An ab initio variationally computed roomtemperature line list for SO
Abstract
Ab initio potential energy and dipole moment surfaces are computed for sulfur trioxide (SO) at the CCSD(T)F12b level of theory and appropriate triplezeta basis sets. These surfaces are fitted and used, with a slight correction for the equilibrium S–O distance, to compute pure rotational and rotationvibraton spectra of SO using the variational nuclear motion program TROVE. The calculations considered transitions in the region 0–4000 cm with rotational states up to . The resulting line list of 349 348 513 transitions is appropriate for modelling room temperature SO spectra. Good agreement is found with the observed infrared absorption spectra and the calculations are used to place the measured relative intensities on an absolute scale. A list of 10 878 experimental transitions is provided in a form suitable for inclusion in standard atmospheric and planetary spectroscopic databases.
I Introduction
Sulfur trioxide (SO) is a stable, planar, symmetric molecule whose electronic ground state is a closed shell. On earth it is a pollutant produced from smokestacks and other industrial exhausts Rawlins et al. (2005). In the atmosphere SO forms sulfuric acid with its association with acid rain, and inside combustion systems it is corrosive. In both cases the reactivity of SO makes it production hard to monitor Fleig et al. (2012). SO is produced naturally on earth as part of volcanic emissions Michaud et al. (2005). It is also thought to be a significant constituent of the atmosphere of Venus Zhang et al. (2010).
The infrared vibrationrotation spectrum of SO (henceforth referred to as SO) has been extensively investigated in a series of papers by Maki and coworkers Kaldor et al. (1973); Ortigoso et al. (1989); Chrysostom et al. (2001); Maki et al. (2001); Barber et al. (2002); Sharpe et al. (2003); Maki et al. (2004). Its “forbidden” rotational spectrum, for which centrifugal distortions can induce transitions, has been investigated using microwave Fouriertransform spectroscopy Meyer et al. (1991). However, although these works provide an extensive list of measured line frequencies, none of them report absolute transition intensities.
There has been limited theoretical work on SO. Dorney et al. Dorney et al. (1973) reported force constants, while Martin Martin (1999) computed an ab initio potential energy surface (PES) using coupled cluster methods and reported theoretical estimates for the band origins of the lowlying vibrational states. Again, neither of these works considered transition intensities.
The lack of any absolute transition intensities for SO places severe limitations on the use of its infrared spectrum for remote sensing applications or inclusion of this data in standard atmospheric and planetary spectroscopic databases Rothman et al. (2009); JacquinetHusson et al. (2011). In this work we report the calculation of a new ab initio PES and associated dipole moment surfaces (DMS) for SO. These are used to not only produce theoretical spectra for SO, but also to place the relative intensity measurements of its infrared spectrum on an absolute scale.
Ii The potential energy and dipole moment surfaces
The ab initio PES was computed using the recentlyproposed explicitly correlated F12 singles and doubles coupled cluster method including a perturbational estimate of connected triple excitations, CCSD(T)F12b Adler et al. (2007), in conjunction with the corresponding F12optimized correlation consistent basis sets, namely the valence correlationconsistent functions augccpVTZF12 and augccpV(T+d)ZF12 for oxygen and sulfur, respectively Yousaf and Peterson (2009). We also utilized the OptRI Yousaf and Peterson (2008) ccpV5Z/JKFIT Weigend (2002) and augccpwCV5Z/MP2FIT Hättig (2005) auxiliary basis sets for evaluating the manyelectron integrals, the exchange and Fock operators, and the remaining electron repulsion integrals, respectively. The value of the geminal Slater exponent was chosen as 1.2. In ab initio calculations Molpro.2010 Werner et al. (2010) was employed. We used a large grid of 13 000 points with the stretching modes ranging from 1.1 to 2.0 Å, interbond angles between 70 and 123, and the inversion angle between 0 and 50. This grid was sufficient to cover energies up to 40 000 cm above the minimum.
This PES was then represented in an analytical form given by the expansion Yurchenko et al. (2005)
(1)  
in the coordinates :
(2)  
(3)  
(4)  
(5) 
where
(6) 
and
(7) 
The same form has been used to represent the PESs of NH, PH, SbH, NH, and BiH Lin et al. (2002); Yurchenko et al. (2006); Ovsyannikov et al. (2008); Yurchenko et al. (2008). The potential parameters were obtained through a least squares fit to the ab initio points with an rootmeansquares (rms) error of 0.067 cm. Weight factors were set using the expression suggested by Partridge and Schwenke Partridge and Schwenke (1997):
(8) 
where , and is the ab initio energy at the th geometry (in cm), measured relative to the equilibrium energy. The ab initio energy is weighted by the factor in the PES fitting; these weight factors favor the energies below 16 000 cm. The ab initio equilibrium geometry as obtained from the fitting is 1.42039 Å. This can be compared with an experimentally derived value 1.41732 Å Ortigoso et al. (1989). We found that the experimental rotational energies of SO Maki et al. (2001) are best described by the latter value. Therefore we decided to use this value of in all our calculations in place of the ab initio value. It is known however that replacing the equilibrium structure may cause undesirably large changes to the rovibrational energies. In order to minimize this effect the following procedure was employed.
The idea is to expand the ab initio PES around some reference value and than remove the linear terms in the PES expansion and replace the expansion center with . In doing this we expect the change of the shape in the PES to partly compensate the effect from replacing with on the vibrational energies. We choose the reference center , where , i.e. on the opposite side from and at the same distance from .
This is the only adjustment to the shape of ab initio PES utilized. Table 1 compares rotational levels obtained by Maki et al. Maki et al. (2001) and those computed using the equilibriumadjusted PES used in this work. The agreement is very good.
The DMS were calculated using the same level of theory as the PES and on the same grid of 13 000 points. The ab initio values were then expressed analytically using the symmetrized molecular bond (SMB) representation described in detail in Ref. Yurchenko et al. (2011). The resulting dipole moment parameters obtained through a least squares fit reproduce the ab initio data with an rms error of 0.00013 D. In these fittings the same factors defined by Eq. (8) were used to weight the geometries according to the corresponding energies. To our knowledge there are no experimental or ab initio dipole moment data in the literature that we could use to validate our DMS against. However, our experience of working dipole moments for different systems LynasGray et al. (1995); Yurchenko et al. (2011); Lodi et al. (2011); Yurchenko et al. (2013) shows that ab initio intensities in most cases are competitive with experimental measurements. The quality of the relative intensities calculated using our DMS is discussed below.
Both the potential energy and dipole moments functions used in the present work are given as supplementary data.
Obs.  TROVE  

2  0  2.0912  2.0916 
3  3  2.6115  2.6119 
4  3  5.3998  5.4006 
4  0  6.9707  6.9718 
5  3  8.8852  8.8864 
6  6  8.3548  8.3559 
6  3  13.0675  13.0694 
7  6  13.2342  13.2360 
7  3  17.9467  17.9493 
8  6  18.8106  18.8132 
8  3  23.5228  23.5263 
8  0  25.0935  25.0972 
9  9  17.2297  17.2319 
9  6  25.0838  25.0874 
9  3  29.7958  29.8002 
10  10  17.2297  17.2319 
10  9  24.2002  24.2035 
10  6  32.0539  32.0584 
10  3  36.7655  36.7709 
10  0  38.3360  38.3417 
20  18  89.8252  89.8372 
20  15  107.0973  107.1122 
20  12  121.2253  121.2425 
20  9  132.2114  132.2304 
20  6  140.0574  140.0777 
20  3  144.7645  144.7856 
20  0  146.3334  146.3548 
80  78  1195.6589  1195.8085 
Iii Rovibrational calculations
iii.1 Basis set convergence and Hamiltonian optimisation
Rovibrational calculations were performed with the program TROVE Yurchenko et al. (2007) adapted to work in the (M) molecular permutationinversion group appropriate for SO. In order to achieve results of high accuracy as well as minimising the requirement for computational resources, it is necessary to optimise the size of the Hamiltonian matrix. This involves preliminary truncation of the basis set, as well as limiting the order of both the kinetic and potential components of the Hamiltonian expansion. TROVE employs a polyad number truncation which controls the size of the basis set. For SO the polyad number is given by
(9) 
where are the quanta associated with 1D basis functions, , whose product gives our vibrational basis set Yurchenko et al. (2007). Each of these basis functions is associated with an internal coordinate , and only functions for which are included in the primitive basis set. Initial tests were carried out to measure the degree of convergence using different values for , and the orders of the kinetic and potential energy expansions. In this work we use a kinetic energy expansion of order 4, and a potential energy expansion of order 8; using a kinetic energy expansion order of 6 requires a more expensive calculation where convergence is already observed to within 0.001 cm when expanding to fourth order. In the present study, we find that the convergence is more sensitive to , and we obtain convergence to within 0.1 cm when is 12 or 14 (see table 2), therefore we use a basis set based on = 12.
Obs. Maki et al. (2001)  = 10  = 12  = 14  
1064.92  1065.83  1065.75  1065.74  
497.57  498.48  498.48  498.48  
1391.52  1387.63  1387.45  1387.43  
530.09  528.61  528.59  528.58  
2(=0)  2766.40  2759.61  2759.12  2758.75 
2(=2)  2777.87  2770.70  2770.29  2769.95 
2  995.02  995.43  995.35  995.35 
2(=0)  1059.81  1057.10  1056.50  1056.44 
2(=2)  1060.45  1057.86  1057.38  1057.33 
+ (=1)  1027.90  1027.58  1027.35  1027.33 
+ (=1)  1593.69  1593.82  1593.36  1593.30 
3(=1)  1589.81  1587.64  1586.46  1586.30 
3(=3)  1591.10  1587.61  1586.43  1586.27 
+  1560.60  1565.51  1565.33  1565.32 
+ 2(=0)  1557.88  1556.38  1555.59  1555.47 
+ 2(=2)  1558.52  1557.12  1556.45  1556.37 
2 + (=1)  1525.61  1524.81  1524.48  1524.46 
3  1492.35  1449.81  1490.76  1490.76 
+ (=1)  1884.57  1881.82  1881.53  1881.51 
3(=1)  4136.39  4138.88  4126.78  4125.92 
As well as using the polyad number to truncate the size of the basis, we employ a further truncation technique by specifying an upper limit for the eigenvalue calculations, i.e. construct the basis set such that it provides energy values up to a limit of . This is based on an estimation whereby eigenvalues of our 1D basis functions are summed together before they are considered for matrix element calculations, and the active space is constructed using basis functions whose eigenvalues sum together to have . For the present study we use = 10 000 cm.
These precautions are particularly important for the SO molecule, as its larger mass (compared to, for example, XH systems) gives rise to small rotational constants, which in turn requires calculations up to high value to ensure adequate coverage of transitions for a given temperature. This means that any unnecessary basis functions will prove computationally expensive. In addition to this basis set minimisation, we can reduce the size of the Hamiltonian further by making use of group theory. SO has (M) molecular group symmetry and the spin0 Bosons which make up the constituent atoms of this molecule allow the molecular rovibrational wavefunctions to exhibit the symmetry of only two of the six irreducible representations of this group in order to satisfy the Pauli Principle; namely the A and A representations. This reduces both the number of Hamiltonian matrices we need to consider and, since symmetry Hamiltonian matrices are larger, their size.
The two other factors which are important in our spectral calculations are i) the wavenumber range of the desired synthetic spectrum, and ii) the temperature at which we wish to simulate it. The quality of a computed spectrum will become sensitive to as the temperature increases; we need to ensure that we calculate all energy states that are significantly populated for the given temperature. This can be checked using the temperaturedependent partition function:
(10) 
where is the total degeneracy of the rovibrational state with energy , with the sum running over all energies at the absolute temperature , and is Boltzmann’s constant. The total degeneracy is given by times the nuclear spin degeneracy, which for the present case of SO is simply 1 for both the A and A symmetries, given that the nuclear spin of O and S are zero. For a given temperature, we can determine the contribution of various states to the value of . We can then check that converges to a specific value as tends to infinity; as increases we require a greater coverage of higherlying energy states. For = 298.15 K we find that converges to better than 1 % at , with a value of . Therefore calculations spanning all ’s up to 85 should be sufficient for simulating spectra at this temperature. Figure 1 shows the value of as a function of all energy levels having quantum number up to a maximum value, , at an absolute temperature of = 298.15 K. As we include energies in the summation for increasing values of we see that the associated energy levels contribute less and less to the value of , until it converges to a limit.
An ab initio PES was previously published by Martin based on coupled cluster methods Martin (1999). As an intial test of the TROVE procedure we used the quartic force field by Martin to compute fundamental term values, and to test our convergence procedures, using a polyad truncation scheme of = 16. We found discrepancies between our results and the values published, particularly in the value of the fundamental term value. We made a substitution to the symmetryadapted force constant associated with this vibration, taking a scaled value from a previously published forcefield Dorney et al. (1973), which resulted in a fair improvement. Table 3 illustrates the differences between the values and our final converged result for the fundamental vibrations.
Obs. Maki et al. (2001)  Martin  Martin ^{a}  This work  

1064.92  1063.36  1064.22  1065.75  
497.57  428.36  487.10  498.48  
1391.52  1386.81  1386.85  1387.45  
530.09  527.35  527.32  528.59 

Martin, with a substituted value Dorney et al. (1973).
iii.2 Intensity simulations
The simulation of absorption spectra can be broken down into two main parts: first, the eigenfunctions and eigenvalues of the numerically constructed Hamiltonian matrix are calculated by a diagonalisation procedure; second, these eigenfunctions are used to compute transition dipoles, line strengthes , Einstein coefficients and intensities for allowed transitions. For SO, the rigorous selection rules determining allowed electric dipole transitions are , and symmetry selection rules . The intensity of a transition between given states is given by the formula
(11) 
where is the transition intensity for a transition from state with energy to state with energy , with . is the partition function. The value of must be converged for the absolute temperature , and with respect to the wavenumber range in which we wish to simulate our spectrum. is the line strength, which is defined by the following integration:
(12) 
for a transition between initial state with rovibrational wavefunction and final state with wavefunction . Here, is the nuclear spin statistical weight factor, and is the electronically averaged component of the molecular dipole moment along the spacefixed axis . The quantum numbers and are projections of the total angular momentum (in units of ) on the laboratory fixed axis, for the initial and final states, respectively.
Maki et al. Kaldor et al. (1973); Ortigoso et al. (1989); Chrysostom et al. (2001); Maki et al. (2001); Barber et al. (2002); Sharpe et al. (2003); Maki et al. (2004) reported an extensive highresolution study of a number of fundamental, combination and overtone bands of SO. Their principle aim was to obtain accurate wavenumber measurements, but relative intensities were also measured. In the present work we convert these data into absolute intensities by normalizing to the theoretical intensities obtained with TROVE at = 298.15 K as described below.
The measurements available to us cover three spectral regions: 405–708 cm (focusing on , , 2, +, +, , 2 and ), 1200–1680 cm(), and 25003280 cm (). Each measurement was made at different values of pressure. For the 405–708 cm window, the measurements were performed at 0.409 and 2.04 Torr, and for the measurements between the 1200–1680 cm window 0.16 Torr and 0.7 Torr was used. 560 lines were measured at 0.7 Torr, however 439 of these had relative intensity values which were negative. We therefore did not use this higher pressure measurement at all.
In Table 4 we compare the numbers of lines identified in each measurement to the numbers of lines computed using TROVE. The latter numbers are the subject of the following selection criteria: , intensity cutoff, cm/molecule, and the wavenumber window, 0–4000 cm. Experimental lines with negative relative intensities were left out of the analysis.
To normalize the experimental intensities, the experimental relative data from each spectral window and each different pressure were scaled to match the theoretical values computed at = 298.15 K. The scaling factors obtained through a minimization procedure using all selected experimental lines with nonzero intensity (see Table 5) are 6.571 cm/molecule (405–708 cm, 0.409 Torr), 1.838 cm/molecule (405–708 cm, 2.04 Torr), 4.823 cm/molecule ( 1200–1680 cm, 0.16 Torr), and 1.328 cm/molecule (25003280 cm, 4.99 Torr). With these factors the ‘experimental’ intensities match the theoretical values reasonably good, for example for 405–708 cm the agreement is within about cm/molecule and cm/molecule, at 0.409 and 2.04 Torr, respectively.
Having the absolute intensities derived, band intensities were estimated for each experimental band as the sum of individual line intensities. In Table 5 these ‘experimental’ band intensities (cm/molecule) are compared to the theoretical values obtain by summing intensities (a) from all TROVE lines from a given window and (b) only from lines with experimental counterparts present. This was done separately for each spectral range, and each measurement pressure therein. In Table 5 these quantities are referenced to as and for the ‘total’ and ‘reduced’ band intensities, respectively, and compared to . The ratio to also shown in Table 5 demonstrates the good quality of the procedure employed as well as of our dipole moment. For example at 0.409 Torr, the differences between ‘experimental’ and theoretical band intensities are within about 20% for all bands from the 405–708 cm region with the exception of  (see also discussion below). It should be stressed here that only one scaling factor for all eight bands from this window was applied at a given pressure. The difference between two theoretical band intensities and gives a measure of the missing experimental transition data. According to Table 5 even stronger bands miss more that 50% of the total intensity.
In Table 5 we also show theoretical values of vibrational transition moments defined as
(13) 
where
(14) 
and and are the vibrational eigenfunctions of the ‘initial’ and ‘final’ states, respectively, variationally computed using TROVE and is the component of the molecular dipole moment along the molecularfixed axis .
Figure 2 presents an overview of the simulated spectrum ( = 298.15 K) with TROVE and experimental absorption spectra of SO for the whole simulation range up to 4000 cm. It reveals the gaps and limitations of the available experimental data. Our intensities based on the ab initio DMS are in very good qualitative agreement with experiment. Figure 3 shows the ‘forbidden’ rotational band as a stick spectrum. It should be noted that the microwave measurements from Ref. Meyer et al. (1991) do not have any intensities reported. In Figure 4 all eight bands from the 405–708 cm region are combined into one graph for each pressure to illustrate the quality of the corresponding experimental data. This figure suggests that the 0.409 Torr data are generally more reliable. This is also reflected by the ratio values from Table 5, which are significantly closer to 1 at the lower pressure. Of the data present for two conditions we therefore place preference on scaled intensity values obtained at the lower pressure. Finally, Figure 5 presents a detailed comparison for the all bands from the three spectral regions studied in this work in the form of stick diagrams.
Band  Obs.  Calc.  Pressure  Pressure  TROVE 
  497.57  498.48  773  1265  5422 
  530.09  528.59  996  2052  12195 
  534.83  537.16  0  69  15147 
+   530.33  528.87  84  571  12477 
2   497.45  496.88  112  704  7171 
+   497.81  498.76  47  602  27182 
2  530.36  528.79  116  775  31096 
2  529.72  527.91  39  455  13718 
  1391.52  1387.45  2014  –  14441 
2  2777.87  2770.29  1527  –  18115 
  –  –  25  –  3439 

Bands within the 405–708 cm window, measured at 2.04 Torr (Maki et al. Maki et al. (2001)).
Band  
2.987  1.537  3.705  2.915  1.559  0.98  1.01  773  1265  0.158  
4.258  3.112  5.949  4.310  3.149  1.01  1.01  995  2052  0.200  
  0.116  0.470  0.661  0.101  0.411  0.88  0.88  112  704  0.221 
  0.062  0.322  0.528  0.052  0.251  0.84  0.78  84  571  0.199 
  0.026  0.260  0.581  0.022  0.215  0.84  0.83  47  602  0.223 
( = 2)   0.112  0.589  0.873  0.102  0.485  0.91  0.82  116  769  0.283 
( = 0)   0.026  0.222  0.405  0.015  0.179  0.57  0.81  38  454  0.196 
  –  0.009  0.101  –  0.003  –  0.29  –  69  0.039 
39.490  –  44.440  39.490  –  –  –  2014  –  0.321  
0.093  –  0.119  0.093  –  –  –  1527  –  0.012 
Iv Conclusions and discussions
The misplaced theoretical bands in Figures 4 and 5 indicate that our ab initio PES of SO requires improvement (see also band centers in Table 4). The theoretical frequencies have an rms deviation of 0.91 cm when compared with the experimental data, which is relatively small compared to the deviation for of 4.07 cm. This is to be expected since our PES is purely ab initio computed at a modest level of theory. We are planning to refine this surface by fitting to all experimental data available.
Table 5 outlines the quality of the intensity scaling procedure, in which the relative values of the experimental intensities where converted to absolute values (cm/molecule). For the 0.409 Torr and 2.04 Torr measurements our comparisons mostly agree to within 20%, with the exception of the  band measured at 0.409 Torr which shows nearly a 50% difference, and the  band measured at 2.04 Torr with 80% uncertainty. The latter can probably be attributed to both the small number of lines in each case, and residual errors in the transition dipole. In the case of the  band the comparison with the 2.04 Torr experiment yields a better value for / than for the 0.409 Torr measurement, which suggests that the number of lines available at 0.409 Torr is too low (see also Figure 4) The significance of the results presented in Table 5 and illustrated in Figure 4 is that these give an estimation on the quality of the ab initio dipole moment surface as well as of the experimental data. Based these numbers we can place a lower estimate on the quality uncertainty for our intensities for each band, for example the experiment and theory for the and bands agree at least to within 3% for both pressure measurements, while it is only 13% for  , and between 17% 18% for the remaining bands.
Our complete roomtemperature line list for SO containing 349 348 513 transitions can be accessed online at www.exomol.com in the ExoMol format described in Tennyson et al. (2013). It includes the transition energies, Einstein coefficients , and absorption intensities estimated for = 298.15 K. Additionally, a list of 10 878 experimental transitions with absolute intensities obtained for = 296 K is included into the supplementary part of this paper in a form suitable for standard atmospheric and planetary spectroscopic databases.
Our future work will be focused on the development of a hot line list for SO for high temperature industrial applications as well as for modelling molecular opacity in atmospheres of (exo)planets and cool stars as part of the ExoMol project Tennyson and Yurchenko (2012) (see www.exomol.com).
Acknowledgment
We thank Alexander Fateev for stimulating our interest in this molecule and for many helpful discussions, and Jeff Barber for supplying his experimental result. This work was supported by grant 10442 from energinet.dk under a subcontract from the Danish Technical University and and the ERC under Advanced Investigator Project 267219.
V References
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