ExoMol line lists -- XXII. The rotation-vibration spectrum of silane up to 1200 K
A. Owens A. Yachmenev W. Thiel J. Tennyson S. N. Yurchenko
physics.chem-ph astro-ph.EP astro-ph.SR




Simple models for multivariate regular variations and the Hüsler-Reiss Pareto distribution
    Zhen Wai Olivier Ho
[Univ. Bourgogne Franche-Comté, Laboratoire de Mathématiques de Besançon, UMR 6623, 16 route de Gray, 25030 Besancon, France. E-mail: zhen_wai_olivier.ho@univ-fcomte.fr] 
Clément Dombry 
[Univ. Bourgogne Franche-Comté, Laboratoire de Mathématiques de Besançon, UMR 6623, 16 route de Gray, 25030 Besancon, France. E-mail: clement.dombry@univ-fcomte.fr] 

    
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A variationally computed ^28SiH_4 rotation-vibration line list applicable for temperatures up to T=1200K is presented. The line list, called OY2T, considers transitions with rotational excitation up to J=42 in the wavenumber range 0–5000cm^-1 (wavelengths λ> 2 μm). Just under 62.7 billion transitions have been calculated between 6.1 million energy levels. Rovibrational calculations have utilized a new `spectroscopic' potential energy surface determined by empirical refinement to 1452 experimentally derived energy levels up to J=6, and a previously reported ab initio dipole moment surface. The temperature-dependent partition function of silane, the OY2T line list format, and the temperature dependence of the OY2T line list are discussed. Comparisons with the PNNL spectral library and other experimental sources indicate that the OY2T line list is robust and able to accurately reproduce weaker intensity features. The full line list is available from the ExoMol database and the CDS database.



molecular data – opacity – planets and satellites: atmospheres – stars: atmospheres – ISM: molecules.




§ INTRODUCTION


 The possibility of silicon-based life existing elsewhere in the Universe is an idea very much rooted in the realms of science fiction. Yet the continued discovery of exoplanets and the exhaustive search for potential biosignature gases <cit.> has renewed interest in the simplest silicon-hydrogen compound, silane (SiH_4). A large number of high-resolution studies on the infrared (IR) absorption spectrum of SiH_4 and its isotopomers have been reported since the first measurements by <cit.>. Rotation-vibration transitions have been detected around the carbon star IRC +10216 <cit.>, and in the atmospheres of Jupiter <cit.> and Saturn <cit.>. Although deemed unlikely, silane has already been considered in the context of H_2-dominated atmospheres on rocky exoplanets <cit.>.

 Despite efforts to document the spectrum of SiH_4, no complete line list exists for this molecule. The PNNL spectral library <cit.> contains IR absorption cross-sections at a resolution of around 0.06cm^-1 for the 600–6500cm^-1 range. Another useful resource is the Spherical Top Data System (STDS) <cit.>, however, a significant portion of the measured transitions and intensities are from unpublished work which makes it difficult to assess the reliability of the data. Theoretical ^28SiH_4 spectra are available from the TheoReTS database <cit.> for a temperature range of 70–300K but the calculations that they are based on are again from unpublished work. A room temperature line list has been produced for the 750–1150cm^-1 region from analysis of the ν_2 and ν_4 bands <cit.>. Whilst absolute line intensities have been determined by <cit.> for a large number of P-branch transitions of the ν_3 band up to the rotational quantum number J=16.

 Previously, we generated potential energy and dipole moment surfaces for silane using state-of-the-art ab initio theory <cit.>. Both surfaces were rigorously evaluated and showed good agreement with experimental results. Computed fundamental term values of ^28SiH_4 using the CBS-F12^ HL potential energy surface (PES) possessed a root-mean-square (rms) error of 0.63cm^-1 compared to experiment. The dipole moment surface (DMS), although tending to marginally overestimate the strength of line intensities, reproduced band shape and structure well.

 Building on this work, we present a comprehensive rotation-vibration line list of ^28SiH_4 suitable for elevated temperatures. The line list, called OY2T, has been computed variationally and utilizes a new `spectroscopic' PES which has been determined by rigorous empirical refinement. The OY2T line list has been produced for the ExoMol database <cit.>, which is providing important molecular spectroscopic data to help characterize exoplanet and other hot atmospheres. Examples of the application of these line lists include: the early detection of water in HD 189733b <cit.> and HD 209458b <cit.> using the BT2 line list <cit.>; the tentative identification of HCN in the atmosphere of super-Earth 55 Canri e <cit.> and TiO in the atmosphere of hot Jupiter WASP-76 b <cit.>; various studies using the 10to10 line list <cit.> to detect methane in exoplanets <cit.>, the bright T4.5 brown dwarf 2MASS 0559-14 <cit.>, and to make detailed line assignments in the near-infrared spectra of late T dwarfs <cit.> in conjunction with the BYTe ammonia line list <cit.>. Conversely, a line list for the diatomic NaH molecule <cit.> was able to rule out the tentative detection of this species in the atmosphere of a brown dwarf.

 Silane is the second five-atom molecule to be treated within the ExoMol framework <cit.> after methane and the 10to10 line list <cit.>, which demonstrated the need to consider a very large number of transitions to correctly model the opacity at elevated temperatures <cit.>. Since the 10to10 line list, several key developments have taken place in our nuclear motion code trove <cit.> that significantly ease the computational burden of theoretical line list production. These include an automatic differentiation method to construct the rovibrational Hamiltonian, the implementation of curvilinear internal coordinates, and a novel vibrational basis set truncation approach which will be discussed later on.

  The paper is structured as follows: In Sec. <ref> we detail the empirical refinement of the CBS-F12^ HL PES, the DMS and intensity simulations, and the variational calculations. In Sec. <ref>, the OY2T line list is presented and we discuss the temperature-dependent partition function of SiH_4, the format of the OY2T line list, the temperature dependence of the OY2T line list, and comparisons with the PNNL spectral library and other experimental data. Concluding remarks are given in Sec. <ref>.



§ METHODS





 §.§ Potential energy surface refinement


 The CBS-F12^ HL PES <cit.> was constructed from extensive, explicitly correlated coupled cluster calculations with extrapolation to the complete basis set limit, and incorporated additional higher-level energy corrections to account for core-valence electron correlation, higher-order coupled cluster terms beyond CCSD(T), and scalar relativistic effects. Although impressive in its accuracy, orders-of-magnitude improvements in predicted transition frequencies can be achieved by refining the PES to experiment. Furthermore, improved energy levels means better wavefunctions and more reliable intensities.

 The refinement was carried out using an efficient least-squares fitting procedure <cit.> implemented in trove. The procedure works by assuming the CBS-F12^ HL PES, V_ CBSF12^HL, is already a reasonable representation such that the effect of the refinement can be treated as a perturbation Δ V, i.e.

    V^'=V_ CBSF12^HL+Δ V ,

where V^' is the refined PES. The correction Δ V is expanded in terms of nine vibrational coordinates ξ={ξ_1,ξ_2,…,ξ_9} according to the formula,

    Δ V = ∑_ijk…Δf_ijk…{ξ_1^iξ_2^jξ_3^k…}^A_1 ,

where the coefficients Δf_ijk… are corrections to the original PES expansion parameters f_ijk…. The expansion terms {ξ_1^iξ_2^jξ_3^k…}^A_1 are symmetrized combinations of the vibrational coordinates ξ={ξ_1,ξ_2,…,ξ_9} and transform according to the A_1 irreducible representation of the T_d(M) molecular symmetry group <cit.>. Further details on the vibrational coordinates and analytic representation of the CBS-F12^ HL PES can be found in <cit.>.

  The new perturbed Hamiltonian, H^'=H+Δ V, is then diagonalized using a basis set of eigenfunctions from the initial unperturbed Hamiltonian H eigenvalue problem. Each iteration utilizes the previous “unperturbed” basis set in this way until a PES of desirable quality is reached. To ensure the consistency of the refined surface, the expansion parameters f_ijk… are simultaneously fitted to both the experimental data and the original ab initio dataset <cit.>. This stops any unrealistic distortion of the PES in regions not sampled by experiment. It also means each expansion parameter can be adjusted irrespective of the amount, or distribution, of experimental energies used in the refinement.

 For SiH_4, a total of 1452 experimental term values up to J=6 were used to refine the CBS-F12^ HL PES. This included 53 vibrational J=0 band centres from <cit.>, and 1399 J>0 energies from the STDS <cit.>. Of the 106 expansion parameters of the CBS-F12^ HL PES, 104 were varied and the results of the refinement are shown in Fig. 1, where we have plotted the fitting residuals Δ E( obs-calc)=E_ obs-E_ calc. Here, E_ obs and E_ calc are the observed and calculated energies, respectively. Overall, the experimental energy levels are reproduced with a rms error of 0.454cm^-1. This error is slightly larger than what is achievable with refinement, however, this was intentional. The energy levels from the STDS are derived from an effective Hamiltonian model based on some unpublished work. It is therefore hard to assess the reliability of the data and how much it can be trusted in the refinement. Despite this, because the CBS-F12^ HL PES is reliable we could depend on it more and ensure the refined PES did not deviate too far away from the original surface.



 Pure rotational energies were given the largest weights in the refinement, while J=0 and rovibrational term values were weighted two orders of magnitude smaller with values between 1–10 based on their energy. Since the weights are normalized in the fitting, relative weighting is more important than absolute values (see <cit.> for further details). Although we have determined a new `spectroscopic' PES of silane, the accuracy of the refined PES is only guaranteed with the computational setup employed in this study. This is usually the case in theoretical line list production using programs that do not treat the kinetic energy operator exactly, and we therefore do not recommend this PES for use otherwise.



 §.§ Dipole moment surface and line intensities


 The electric DMS utilized for the present study was computed at the CCSD(T)/aug-cc-pVTZ(+d for Si) level of theory. Further details on the analytic representation and performance of the DMS can be found in <cit.>, where it was shown that the DMS marginally overestimates the strength of line intensities. This discrepancy is very minor though as we will see in Sec. <ref>.

 To simulate absolute absorption intensities we used the expression,

    I(f ← i) = A_if/8π cg_ns(2 J_f+1)exp(-E_i/kT)/Q(T) ν_if^2[1-exp(-hcν_if/kT)] ,

where A_if is the Einstein A coefficient of a transition with wavenumber ν_if (in cm^-1) between an initial state with energy E_i, and a final state with rotational quantum number J_f. Here, k is the Boltzmann constant, T is the absolute temperature, h is the Planck constant and c is the speed of light. The nuclear spin statistical weights are g_ns={ 5,5,2,3,3} for states of symmetry { A_1,A_2,E,F_1,F_2}, respectively. The partition function for a given temperature Q(T) will be discussed in Sec. <ref>. All transitions obey the symmetry selection rules

    A_1 ↔ A_2,  E ↔ E,  F_1 ↔ F_2 ,

and the standard rotational selection rules,

    J-J=0,± 1,  J+J 0 ,

where  and  denote the upper and lower state, respectively. Note that the ExoCross code (Yurchenko, Al-Refaie & Tennyson (in preparation); available at <https://doi.org/10.5281/zenodo.400748>) was employed for all spectral simulations.



 §.§ Variational calculations


 The computer program trove was used for all rovibrational calculations. Since the methodology of trove is well documented <cit.> and we have previously reported calculations on SiH_4 <cit.>, we summarise only the key aspects relevant for this work.

 The rovibrational Hamiltonian was constructed numerically using an automatic differentiation method <cit.>. The Hamiltonian was represented as a power series expansion around the equilibrium geometry in terms of nine, curvilinear internal coordinates, with the kinetic and potential energy operators truncated at 6th order. Atomic mass values were used throughout. A multi-step contraction scheme was used to build the vibrational basis set, the size of which was controlled by the polyad number

    P = 2(n_1+n_2+n_3+n_4)+n_5+n_6+n_7+n_8+n_9 ≤ P_max .

The quantum numbers n_k for k=1,…,9 are related to primitive basis functions ϕ_n_k, which are obtained by solving a one-dimensional Schrödinger equation for each kth vibrational mode with the Numerov-Cooley method <cit.>. Multiplication with symmetrized rigid-rotor eigenfunctions |J,Γ_rot,n⟩ produces the final basis set for J>0 calculations. Here, the label Γ_rot is the rotational symmetry and n is a multiplicity index used to count states within a given J (see <cit.>).

 In calculations we set P_max=12, which resulted in 19,237 vibrational basis functions corresponding to energies up to h c · 15,000cm^-1. Describing high rotational excitations with such a large basis set can quickly become computationally intractable (rovibrational matrices scale linearly with J). It was therefore necessary to reduce the number of basis functions. To do this we employed a new, basis set truncation approach (Yurchenko et al. (in preparation)) based on vibrational transition moments, which are relatively inexpensive to compute in trove. The vibrational transition moment between two states is defined as,

    μ_if = √(∑_α=x,y,z|⟨Φ^(f)_vib|μ̅_α|Φ^(i)_vib⟩|^2) ,

where |Φ^(i)_vib⟩ and |Φ^(f)_vib⟩ are the initial and final state vibrational eigenfunctions, respectively, and μ̅_α is the electronically averaged dipole moment function along the molecule-fixed axis α=x,y,z.

 After computing J=0 energies, all possible transition moments were calculated for a lower energy threshold of h c · 8000cm^-1 (same as for the final OY2T line list). Once known, we estimated the vibrational band intensity at an elevated temperature (e.g. 1500K) for every transition moment. Each vibrational energy level, and consequently each J=0 basis function, was assigned a band intensity value which was simply the largest computed for that state when acting as |Φ^(i)_vib⟩ in Eq. (<ref>). The J=0 basis set was then pruned by removing basis functions and energy levels above h c · 8000cm^-1 with band intensity values smaller than 1× 10^-21cm/molecule. This is around three orders of magnitude smaller than the largest computed value. The band intensity threshold was chosen solely on a computational basis so that the rovibrational calculations were manageable with the resources available to us. The resulting pruned basis set contained 3817 functions (corresponding to energies up to h c · 15,000cm^-1) and was multiplied in the usual manner with symmetrized rigid-rotor functions for J>0 calculations.

 An advantage of this truncation approach is that the accuracy of the vibrational energy levels and respective wavefunctions computed using a larger value of P_max is retained. Naturally we will lose some information on weaker lines involving states above h c · 8000cm^-1 and it is hard to quantify the effect without more rigorous calculations. Removing functions from the basis set will also influence predicted rovibrational energies but the errors introduced, and the basis set convergence error, can to some extent be compensated for by refining the PES with the pruned basis set, which is what we have done. The resulting `effective' PES is only reliable with the same computational setup but this is usually the case in theoretical line list production with trove.

 The OY2T line list was computed with a lower state energy threshold of h c · 8000cm^-1 and considered transitions up to J=42 in the 0–5000cm^-1 range. To improve the accuracy of the line list we employed an empirical basis set correction (EBSC) <cit.>, which is essentially a shift of the band centres to better match experiment. Recall that in the multi-step contraction scheme used by trove, the J=0 eigenfunctions form the final basis set and the vibrational part of the Hamiltonian is diagonal with respect to this basis set. It is possible then to replace the diagonal elements with accurate experimental values for the J=0 energies when constructing the Hamiltonian matrix. For the OY2T line list, only the four fundamentals, listed in Table 1, were replaced as these wavenumbers are well established.





§ RESULTS





 §.§ Partition function of silane



 The temperature-dependent partition function Q(T) is defined as,

    Q(T)=∑_i g_i exp(-E_i/kT) ,

where g_i=g_ ns(2J_i+1) is the degeneracy of a state i with energy E_i and rotational quantum number J_i. Summing over all computed rovibrational energy levels, in Fig. 2 we have plotted the convergence of Q(T) as a function of J for different temperatures. At T=1200K the partition function is converged to around 0.1%. Our calculated room temperature partition function Q(296  K)=1532.93 is in excellent agreement with a value of Q(296  K)=1533.00 from the TheoReTS database <cit.>. It also agrees well with an approximate estimate of Q≈ Q_rot× Q_vib=1527.36 used in our previous study <cit.>, where Q_rot and Q_vib are the rotational and vibrational partition function, respectively.



 Given that the OY2T line list has been computed with a lower energy threshold of h c · 8000cm^-1, it is informative to  study a reduced partition function Q_ limit, which only considers energy levels up to h c · 8000cm^-1 in the summation of Eq. (<ref>). In Fig. 3 we plot the ratio Q_ limit/Q with respect to temperature, and this can provide a measure of completeness of the OY2T line list. At T=1200K, the ratio Q_ limit/Q=0.94 and we recommend this as a `soft' temperature limit to the OY2T line list. Using the line list above this temperature will result in the progressive loss of opacity, however, it is possible to estimate the missing contribution with the ratio Q_ limit/Q <cit.>. Note that our full partition function evaluated on a 1K grid from 70–1400K is provided as supplementary material.





 §.§ OY2T line list format


 A complete description of the ExoMol data structure along with examples was recently reported by <cit.>. The  file contains all computed rovibrational energies (in cm^-1). Each energy level possesses a unique state ID with symmetry and quantum number labelling as shown in Table 2. The  files, which are split into frequency windows so that they are easier to handle, contain all computed transitions with upper and lower state ID labels, and Einstein A coefficients. An example from a  file for the OY2T line list is given in Table 3.







 §.§ Validation of the OY2T line list



 The OY2T line list contains nearly 62.7 billion (62,690,449,078) transitions between 6.1 million (6,142,521) energy levels. The distribution of lines and energies is illustrated in Fig. 4, where we have plotted the total number computed for each value of J. The density of transitions is largest between 18≤ J ≤ 26 but this drops off relatively smoothly and by J=42 we have nearly calculated all possible transitions for our computational setup (e.g. lower energy threshold of h c · 8000cm^-1, wavenumber range of 5000cm^-1, pruned rovibrational basis set, etc.). The decrease in the number of energy levels after J=34 is a result of the upper energy threshold of h c · 13,000 cm^-1



 The temperature dependence of the OY2T line list is shown in Fig. 5, where we have simulated integrated absorption cross-sections at a resolution of 1cm^-1 using a Gaussian profile with a half width at half maximum (hwhm) of 1cm^-1. Very weak intensities can be seen to grow several orders of magnitude stronger as the temperature increases. This smoothing of the spectrum is a result of vibrationally excited states becoming more populated and causing the rotational band envelope to broaden.



 A good benchmark of the OY2T line list is to compare with the PNNL spectral library <cit.>. In Fig. 6 and Fig. 7 we have simulated cross-sections at a resolution of 0.06cm^-1 using a Gaussian profile with a hwhm of 0.135cm^-1. The experimental PNNL silane spectrum was measured at a temperature of 25^∘C with the dataset subsequently re-normalized to 22.84^∘C (296K). It is of electronics grade silane gas, which is composed of ^28SiH_4 (92.2%), ^29SiH_4 (4.7%), and ^30SiH_4 (3.1%). We have therefore scaled our computed ^28SiH_4 cross-sections by 0.922 to ensure a reliable comparison. As expected <cit.>, the OY2T cross-sections are marginally stronger but this is very slight. Overall the agreement is very encouraging, particularly the performance of the OY2T line list for weaker bands (see the right-hand panels of Fig. 7) as we are missing contributions from ^29SiH_4 and ^30SiH_4.





 Finally, we compare absolute line intensities of the OY2T line list with measurements of the ν_3 band from <cit.>, and with the 750–1150cm^-1 region line list from <cit.>. Intensities were computed using Eq. (<ref>) at T=296K in Fig. 8, and at T=298K in Fig. 9. As was previously found when validating the DMS <cit.>, there is good agreement with the results of <cit.>, where transition intensities up to J=16 were recorded at a resolution of 0.0011cm^-1 with an estimated experimental measurement accuracy of 10%. For the ^28SiH_4 line list from <cit.> the agreement is also pleasing. Their line list contains 3512 transitions up to J=27 with the intensities determined from an analysis of 787 of the measured lines. Again the OY2T line intensities are slightly stronger in places but overall the band structure is well matched and weaker intensity features are accurately accounted for. It is worth noting that the OY2T line list is up to J=42 which would explain some of the additional spectral features in both comparisons.







§ CONCLUSION



 A comprehensive rotation-vibration line list for ^28SiH_4 has been 
presented. The OY2T line list covers the 0–5000cm^-1 region and 
includes transitions up to J=42. Analysis of the temperature-dependent 
partition function suggests that the OY2T line list can be confidently used for 
temperatures below T=1200K. Applications above this temperature are likely 
to result in the loss of opacity. Comparisons with the PNNL spectral library and 
other experimental sources showed that the OY2T line list is robust and able to 
accurately reproduce the intensity features of weaker bands. The OY2T line list 
can be downloaded from the ExoMol database at <www.exomol.com> or the CDS 
database at <http://cdsarc.u-strasbg.fr>.

The line list OY2T will be useful for modelling of absorption of SiH_4 in 
atmospheres of exoplanets. Its quality (completeness and accuracy) should be 
sufficient, at least in principle, to detect silane in an atmosphere of a hot 
exoplanet from the transit spectroscopic observations, 
when combined with a proper atmospheric and radiative transfer models and 
providing that the abundance of SiH_4 is sufficiently large to be detectable. 
However for high resolution detection techniques such as the high-dispersion 
spectroscopy developed by <cit.>, our line positions might not be 
sufficiently accurate. This technique is based on the Doppler shifts of a large 
number of spectroscopic lines of a given species, which are cross-correlated to 
the reference lab data on the line positions. The required resolution of the 
line positions should be as high, R ≥ 100,000. We do have a method to 
address this problem, which is to replace the theoretical energy levels in a 
synthetic line list with experimentally derived ones, usually obtained 
with the MARVEL (measured active rotational-vibrational energy levels) procedure
<cit.>. The most recent example of is the hybrid line list 
for H_3^+ MiZATeP <cit.>.  The experimental spectroscopic data 
available for SiH_4 should be sufficient to generate such hybrid line list for 
this molecule in the future.


 Natural extensions to the OY2T line list would be the consideration of a larger 
frequency range, an increased lower energy threshold in computations, and the 
inclusion of higher rotational excitations. These issues, although 
computationally challenging, are straightforward to address but will only be 
done if there is a demand for such work. It would be useful to derive a 
consistent set of normal mode quantum numbers  v_k for the OY2T line 
list. These are commonly used in high-resolution spectroscopic applications and 
could be easily incorporated by updating the  file. Work in this 
direction is underway and any updates to the OY2T line list will be released on 
the ExoMol website. It may also be worthwhile to explore a more compact 
representation <cit.> of the OY2T line list given the huge number of 
lines that have been generated. Certainly we are in a position to produce 
temperature-dependent cross-sections for a fixed resolution if requested.







§ ACKNOWLEDGMENTS

This work was part of ERC Advanced Investigator Project 267219. We also acknowledge support from FP7-MC-IEF project 629237, COST Action CM1405 MOLIM, and the Max Planck Computing and Data Facility (MPCDF).

mn2e


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