++ code.

For a further investigation of the corresponding accuracy issues, the ab initio PES can be used to derive effective spectroscopic models as it was done in the case of methane [114] [115] [116]. This will permit to accurately compute physically meaningful values of the resonance coupling parameters from ab initio surface for advanced analyses of high-resolution spectra.

SUPPLEMENTARY MATERIAL

See supplementary material for the ab initio and the empirically optimized PES in orthogonal coordinates.

Figure 1 .

 1 Figure 1. Vectors of Jacobi coordinates for the acetylene molecule.

Figure 2 .

 2 Figure 2. One-dimensional PES sections for angular symmetrized coordinates S3, S6, and torsion sections for

Figure 3 .

 3 Figure 3. The basis set convergence of angular functions versus the parameter Lmax. The difference between angular energy levels calculated for nearby values of Lmax is depicted with different color codes. The descrepancies between two largest basis sets shown with black crosses indicate the convergence of the order of 0.1-0.01 cm -1 in the vibrational energy range up to about 8000-10000 cm -1 .

Figure 4 .

 4 Figure 4. Convergence of angular energy levels Ea at J=20, 0, 0 p   . The descrepancies Ea between two calculations using increasing numbers Na of angular basis functions are depicted : Ea (Na=1800) -(Na=1200) up to 6800 cm -1 in black crosses; Ea (Na=2400) -(Na=1800) up to 8000 cm -1 in red circles; Ea (Na=3600) -(Na=2400) up to 8700 cm -1 in red circles. The values of Lmax=28 are held fixed for all calculations.

Figure 5 .

 5 Figure 5. Convergence of vibrational (J=0) energy levels. The radial basis sets V(n,m) contains n radial functions for the CC bond and m radial functions for the CH atom groups. The differences between radial energy levels calculated for successive numbers of basis functions are depicted with different color codes. The descrepancies between two largest basis sets shown with blue crosses indicate the convergence of the order of 0.1-0.01 cm -1 in the vibrational energy range up to about 7500 -10000 cm -1 .

Figure 6

 6 Figure 6 One-dimensional radial corrections along the CC bond (solid line). The contributions ΔQ = CCSDTQ -CCSDT(Q) -the rigorous account for connected quadruple excitations in dynamic electron correlation -are depicted with black crosses, while the ΔAC6Z =(ACV6Z-ACV5Z) , ΔQ = CCSDTQ(P) -CCSDTQ, ΔP = CCSDTQP -CCSDTQ(P) corrections calculated at the CCSD(T) level are are depicted with circles, triangles and stars, respectively. The sum of all corrections ΔQ+P+6Z = ΔQ + Δ(P) + ΔP+ ΔAC6Z is depicted with solid line.

22 #

 22 473#) Rel.= relativistic; T(Q) = CCSDT(Q) -CCSD(T) correlation; QP+6Z = quadruple ΔQ+6Z correction (see the text for 474 explanation). 475 #) Experimental minus calculated levels corresponding to the full ab initio PES(ACVZ6 + 4Corr).

  m l m . The set of basis functions includes all possible functions of the form (14) such that 12 max l l L  [84].

  same number of functions with two components, but these functions themselves are different. In the case 1, 0 p   , the functions have the form

.

  Since the basis fubctions in each of the four blocks , energy levels in different blocks are not degenerate (except for possible accidental coincidence) at J = 0. If we remove the Coriolis interaction term

Figure 7 Figure 7 .

 77 Figure 7. Scheme of corner matrix elements for J=3. The basis is formed by eigenfunctions for each m1+m2. Nonzero matrix elements occur for the functions of neighboring blocks.

Figure 8 .

 8 Figure 8. Comparison of the J-dependence of (e-f) splittig for the states (1001 1 0 0 ) 1 g (left) and (0000 0 3 1 ) 1 u (right) between our calculated and empirical values reported by MARVEL study[23]. At the scale of the upper panes, our theoretical values depicted with square, and empirical MARVEL values depicted with red circles nearly coinside. At the blown up scale of the lower panes the difference δ = (MARVELtheoretical ) is depicted by circles on the lower panel with the estimated uncertainty of empirical values shown as the error bars.

[ 23 ]Figure 9 .

 239 Figure 9. Differences of the (e-f) splittings predicted in the present study and the MARVEL database [23], for selected Π type states ( K = 1 ).

Figure 10 .

 10 Figure 10. Differences of the (e-f) splittings predicted in the present study and the MARVEL database [23], for selected Δ type states ( K = 2 ).

  1 1 ) 2 u state, both derivatives are smooth, except for a small feature in the interval J=25-28 for the empirical curve. The error bars at the lower pane of this figure show that empirical splitting are quite accurate whereas the theoretical values are somewhat overestimated. At the right-hand side of Figure 11 for the (1000 0 0 1 ) 1 u state, the theoretical derivative is still smooth, whereas the empirical ones are erratic with much larger error bars for the empirical splittings.

Figure 11 .

 11 Figure 11. The second derivatives (Ee-Ef ) with respect to J for the states (0001 1 1 1 ) 2 u (left) and (1000 0 0 1 ) 1 u (right). At the upper panes of the figures, the theoretical values are depicted with square, and the empirical MARVEL values [23] are depicted with red circles. At the blown up scale of the lower panes the difference δ = (MARVEL -theoretical ) is depicted by circles on the lower panel with the estimated uncertainty of empirical values shown as the error bars.

  is used as the angular function. The torsion angle  is the angle between the 1 dR  and 2 dR  planes. It is obvious that the symmetry group used for constructing the PES contains only two generating elements: the permutation of coordinates(12)

		y d	1 exp( ( a d d    	e	))	with the parameter a=1.9 is used as the radial
	function, and	 ( ) sin( f 	 		/ 2)

Table 1 .

 1 RMS and standard deviations of the PES fits in various coordinate sets.

	Set	I	II	III
	Order of expansion	6	7	6 plus
	Torsion function	cos(τ)	cos(τ)	cos(τ /2)
	#Parameters/Initial # Param	304/422	408/771	320/493
	RMS (STD)	0.96 (0.97)	0.57(0.59)	0.99(0.79)

Table 2 .

 2 Equilibrium geometry of the C2H2 molecule, as optimized at various levels of ab initio

	theory
	Coordinates	CCSD(T)/	CCSD(T)/	CCSD(T) /	CCSD(T)/	CCSD(T) /	CCSD(T) /
		pwCVQZ	ACVQZ	ACV5Z	ACV5Z-	CV5Z	CV6Z
					DK**
	re(CC) / Å	1.20341713	1.20403710	1.20305941	1.20278890 1.20289953	1.20262992
	re(CH) / Å	1.06210688	1.06236424	1.06185072	1.06168717 1.06173152	1.06166908
	** Accounting for Douglas-Kroll-Hess relativistic corrections denoted DK in what follows.

Table 3 . Some J=0,1 lower levels with contributions of relativistic, DBOC corrections, and high-order 471 electron correlations up to 3376 cm -1 . 472

 3

	Vib. State	С Emp.	Contributions of corrections # Theoretical levels with corrections
			levels *	Rel.	DBOC T(Q)	QP+6Z ACV5Z	ACV5Z +3corr	ACV5Z +4corr	Exp-Calc##
	0 0 0 0 0 0 0 0	e	2.353286	0.00	0.00	0.00	0.00	2.35	2.35	2.353372	0.00
	0 0 0 1 1 0 0 1	f	614.044355	-1.05	0.34	-5.75	2.82	616.25	609.78	612.59791	1.45
	0 0 0 1 1 0 0 1	e	614.054888	-1.05	0.34	-5.75	2.82	616.26	609.79	612.60838	1.45
	0 0 0 0 0 1 1 1	e	731.506987	0.30	-0.01	-3.30	1.03	733.61	730.60	731.62343	-0.12
	0 0 0 0 0 1 1 1	f	731.516374	0.30	-0.01	-3.30	1.03	733.62	730.61	731.63283	-0.12
	0 0 0 2 0 0 0 0	e	1230.390303	-2.00	0.64	-11.16	4.90	1235.74	1223.22	1228.1242	2.27
	0 0 0 2 0 0 0 0	e	1232.749162	-2.00	0.64	-11.16	4.90	1238.10	1225.58	1230.483	2.27
	0 0 0 1 1 1 -1 0	e	1328.073466	-0.81	0.31	-8.96	3.96	1332.55	1323.09	1327.0474	1.03
	0 0 0 1 1 1 -1 0	e	1330.434308	-0.81	0.31	-8.96	3.96	1334.91	1325.45	1329.4084	1.03
	0 0 0 1 1 1 -1 0	f	1340.550679	-0.79	0.33	-9.18	3.68	1344.96	1335.32	1338.9995	1.55
	0 0 0 1 1 1 -1 0	f	1342.911183	-0.78	0.33	-9.18	3.68	1347.32	1337.68	1341.3598	1.55
	0 0 0 0 0 2 0 0	e	1449.112363	0.58	-0.03	-6.67	1.99	1453.53	1447.41	1449.4051	-0.29
	0 0 0 0 0 2 0 0	e	1451.474822	0.59	-0.03	-6.68	1.99	1455.89	1449.78	1451.7676	-0.29
	0 0 0 3 1 0 0 1	e	1856.947677	-2.90	0.92	-16.47	7.01	1865.25	1846.80	1853.8092	3.14
	0 0 0 3 1 0 0 1	f	1856.968230	-2.90	0.92	-16.47	7.01	1865.27	1846.82	1853.83	3.14
	0 0 0 2 2 1 -1 1	e	1942.356142	-1.84	0.61	-14.47	6.18	1949.69	1933.99	1940.1677	2.19
	0 0 0 2 2 1 -1 1	f	1942.371328	-1.84	0.61	-14.47	6.18	1949.70	1934.00	1940.1828	2.19
	0 0 0 2 0 1 1 1	f	1962.052115	-1.77	0.62	-14.64	5.89	1969.60	1953.82	1959.7053	2.35
	0 0 0 2 0 1 1 1	e	1962.057447	-1.77	0.62	-14.64	5.89	1969.61	1953.82	1959.7105	2.35
	0 1 0 0 0 0 0 0	e	1974.316617	0.29	0.04	-7.09	2.12	1978.58	1971.81	1973.9309	0.39
	0 1 0 0 0 0 0 0	e	1976.656600	0.29	0.04	-7.09	2.12	1980.92	1974.15	1976.2719	0.38
	0 0 0 1 1 2 0 1	e	2050.232293	-0.57	0.28	-12.25	5.17	2057.06	2044.52	2049.6873	0.54
	0 0 0 1 1 2 0 1	f	2050.251456	-0.57	0.28	-12.25	5.17	2057.08	2044.54	2049.7064	0.55
	0 0 0 1 -1 2 2 1	f	2068.152962	-0.53	0.31	-12.60	4.53	2074.97	2062.14	2066.6707	1.48
	0 0 0 1 -1 2 2 1	e	2068.161538	-0.53	0.31	-12.60	4.53	2074.97	2062.15	2066.679	1.48
	0 0 0 0 0 3 1 1	e	2171.514343	0.87	-0.04	-10.14	2.87	2178.28	2168.97	2171.8424	-0.33
	0 0 0 0 0 3 1 1	f	2171.533306	0.87	-0.04	-10.14	2.87	2178.30	2168.99	2171.8615	-0.33
				-3.70	1.16	-21.43	8.75	2497.35	2473.38	2482.1288
				-3.70	1.16	-21.43	8.75	2499.72	2475.74	2484.493
	0 0 0 3 1 1 -1 0	e	2560.594937	-2.73	0.85	-19.47	7.89	2571.18	2549.84	2557.7257	2.87
	0 0 0 3 1 1 -1 0	e	2562.961453	-2.73	0.85	-19.47	7.89	2573.55	2552.20	2560.0924	2.87
	--0 1 0 1 1 0 0 1	e	2575.873197	-0.84	0.40	-13.09	5.03	2582.51	2568.98	2574.0152	1.86
	0 1 0 1 1 0 0 1	f	2575.883903	-0.84	0.40	-13.09	5.03	2582.52	2568.99	2574.0257	1.86
				-2.68	0.89	-19.93	7.84	2594.50	2572.78	2580.6153
	0 0 0 3 1 1 -1 0	f	2586.208355	-2.68	0.89	-19.93	7.84	2596.86	2575.14	2582.9806	3.23
	0 0 0 2 2 2 -2 0	e	2648.014468	-1.64	0.56	-17.58	7.48	2657.79	2639.12	2646.604	1.41
	0 0 0 2 2 2 -2 0	e	2650.383588	-1.64	0.56	-17.59	7.48	2660.16	2641.49	2648.9731	1.41
				-1.63	0.60	-18.00	7.21	2670.77	2651.74	2658.9475
	0 0 0 2 2 2 -2 0	f	2663.556848	-1.63	0.60	-18.00	7.21	2673.14	2654.10	2661.3156	2.24
				-1.56	0.61	-18.20	6.71	2693.90	2674.77	2681.4791
				-1.55	0.61	-18.20	6.71	2696.27	2677.13	2683.8459
	0 1 0 0 0 1 1 1	e	2704.249883	0.58	0.04	-10.60	3.18	2710.74	2700.76	2703.9442	0.31
	0 1 0 0 0 1 1 1	f	2704.259331	0.58	0.04	-10.60	3.18	2710.75	2700.77	2703.9536	0.31
	0 0 0 1 1 3 -1 0	e	2757.797907	-0.34	0.25	-15.59	6.12	2767.08	2751.40	2757.5199	0.28
	0 0 0 1 1 3 -1 0	e	2760.168254	-0.34	0.25	-15.59	6.12	2769.45	2753.77	2759.8905	0.28
				-0.30	0.31	-16.13	5.36	2792.94	2776.82	2782.1796
	0 0 0 1 1 3 -1 0	f	2786.022355	-0.30	0.31	-16.13	5.36	2795.31	2779.19	2784.5484	1.47
	0 0 0 0 0 4 0 0	e	2880.220077	1.13	-0.04	-13.67	3.71	2889.48	2876.90	2880.6153	-0.40
	0 0 0 0 0 4 0 0	e	2882.592018	1.13	-0.04	-13.67	3.71	2891.85	2879.27	2882.9871	-0.40
	0 1 0 1 1 1 -1 0	e	3281.899025	-2.57	0.84	-17.44	4.51	3296.16	3276.99	3281.5016	0.40
	0 1 0 1 1 1 -1 0	e	3284.244150	-2.57	0.84	-15.10	4.51	3296.17	3279.34	3283.8474	0.40
	0 0 1 0 0 0 0 0	e	3294.839579	-0.35	0.14	-8.87	2.69	3300.99	3291.91	3294.606	0.23
	0 0 1 0 0 0 0 0	e	3297.184318	-0.35	0.14	-10.52	4.34	3303.34	3292.61	3296.9503	0.23
				-0.58	0.40	-15.08	4.27	3309.51	3294.25	3298.5257
	0 1 0 1 1 1 -1 0	f	3302.982999	-0.58	0.40	-16.72	5.91	3311.86	3294.96	3300.8733	2.11
	1 0 0 0 0 0 0 0	e	3372.838987	0.53	-0.48	-4.44	3.25	3374.30	3369.91	3373.1654	-0.33
	1 0 0 0 0 0 0 0	e	3375.186841	0.53	-0.48	-6.76	5.58	3376.64	3369.92	3375.5051	-0.32

Table 4 . Vibrational levels for the 12 C2H2 isotopologue up to 8-th polyad.

 4

	Vibr State		Empirical	E"*	Calc	E"-Calc E"-Calc +	E"-Calc ++
			M.Herman				2 adjusted	7 adjusted
			et all.*				param	param
	0001100	0,1,1,g	612.871	614.048 612.5976	1.450	0.321	0.212
	0000011	0,1,1,u	730.332	731.509 731.6228	-0.114	0.006	0.018
	Polyad 1					1.029	0.227	0.150
	0002000	0,2,0,g	1230.39	1230.390 1228.123	2.267	0.064	-0.110
	0002200	0,2,2,g	1233.52	1235.873 1233.287	2.586	0.354	0.175
	000111-1	0,2,0,u1	1328.074	1328.074 1327.046	1.028	0.021	0.122
	0001-111	0,2,0,u2	1340.552	1340.552 1338.999	1.553	0.546	0.304
	0001111	0,2,2,u	1347.52	1349.873 1348.61	1.263	0.256	0.241
	0000020	0,2,0,g	1449.112	1449.112 1449.403	-0.291	-0.057	-0.120
	0000022	0,2,2,g	1463.016	1465.369 1465.648	-0.278	-0.039	-0.031
	Polyad 2					1.561	0.267	0.178
	0003100	0,3,1,g	1855.72	1856.897 1853.808	3.089	-0.162	-0.324
	0003300	0,3,3,g	1861.93	1865.460 1861.995	3.465	0.154	-0.043
	000221-1	0,3,1,u	1941.179	1942.356 1940.166	2.190	0.088	0.114
	0002011	0,3,1,u	1960.874	1962.051 1959.709	2.342	0.261	0.016
	0002211	0,3,3,u	1972.56	1976.090 1973.699	2.391	0.285	0.257
	0100000	1,3,0,g1	1974.316	1974.316 1973.92	0.396	0.374	0.265
	0001-122	0,3,1,g	2049.059	2050.236 2049.685	0.551	-0.337	-0.214
	0001120	0,3,1,g	2066.99	2068.167 2066.677	1.490	0.596	0.337
	0001122	0,3,3,g	2084.81	2088.340 2087.343	0.997	0.111	0.171
	0000031	0,3,1,u	2170.343	2171.520 2171.84	-0.320	0.029	-0.087
	0000033	0,3,3,u	2198.09	2201.620 2202.046	-0.426	-0.067	-0.069
	Polyad 3					1.942	0.275	0.204
	000311-1	0,4,0,u1	2560.6	2560.600 2557.722	2.878	-0.242	-0.127
	000331-1	0,4,2,u	2561.526	2563.879 2560.613	3.267	0.090	0.065
	0101000****	1,4,1,g	2574.7	2575.877 2574.007	1.869	0.698	0.430
	0003-111	0,4,0,u2	2583.845	2583.845 2582.978	0.867	0.124	-0.165
	0003111	0,4,2,u	2589.683	2592.036 2588.881	3.155	0.037	-0.126
	000222-2	0,4,0,g1	2648.018	2648.018	2646.6	1.418	-0.557	-0.339
	0002-222	0,4,0,g2	2661.188	2661.188 2661.313	-0.125	0.259	0.114
	0002220	0,4,2,g	2666.151	2668.504 2666.049	2.455	-0.399	-0.258
	0100011	1,4,1,u	2703.076	2704.253 2703.936	0.317	0.416	0.287
	000113-1	0,4,0,u1	2757.798	2757.798 2757.516	0.282	-0.493	-0.413
	0001-133	0,4,2,u	2773.193	2775.546	2775.3	0.246	-0.521	-0.404
	0001-131	0,4,0,u2	2783.651	2783.651 2784.546	-0.895	0.680	0.326
	0001131	0,4,2,u	2795.5	2797.853 2796.443	1.411	0.629	0.380
	0000040	0,4,0,g1	2880.22	2880.220 2880.611	-0.391	0.064	-0.149
	0000042	0,4,2,g	2894.069	2896.422 2896.816	-0.393	0.068	-0.089
	Polyad 4					1.723	0.421	0.276
	010111-1	1,5,0,u1	3281.899	3281.899 3281.467	0.432	-0.159	0.317
	0010000	1,5,0,u1	3294.839	3294.839 3294.553	0.286	-0.175	0.337
	0101-111	1,5,0,u2	3300.637	3300.637 3298.519	2.118	1.070	0.634
	0101111	1,5,2,u	3307.713	3310.066 3308.318	1.749	0.701	0.496
	1000000	1,5,0,g1	3372.849	3372.849 3373.099	-0.250	-0.264	0.617
	0100020	1,5,0,g1	3420.388	3420.388 3420.235	0.153	0.367	0.142
	0100022	1,5,2,g	3434.113	3436.466 3436.276	0.190	0.410	0.246
	Polyad 5					1.065	0.543	0.435
	000511-1	0,6,0,u1	3818.44	3818.440 3813.835	4.605	-0.456	-0.067
	000531-1	0,6,2,u	3820.24	3822.593 3817.577	5.016	-0.099	0.183
	000511-1	0,6,0,u2	3850.32	3850.320 3845.604	4.716	-0.306	-0.391
	000511-1	0,6,2,u	3855.82	3858.173 3853.419	4.754	-0.285	-0.287
	0011100	1,6,1,u	3882.41	3883.587 3881.789	1.798	0.051	0.355
	0011100	1,6,1,u	3898.34	3899.517 3897.966	1.551	-0.007	0.409
	0200000	2,6,0,g1	3933.897	3933.897 3935.481	-1.584	0.699	0.466
	1001100	1,6,1,g	3970.05	3971.227 3970.284	0.943	-0.153	0.611
	0010011	1,6,1,g	4002.44	4003.617 4003.213	0.404	-0.206	0.152
	0010011	1,6,1,g	4016.71	4017.887 4017.929	-0.042	-0.342	0.227
	1000011	1,6,1,u	4092.345	4093.522 4094.005	-0.483	-0.377	0.487

-1

. Level 0008000 0,8,0,g1 appears twice. Probably, level 0008200 0,8,2,g has wrong wavenumber. ****) Wrong energy level identification

29 577 Table 5 .

 5775 The differences between the empirical and calculated splittings presented as an expansion to 578 the third order in cm -1 . 579

	Energy E,		State	C0	C1	C2	C3	RMS,	Adj. R-	# of	J
	cm -1								cm -1	Square	Points	removed
											from fit
	1328.073466		(0001 1 1 -1 ) 0 u	0.18086(14)	0.000012(40)	0.0000038(31)	-0.000000487(69)	0.00020	0.99429
	2757.797907		(0001 1 3 -1 ) 0 u	0.73712(24)	-0.000130(39)	0.0000099(14)		0.00018	0.95786	1,3,12
	4673.631058		(1001 1 1 -1 ) 0 u	0.25624(73)	-0.000138(113)	-0.0000181(36)		0.00137	0.95042
	6623.139603		(1101 1 1 -1 ) 0 u	0.88059(98)	0.000578(302)	-0.0001100(249)	-0.00000151(58)	0.00099	0.99856
	731.506987		(0000 0 1 1 ) 1 u	0.000028(105)	0.000011(27)	0.0000000(19)	0.000000034(39)	0.00009	0.9361
	2171.514343		(0000 0 3 1 ) 1 u	0.000045(120)	-0.000031(18)	0.00000581(61)		0.00017	0.97627	3,24,29,30
	614.044355		(0001 1 0 0 ) 1 g	0.0000089(893)	0.000014(25)	0.0000031(18)	0.000000031(39)	0.00010	0.99263
	2050.232293		(0001 1 2 0 ) 1 g	-0.000257(417)	0.00018(11)	0.0000001(85)	0.00000086(18)	0.00047	0.99655
	2068.161538		(0001 -1 2 2 ) 1 g	0.000095(743)	-0.000047(204)	-0.000011(15)	-0.00000020(32)	0.00083	0.96951
	1962.057447		(0002 0 1 1 ) 1 u	-0.000676(465)	0.00027(16)	-0.000096(15)	-0.00000077(38)	0.00044	0.99935
	1942.356142		(0002 2 1 -1 ) 1 u	-0.000030(104)	0.000051(33)	0.000077(28)	-0.000000233(68)	0.00011	0.99995
	4017.877611		(0010 0 1 1 ) 1 g	0.00024(154)	0.00032(42)	-0.000119(32)	0.00000190(67)	0.00173	0.98609
	3899.504045		(0011 1 0 0 ) 1 u	0.000378(389)	-0.000004(143)	0.000086(14)	-0.00000062(41)	0.00034	0.99884
	2704.249883		(0100 0 1 1 ) 1 u	-0.000099(39)	0.0000542(29)			0.00009	0.94204
	4141.222136		(0100 0 3 1 ) 1 u	0.00055(28)	-0.000065(47)	0.0000064(16)		0.00043	0.81948
	4003.600972		(0101 1 2 0 ) 1 g	0.0093(23)	-0.00172(50)	0.000147(23)		0.00285	0.89821
	7512.125734		(0200 0 5 1 ) 1 u	0.00008(116)	0.00016(38)	0.0000097(334)	0.00000092(84)	0.00112	0.97459
	5819.194833		(0202 2 1 -1 ) 1 u	-0.00170(91)	0.00058(30)	0.000042(29)	0.00000113(78)	0.00072	0.99721
	4093.511896		(1000 0 1 1 ) 1 u	0.00030(12)	-0.0000080(172)	0.0000040(5)		0.00014	0.97666	1,3,10,25
	3971.220032		(1001 1 0 0 ) 1 g	-0.00132(58)	0.000315(34)			0.00143	0.7568	30
	5270.885653		(1002 2 1 -1 ) 1 u	0.0022(11)	-0.000095(212)	0.0001127(87)		0.00111	0.99398	1,4
	6054.402350		(1100 0 1 1 ) 1 u	-0.00086(100)	0.00020(33)	0.000064(29)	0.00000036(75)	0.00097	0.99454
	7468.787187		(1100 0 3 1 ) 1 u	0.0019(11)	-0.00077(31)	0.000174(23)	-0.00000203(48)	0.00125	0.99737
	1465.378177		(0000 0 2 2 ) 2 g	0.00051(30)	-0.000183(77)	0.0000130(54)	0.00000021(11)	0.00026	0.995
	2896.436694		(0000 0 4 2 ) 2 g	0.00062(55)	-0.000303(81)	0.0000309(26)		0.00046	0.98771	2,3,7
	1349.881320		(0001 1 1 1 ) 2 u	-0.00013(18)	0.000046(46)	-0.0000075(33)	0.000000568(67)	0.00015	0.99686
	2775.559135		(0001 1 3 1 ) 2 u	0.00031(31)	-0.000099(51)	0.0000066(18)		0.00030	0.75369	22
	1235.874392		(0002 2 0 0 ) 2 g	0.0091(12)	-0.00582(39)	0.000967(37)	-0.0000238(10)	0.00072	0.99947
	2563.894400		(0003 3 1 -1 ) 2 u	-0.00029(219)	-0.00059(44)	0.000157(19)		0.00223	0.9834
	4743.370547		(0010 0 2 2 ) 2 u	0.00002(272)	0.000074(741)	-0.000019(56)	0.0000027(12)	0.00217	0.97188
	4627.107147		(0011 1 1 1 ) 2 g	0.00038(105)	-0.00017(29)	0.0000094(229)	0.00000050(52)	0.00080	0.95238
	4511.583389		(0012 2 0 0 ) 2 u	0.00303(88)	-0.00154(20)	0.0001872(98)		0.00078	0.99478
	4816.554911		(1000 0 2 2 ) 2 g	0.00010(107)	-0.00022(28)	0.000022(19)	0.000000068(401)	0.00092	0.96016
	4694.404604		(1001 1 1 1 ) 2 u	-0.00039(89)	-0.00012(14)	0.0000127(50)		0.00110	0.71164
	2201.641273		(0000 0 3 3 ) 3 u	0.00020(21)	-0.000027(12)			0.00045	0.15048	27,28
	2088.358695 580		(0001 1 2 2 ) 3 g	0.00087(75)	-0.000100(42)			0.00164	0.15186	30
	581	93 sets of differences for e and f vibrational-rotational energy levels of the acetylene molecule
	582	were considered when compiling Table 5. Some of these bands are shown in Figures 9, 10, and
	583	also in the Supplementary material files. Each group consists of a set of differences between e
	584	and f levels for different values of the total angular momentum J (our calculations of the VR
	585	energy levels of the acetylene molecule were carried out for values 0  J  30). These 93 groups
	586	consisted of: 9 difference sets with K = 0 ( state), 47 difference sets with K = 1 ( state), 26
	587	difference sets with K = 2 ( state), and 11 difference sets with K = 3 (  state). Not all
	588	considered sets were used while compiling Table 5.
	589
	590
	591
	592
	593

Table 6 . Vibrational levels of the C2D2 isotopologue up to 6500 cm -1 , in cm -1 units.

 6

	Vibr State		Empirical*	E" *	Calc	E"-Calc	E"-Calc++
							7 adjusted
							param
	0001100	0,1,1,g	511.532	512.380	510.9645	1.415	0.294
	0000011	0,1,1,u	538.636	539.484	539.3923	0.091	0.098
	0002000	0,2,0,g	1024.81	1024.810	1022.521	2.289	0.171
	000111-1	0,2,0,u	1041.49	1041.490	1040.099	1.391	0.305
	0001-111	0,2,0,u	1048.66	1048.660	1048.8	-0.140	0.460
	0000020	0,2,0,g	1070.86	1070.860	1070.86	0.000	-0.017
	000221-1	0,3,1,u	1552.75	1553.598	1551.168	2.429	0.343
	0002011	0,3,1,u	1562.43	1563.278	1560.912	2.365	0.345
	0001-122	0,3,1,g	1574.82	1575.668	1574.387	1.281	0.206
	0001-122	0,3,1,g	1585.38	1586.228	1584.628	1.599	0.499
	0000031	0,3,1,u	1604.54	1605.388	1605.346	0.042	0.022
	0100000	1,0,0,g	1764.796	1764.796	1764.757	0.039	-0.132
	0000040	0,4,0,g	2136.81	2136.810	2132.263	4.547**	4.440**
	0101100	1,1,1,g	2272.401	2273.249	2271.788	1.460	0.183
	0100011	1,1,1,u	2305.079	2305.927	2305.649	0.277	0.136
	0010000	1,0,0,u	2439.244	2439.244	2439.526	-0.282	0.010
	0000051	0,4,1,g	2659.35	2660.198	2663.045	-2.847**	-3.116**
	1000000	1,0,0,g	2705.160	2705.160	2705.174	-0.014	-0.089
	0102000	1,2,0,g	2783.132	2783.132	2780.78	2.352	0.119
	010111-1	1,2,0,u	2803.959	2803.959	2802.443	1.516	0.297
	0101-111	1,2,0,u	2811.050	2811.050	2809.222	1.828	0.591
	0100020	1,2,0,g	2838.692	2838.692	2838.556	0.136	0.009
	0011100	1,1,1,u	2945.17	2946.018	2944.963	1.055	0.361
	0010011	1,1,1,g	2972.70	2973.548	2973.748	-0.201	0.198

*) Empirical levels from work

ACKNOWLEDGMENTS

The support of RNF-anr grant (Grant No. 22-42-09022) is acknowledged. Support from the French ANR TEMMEX project (Grant 21-CE30-0053-01) is acknowledged. The authors thank to Oleg Lyulin for a useful discussion. A.T. acknowledges support by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the ELTE New National Excellence Program (ÚNKP-21-5) financed by the Hungarian Ministry of Innovation and Technology and the Hungarian NKFIA Fund.