ExoMol molecular line lists - XXVII: spectra of C2H4

A new line list for ethylene, $^{12}$C$_2$$^1$H$_4$ is presented. The line list is based on high level ab initio potential energy and dipole moment surfaces. The potential energy surface is refined by fitting to experimental energies. The line list covers the range up to 7000 cm$^{-1}$ (1.43 $\mu$m) with all ro-vibrational transitions (50 billion) with the lower state below 5000 cm$^{-1}$ included and thus should be applicable for temperatures up to 700 K. A technique for computing molecular opacities from vibrational band intensities is proposed and used to provide temperature dependent cross sections of ethylene for shorter wavelength and higher temperatures. When combined with realistic band profiles (such as the proposed three-band model), the vibrational intensity technique offers a cheap but reasonably accurate alternative to the full ro-vibrational calculations at high temperatures and should be reliable for representing molecular opacities. The C$_2$H$_4$ line list, which is called MaYTY, is made available in electronic form from the CDS.


INTRODUCTION
Hydrocarbons are an important class of molecules for planetary atmospheres. Methane in particular has been detected in many places in the solar system including: the atmospheres of Jupiter (Gladston et al. 1996;Atreya et al. 2003), Saturn (Guerlet et al. 2009), Mars (Atreya et al. 2007), Uranus and Neptune (Lunine 1993) as well as in exoplanetary atmospheres (Swain et al. 2008;Beaulieu et al. 2011). Methane is thought to be a key biosignature (Sagan et al. 1993). In atmospheres with an abundance of methane, chemical reactions initiated by photolysis of C-H bonds leads to the formation of larger hydrocarbons (Gladston et al. 1996;Guerlet et al. 2009;Hu & Seager 2014). Particularly important are the C 2 H n hydrocarbons: acetylene, ethylene and ethane. These molecules have been detected (along with propane, C 3 H 6 ) in the atmospheres of the solar system gas giants (Gladston et al. 1996;Atreya et al. 2003;Guerlet et al. 2009;Lunine 1993). They have also been observed in the atmosphere of Saturn's largest moon Titan (Niemann et al. 2005) which has lakes of liquid hydrocarbons (Stofan et al. 2007). Hydrocarbons were even detected by the Cassini probe in plumes from Enceladus (Waite et al. 2006). Ethylene, the focus of this work, is well-known in the in the circumstellar envelope of IRC+10216 (Betz 1981;Fonfria et al. 2017) and is thought to be important in the atmospheres of exoplanets (Tinetti et al. 2013).
The ro-vibrational energy levels of ethylene have been the focus of multiple theoretical works in this decade. This is due to both its importance and because it is one of the few 6-atom molecules which is relatively rigid: the barrier to rotation of the CH 2 groups is 23 000 cm −1 and involves breaking the π bond (Krylov et al. 1998). This makes ethylene an ideal candidate to develop theoretical methods for medium sized molecules. Avila & Carrington (2011) calculated vibrational energies of C 2 H 4 up to 4100 cm −1 using a basis pruning scheme and the Lanczos algorithm for obtaining the eigenvalues. This was carried out using the quartic force field potential energy surface (PES) of Martin et al. (1995). Carter et al. (2012) then built upon this work by calculating the ro-vibrational energies up to J = 40 and transition intensities using a dipole moment surface (DMS) computed at the MP2/aug-cc-pVTZ level of theory. The ethylene molecule was also used as a test system to develop a new pruning approach by the same group (Wang et al. 2015). A new C 2 H 4 PES (Delahaye et al. 2014) and DMS (Delahaye et al. 2015) was recently constructed which gives even more accurate energies and intensities. A high temperature line list was subsequently constructed using these surfaces by Rey et al. (2016).

ExoMol XXVII: Line list for Ethylene 3
In this work we present new ab initio potential energy and dipole moment surfaces for ethylene and use them to compute a line list for elevated temperatures as part of the ExoMol database project (Tennyson & Yurchenko 2012;Tennyson et al. 2016). We name this line list MaYTY. Compared to the line list of Rey et al. (2016) we slightly increase the applicable frequency range and include many more weak transitions (50 billion here compared to 60 million previously) which are important for total opacity.
Rovibrational energy levels were computed variationally using a refined PES with the TROVE program suite (Yurchenko et al. 2007;. Ethylene is the first 6 atom molecule in the Exomol database and the largest for which we have computed a line list so far. We also propose a new procedure for computing molecular opacities from vibrational transition moments only. Similar J = 0 approaches are very common in simulating spectra of large polyatomic molecules (Jornet-Somoza et al. 2012), where either very simple band profiles (e.g. Lorentzian) or sophisticated functional forms (such as the narrow band approach of Consalvi & Liu (2015)) are used. Here we develop a three-band model based on three fundamental bands of C 2 H 4 (one parallel and two perpendicular), which also represent its three dipole moment components. This J = 0-effort approach has allowed us to significantly extend the temperature as well as the frequency range of our line list and should be also useful for larger polyatomic molecules.
The paper is organised as follows: In Section 2 we give details of our PES and DMS along with our variational calculations and how transition intensities were calculated. In Section 3 we give details of the MaYTY line list and compare with experimental data. The new procedure for generating opacities from vibrational band intensities is discussed in Section 4. We present conclusions in Sections 5.

Potential Energy Surface and Refinement
An initial potential energy surface was constructed from ab initio quantum chemistry calculations. The explicitly correlated coupled cluster method CCSD(T)-F12b (Adler et al. 2007) was used with the F12-optimised correlation consistent polarized valence cc-pVTZ-F12 basis set  in the frozen core approximation. A Slater geminal exponent of β = 1.0 a −1 0 was used (Hill et al. 2009). For the resolution-of-the-identity approximation to many-electron integrals we utilized the OptRI (Yousaf & Peterson 2008) basis set, specifically matched to the cc-pVTZ-F12. The additional many-electron integrals arising in the explicitly correlated methods are calculated using the density fitting approach, for which we employed cc-pV5Z/JKFIT (Weigend 2002) and aug-cc-pwV5Z/MP2FIT (Hättig 2005) auxiliary basis sets. All calculations were carried out using MOLPRO2012 (Werner et al. 2012).
Electronic energies were calculated on a grid of 120 000 molecular geometries for energies of up to hc · 40 000 cm −1 above the equilibrium geometry value. Up to eight of the twelve internal coordinates were varied at once. The twelve coordinates used to represent the PES are: ξ 1 = r 0 − r eq 0 for the C-C bond stretching coordinate; ξ j = r i − r eq 1 j = 2, 3, 4, 5 for each of the C-H i (i = 1..4) bond stretching coordinates; ξ k = θ i − θ eq 1 k = 6, 7, 8, 9, for each of the C-C-H i (i = 1..4) valence angle bending coordinates; ξ 10 = π − β 1 and ξ 11 = β 2 − π where β 1 and β 2 are the two H-C-H book-type dihedral angles; and ξ 12 = 2τ − β 1 + β 2 where τ is the dihedral angle between the two cis hydrogens. For clarity, the angular internal coordinates are shown on Fig. 1. Values of r eq 0 = 1.331Å, r eq 1 = 1.081Å and θ eq 1 = 121.45 • have been used.
The ab initio energies were least squares fit to an analytical form consisting of long and short range parts as where f damp is a damping function to remove the contribution of the short range component at geometries where the internal coordinates are far from their equilibrium values and has 12 Γ C ijk··· (6) where Γ ≡ A g produces symmetrized combinations of different permutations of the coordinates in the D 2h (M) molecular symmetry group, C ijk··· are expansion parameters, and Morse oscillator functions describe the stretching coordinates ξ ′ i = 1−exp(−a i ξ i ) with a 1 = 1.88139 A −1 and a 2..5 = 1.79890Å −1 . The product was limited to a maximum of 8 coordinates coupled at the same time with the sum of powers i + j + · · · + t 8. A total of 1269 terms were used in the sum.
The constants of the long range function in Eq. (5) and the expansion parameters of the short range potential in Eq. (6) were found by least squares fitting to the ab initio energies.
Weight factors for energies were used as proposed by Partridge & Schwenke (1997) where hcẼ i is the electronic energy at i-th geometry, V top = 30 000 cm −1 , V lim = 15 000 cm −1 , and N i is the normalisation constant. A weighted root-mean square (rms) error of 3.2 cm −1 was obtained for energies up to hc · 40, 000 cm −1 . Expansion parameters and the explicit forms of the symmetrised products in Eq. (6) are given in a Fortran 90 subroutine in the supplementary information.
To improve the accuracy of nuclear motion calculations, the PES was refined using experimental data. Refinement was carried out using a least-squares fitting procedure as implemented in TROVE (Yurchenko et al. 2011a) with the pruned basis set (see below) in a very similar manner to that described in a recent paper from our group (Owens et al. 2017).
Due to the large number of parameters used for the analytical representation of the PES and the size of the eigenfunctions for ethylene, only parameters in Eq. (6) with exponents summing to 2 were allowed to vary. This includes linear (ξ i ), harmonic (ξ 2 i ) and mixed terms (ξ i ξ j ) for a total of 21 parameters. Refinement was carried out in two stages. First, 109 experimental vibrational J = 0 band centres taken from Georges et al. (1999) were used.
This gave an initial refinement. Then, 21 rotational-vibrational J = 1 energies from the HITRAN database (Gordon & et al. 2017) were added and the refinement restarted. Pure rotational energies were given the largest weights in the refinement of order 10 4 followed by J = 1 rotational-vibrational levels of order 10 3 and finally vibrational energies of order 0-10 3 depending on the reported accuracy of these levels. Weights are normalised during the refinement and so only relative values are important (Yurchenko et al. 2011a).
The refined PES was found to give accurate values for a further 155 J = 2, 3 and 4 energy levels which were included, but further iterations of refinement did not give improved values. This is due to both the size of the least-squares fitting problem and that added rotationalvibrational levels were from the same vibrational bands as the J = 1 energies. The difference between all observed energy levels used and the values given by our refined PES is shown in Fig. 2. The vibrational energies with observed−calculated errors of > 4 cm −1 were retained in the refinement to still provide some constraint to these states but were given relative weightings of a thousand times less than the HITRAN vibrational energies.
For the refined surface we obtained an rms error of 2.73 cm −1 for the vibrational energies(reduced to 1.95 cm −1 when bands which were given weights of zero in the refinement were excluded) compared to the values quoted in Georges et al. (1999). This is a large error but many of the bands included also gave large errors for the global effective Hamiltonian model used by Georges et al. and are of low accuracy. Bands with the largest errors were given a weighing of zero in our fit. For the J = 1 data we obtain an rms error of 0.45 cm −1 and 0.50 cm −1 when all J = 1 − 4 is included respectively. When combined with the vibrational levels we obtain an overall rms of 1.75 cm −1 , which is reduced to 1.27 cm −1 when bands which were given weights of zero in the refinement were excluded.

Dipole Moment Surface
Ab initio calculations for the DMS were carried out at the CCSD(T)-F12b/aug-cc-pVTZ level of theory using the finite field method. The frozen core approximation with a Slater geminal exponent β = 1.0 a −1 0 was employed using the same ansatz and auxiliary basis sets as the explicitly correlated PES calculations. For each of the x, y and z Cartesian components an electric field of strength ±0.001 a.u. was applied and the dipole moment projections µ x , µ y and µ z computed as derivatives of the electronic energy with respect to the field strength using central finite differences. Calculations were carried out at about 93 000 different molecular geometries with energies up to hc · 40 000 cm −1 , with up to six of the twelve internal coordinates varied at once.
The DMS was fitted to an analytical form as follows. The origin of the molecule-fixed xyz coordinate system r O was taken to be the centre of the C 1 -C 2 bond. The z-axis is chosen to be along the C 1 -C 2 bond: where r C 1 denotes Cartesian coordinates of carbon atom C 1 . The x axis is a symmetric combination average of the four normals to the four planes C 1 C 2 H i (i = 1, 2, 3, 4) as given by e x = e 1 + e 2 + e 3 + e 4 ||e 1 + e 2 + e 3 + e 4 || and the y axis is chosen as in the right-handed system. Here the normals are defined using the cross-products of the unit vector e z with the corresponding C-H bond vectors where r H i denotes Cartesian coordinates of hydrogen atoms. The Cartesian axes x, y and z transform according to D 2h (M) as B 3u , B 2u and B 1u irreducible representations (irreps), where µ α (α = x, y, z) are functions of the internal coordinates of the form where Γ produces symmetrized combinations of different permutations of the coordinates in the B 3u , B 2u and B 1u irreps for α = x, y and z, respectively, and F

Variational Calculations
Variational ro-vibrational calculations were carried out using the TROVE program. The TROVE methodology is well documented (Yurchenko et al. 2007(Yurchenko et al. , 2009Yurchenko et al. 2017a;) and has been applied to a variety of molecules as part of the ExoMol project (Yurchenko et al. 2009 Only the specific details used in this work on ethylene will be discussed here.
The ro-vibrational Hamiltonian was constructed numerically via an automatic differentiation method . The Hamiltonian was expanded using a power series in curvilinear coordinates around the equilibrium geometry of the molecule.
The coordinates used were the same as those used to fit the PES.
The kinetic energy operator was expanded to the 6th order and the potential energy operator to the 8th order. The same Morse coordinates as used in Eq. (6) were used for the potential expansion for the stretching coordinates (i = 1 − 5) with the other bending coordinates expanded as ξ i themselves. Atomic masses were used throughout.
A multistep contraction scheme was used to build the vibrational basis set. For each coordinate a one-dimensional Schrödinger equation was solved using the Numerov-Cooley approach (Noumerov 1924;Cooley 1961;Yurchenko et al. 2007) to generate basis functions φ n i (ξ i ) with vibrational quantum number n i . The vibrational basis set functions |v are formed as products of the 1D basis functions The basis set is truncated by the polyad number P via P = n 1 + 2(n 2 + n 3 + n 4 + n 5 ) + n 6 + n 7 + n 8 + n 9 + n 10 + n 11 + n 12 P max .
A value of P max = 10 was used. This is a smaller value than used for previous Exomol line lists (Underwood et  set we used a complete vibrational basis set (CVBS) extrapolation procedure similar to that described by Owens et al. (2015). Variational calculations were carried out with P max = 6, 8 and 10 respectively. From this we estimate that above 4000 cm −1 there are some vibrational levels (typically those with multiple bending modes excited) which are only converged to around 4 cm −1 with a P max = 10 basis. The average convergence error for 0-5 000 cm −1 is estimated to be only 1.5 cm −1 however. It should be noted that these estimates do not account for the fact that the PES refinement procedure described above tends to compensate partly or fully for the basis set convergence errors, even when extrapolating to higher vibrational excitations. Strictly speaking, in order to get a sensible convergence error, one would need to produce a refined PES for all three values of P max = 6, 8 and 10. These estimates do however indicate the possible error of our effective (P = 10) PES if used with larger basis sets or other nuclear motion methods.
To increase the computational efficiency of this step, a new algorithm for sorting and calculating matrix elements of the PES between primitive basis functions was implemented.
This procedure also sets these elements to zero for potential expansion coefficients with values smaller than a tolerance factor. Here we take this as 0.01 (in the units of cm −1 , Angstrom and radian). This procedure led to around a 70 fold speed up for smaller basis test calculations whilst only affecting the accuracy of vibrational states by 0.01 cm −1 , far lower than the error of the ab initio PES. This new 'fast-ci' method will be described fully in a subsequent publication.
Following this procedure, 145 240 vibrational eigenfunctions |Φ (i) vib of C 2 H 4 were obtained with term energiesẼ (J=0) i up to 21 000 cm −1 above the ground state (our post refinement zero-point-energy is 11 022.5 cm −1 ). According to the J = 0-contraction scheme TROVE uses these J = 0 eigenfunctions as the vibrational basis set. However using a basis set of this size for high rotationally excited levels is currently impractical and it was necessary to reduce the number of basis functions. The basis set was further truncated using the same approach based on the vibrational band intensity as described in a recent paper for the silane (SiH 4 ) line list (Owens et al. 2017), which will be referenced to as intensity basis set pruning (IBSP). According to this approach, the vibrational basis functions |Φ (i) vib above some energy threshold,Ẽ (J=0) max , should be truncated with the exception of functions responsible for significant contribution to the absorption opacity (larger than some intensity threshold I max ).
In turn, the absorption contribution is estimated from the intensities of the corresponding bands using these functions as the upper or lower states.
We define the vibrational absorption intensity (cm/molecule) for the band f ← i as The vibrational Einstein coefficient (s −1 ) is given by where the vibrational transition moment µ f i (D) is Here h is Planck's constant, Q (J=0) is the vibrational (J = 0) partition function,Ẽ The vibrational absorption intensities were computed between each state at an elevated temperature of 800 K. For each vibrational state, the largest intensity to or from that state was then associated with that state. The J = 0 basis set was then pruned based on this.
All states up to hc·8000 cm −1 were retained. States with energy above this were discarded if their largest intensity was less than some value I max . Here a value of I max = 1 × 10 −24 cm/molecule was used. This value was chosen to retain as many states as possible (which support intense transitions) whilst making the calculations for high J practical. The resulting pruned vibrational basis contained 13 572 functions corresponding to energies up to hc·12 000 cm −1 . This basis was then used for J > 0 calculations by combining it with symmetrized rigid-rotor functions as described previously (Yurchenko et al. 2009(Yurchenko et al. , 2017a. The pruning procedure based on the J=0-contraction has the advantages that the accuracy of the vibrational energy levels and eigenfunctions computed using the unpruned basis is retained. The errors introduced in pruning the basis for the ro-vibrational levels are compensated for by refining the PES with the pruned basis. Fig. 3 shows vibrational intensities of C 2 H 4 computed using Eq. (12) for T = 500 K as cross sections. Here we compare the total cross sections (no pruning) and the contribution missing due to the intensity-based pruning. The effect of the pruning on the intensities is negligible for the range below 7000 cm −1 (∼ 0.01 %). This is especially important for hot spectra applications, where the completeness of the molecular absorption arguably plays a more important role than the accuracy .

Line Intensities
The eigenvectors from the variational calculation along with the DMS were used to compute Einstein-A coefficients of transitions. These satisfy the rotational selection rules (Bunker & Jensen 1998) J ′ − J ′′ = 0, ±1, and J ′ + J ′′ = 0, where J ′ and J ′′ are the upper and lower values of the total angular quantum number J and symmetry selection rules The absolute absorption intensities are then given by (Bunker & Jensen 1998) where J f is the rotation quantum number for the final state,ν f i is the transition fre- Intensities were computed using a lower energy range of 0 -5000 cm −1 taking into account up to J = 78 for transitions frequencies between 0 and 7000 cm −1 . An intensity cut-off of 10 −50 cm molecule −1 at T = 298 K was used, ensuring that essentially all transitions are taken into account for up to around 700 K (see section 3.1).

Partition Function
The temperature-dependent partition function Q(T ) is defined as where g i = g ns (2J i + 1) is the degeneracy of the state i with energy E i and rotational quantum number J i . Fig. 4 shows the convergence of Q(T ) as a function of J at different temperatures. At 700 K the partition function is converged to 0.02%. In Table 2 we compare the partition function calculated at various temperature with those of literature values. In general agreement between the various sources is good. Our value which increases slightly faster with tempera- ture than those of Rey et al. (2016) is probably due to our more complete treatment of the energy levels. In the supplementary information we provide the partition function between 0 and 1500 K at 1 K intervals.
The current line list was computed with a lower energy threshold of hc · 5000 cm −1 . To assess the completeness of our line list we compute a reduced partition function, Q limit which only takes into account energies up to hc · 5000 cm −1 in Eq. (18). Fig. 4 also shows a plot of the ratio of Q limit /Q. At 700 K the ratio is 0.98 and this temperature can be taken as a soft limit. At higher temperatures opacity will progressively be underestimated (see Section 4).

Line List Format
A complete description of the ExoMol data structure along with examples was reported by Tennyson et al. (2016). The .states file contains all computed ro-vibrational energies (in cm −1 ) relative to the ground state. Each energy level is assigned a unique state ID with symmetry and quantum number labelling as shown in   NẼ gtot J Γtot n 1 n 2 n 3 n 4 n 5 n 6 n 7 n 8 n 9 n 10 n 11 n 12 Γ vib J K τrot Γrot (1 is Ag, 2 is Au, 3 is B 1g , 4 is B 1u , 5 is B 2g , 6 is B 2u , 7 is B 3g and 8 is B 3u ); n 1 -n 12 : TROVE vibrational quantum numbers (QN); see Table 1 Figure 5. Overview of absolute intensities of MaYTY compared to HITRAN data at 296 K.

Validation
The  pure. For comparison with our line list PNNL cross sections were multiplied by 9.28697 × 10 −16 /0.97729 to convert to cm 2 molecule −1 units and account for the 12 C 2 1 H 4 isotopologue.
We simulated the spectrum using a resolution of 0.1 cm −1 using a Voigt profile with a halfwidth half-maximum (HWHM) value of 0.1 cm −1 . The PNNL spectrum allows a comparison to our calculated line list up to around 6200 cm −1 . Fig. 8 shows the temperature dependence of the absorption cross sections for the MaYTY line list simulated using a resolution of 5 cm −1 where again a Voigt profile with a HWHM of 0.1 cm −1 was used. While regions of weak absorption at 296 K increase by an order of magnitude or more as the temperature is increased, the overall band structure of the absorption does not change greatly with temperature. This behaviour contrasts with other molecules, such as methane , whose band shapes show a strong temperature dependence.    Table 3; the only difference with the ro-vibrational .states file is that the statistical weights are all set to 1 according with Eqs. (12) and (13).
The vibrational line list in this format can be used together with ExoCross to generate absorption vibrational intensities using Eq. (12) (instead of its ro-vibrational analogy in Eq. (17)). Another potential application of our extensive vibrational line list for ethylene is to generate spectra using the spectroscopic tool PGOPHER (Western 2017). One of the recent features of PGOPHER is to import band centers and and transition moments from an external vibrational line list.
Here we use the hot vibrational line list for C 2 H 4 to produce temperature-dependent vibrational cross sections by 'broadening' the corresponding band intensity with suitable band profiles. The vibrational cross sections should, at least approximately, conserve the opacity stored in each vibrational band and thus offer an approximate but simple way of simulating molecular opacity.
In fact, it is common in applications involving large polyatomic molecules to use vibrational intensities for modelling molecular absorption, where Lorentzian or Gaussian functions are used as band profiles. There are also more realistic but elaborate alternatives to represent the band profiles, such as, for example the narrow band approach (Consalvi & Liu 2015).
Here we develop a three-band model, where different vibrational bands (perpendicular and parallel) are modelled using three realistic basic shapes, corresponding to three components of the vibrational dipole momentμ α of ethylene. Table 1, only the ν 7 , ν 9 , ν 10 , ν 11 , ν 12 bands are IR active: ν 9 and ν 10 (see Figs. 6) are parallel bands as they possess the same symmetry B 1u as the z component of the molecular dipole moment µ. The perpendicular bands ν 11 and ν 12 are of the type B 2u (corresponding toμ y ), while the perpendicular band ν 7 is of the type B 3u (µ x ). These three band types (B 1u , B 2u and B 3u ) have different shapes, which we use as templates to model all other vibrational bands of C 2 H 4 . We select the three strongest fundamental bands, one for each type: ν 12 (B 1u ), ν 9 (B 2u ) and ν 7 (B 3u ), and use the corresponding ro-vibrational cross sections at different temperatures to construct three temperature-dependent, normalised band profiles as follows. For each temperature and band in question the corresponding cross-sections on a grid of 1 cm −1 are normalised and shifted to have the center atν = 0.

As indicated in
Three profile templates for T = 500 K and T = 1500 K are shown in Fig. 9. The T =  Figure 9. Normalized ν 12 , ν 9 and ν 7 band profiles with a HWHM=0.1 cm −1 . Left: the T = 500 K profiles were generated using the MaYTY line list with the Voigt profile. Right: The T = 1500 K profile was generated using PGOPHER with the Lorentzian line broadening and spectroscopic constants from Bach et al. (1998) and Rusinek et al. (1998).
500 K profiles were generated using the MaYTY line list in conjunction with the Voigt line profile with HWHM=0.1 cm −1 . For the 1500 K temperature case our line list is rotationally incomplete (J max = 78), therefore we used the effective Hamiltonian approach to generate the corresponding band profiles with significantly higher J max = 120. Towards this we employed PGOPHER together with the ν 7 , ν 9 and ν 12 spectroscopic constants from Bach et al. (1998) and Rusinek et al. (1998), and a Voigt line profile with HWHM=0.1 cm −1 .
These profiles are then applied for the vibrational cross sections at the temperature in question by using the symmetry multiplication rule: if Γ i and Γ f are, respectively, the symmetries of the initial and upper states and Γ α is the symmetry of the dipole moment componentμ α , for an IR active bandν (J=0) f i the following relation holds (Bunker & Jensen 1998): Note that the equal sign here (not ∈) is due to D 2h (M) being an Abelian symmetry group.
We thus use this rule to choose between the B 1u , B 2u or B 3u templates when generating cross sections for specific bandsν . This rule, however, does not always hold: a large number of forbidden (and weak) bands have non-zero intensities due to interactions between vibrational states. In such cases we use a simple Lorentzian band profile with HWHM of 60 cm −1 .
An example of vibrational cross sections of C 2 H 4 at T = 500 K and T = 1500 K generated using this methodology is shown in Fig. 10, where they are also compared to the ro-vibrational cross sections. The vibrational cross sections are more complete and also provide larger coverage (here shown up to 10 000 cm −1 ). For example, the lower display on  The methodology of combining realistic band profiles with vibrational intensities can be especially useful for larger polyatomic molecules, where the size of the calculations becomes prohibitive. This requires knowing the ro-vibrational spectra of the three fundamental bands to generate the realistic band profiles, for which we took advantage of having the complete, ro-vibrational line list. In practical applications when this is not accessible, these profiles could be modelled using effective rotational methods, using for example PGOPHER (Western 2017) as we demonstrated , which only requires the corresponding spectroscopic constants of these (up to) three fundamental bands.
The temperature dependent vibrational cross sections can be useful for evaluating opacities of molecules (especially at higher temperatures) when completeness is more important than high accuracy. The approximations used for vibrational intensities are: (i) the rotational and vibrational degrees of freedom are independent and (ii) lower resolution is assumed.
Due to the missing interaction between the rotational and vibrational degrees of freedom,  Figure 11. The vibrational 'top-up' cross sections of C 2 H 4 computed using the vibrational line list with the 3-band model at T = 500 K (dark blue area) and the ro-vibrational intensities generated using the ExoMol line list with the Gaussian line profile of HWHM=1 cm −1 (red line). this vibrational methodology is not capable of reconstructing some forbidden bands, which are caused by this interaction. This is evident in Fig. 10, where some weaker parts are missing. It is important to note that the vibrational band intensity of a given vibrational band computed using Eq. (12) is the same as the corresponding integrated ro-vibrational intensities from Eq. (17), at least if the interaction with other vibrational bands is ignored.
Thus although the vibrational intensity treatment is highly approximate, it should be better for preserving the opacity in simulations.
In line with our 'hybrid'-methodology (Yurchenko et al. 2017b), the generated vibrational cross sections can be now divided into the strong and weak parts, with the latter representing the absorption, missing from our line list due to the vibrational basis set pruning. These 'weak' vibrational bands form absorption 'continuum' cross sections and can be used to compensate for missing absorption when higher temperatures or larger spectroscopic coverage is required. Fig. 11 shows this absorption continuum of C 2 H 4 at T = 500 K up to