TY - JOUR
AU - Tennyson,, Jonathan
AB - Abstract Comprehensive line lists for phosphorus monoxide (31P16O) and phosphorus monosulphide (31P32S) in their X2Π electronic ground state are presented. The line lists are based on new ab initio potential energy (PEC), spin–orbit (SOC) and dipole moment (DMC) curves computed using the MRCI+Q−r method with aug-cc-pwCV5Z and aug-cc-pV5Z basis sets. The nuclear motion equations (i.e. the rovibronic Schrödinger equations for each molecule) are solved using the program duo. The PECs and SOCs are refined in least-squares fits to available experimental data. Partition functions, Q(T), are computed up to T = 5000 K, the range of validity of the line lists. These line lists are the most comprehensive available for either molecule. The characteristically sharp peak of the Q-branches from the spin–orbit split components gives useful diagnostics for both PO and PS in spectra at infrared wavelengths. These line lists should prove useful for analysing observations and setting up models of environments such as brown dwarfs, low-mass stars, O-rich circumstellar regions and potentially for exoplanetary retrievals. Since PS is yet to be detected in space, the role of the two lowest excited electronic states (a4Π and B2Π) are also considered. An approximate line list for the PS X–B electronic transition, which predicts a number of sharp vibrational bands in the near ultraviolet, is also presented. The line lists are available from the CDS (http://cdsarc.u-strasbg.fr) and ExoMol (www.exomol.com) data bases. molecular data – opacity, astronomical data bases: miscellaneous, planets and satellites: atmospheres, stars: AGB and post-AGB, stars: atmospheres 1 INTRODUCTION Several phosphorus-containing molecules have been discovered around evolved stars including PN, HCP, CP and PO (Tenenbaum, Woolf & Ziurys 2007; Milam et al. 2008; De Beck et al. 2013) and PH3 has been detected in the circumstellar envelope of IRC+10216 (Agundez et al. 2014). Other phosphorous-containing species including PS are yet to be detected (De Beck et al. 2013). Models suggest that a variety of phosphorous-bearing species become important in the atmospheres of low-mass stars, brown dwarfs and giant exoplanets at elevated temperatures (Visscher, Lodders & Fegley 2006). PN has been observed in the 3 mm region towards the low-mass star-forming region of L1157 (Yamaguchi et al. 2011, 2012). Additionally, being a primal biogenic element found in all living systems, phosphorus is hence essential to life on Earth (Maciá, Hernández & Oró 1997). Phosphorus is present in nucleic acids, several proteins, and is a fundamental component of the adenosine triphosphate molecule, which is accountable for energy transfer in cells. P-containing molecules are thought to provide important biomarkers in the early Earth (Li, Sun & Chan 2013) and such molecules could play a similar role in exoplanets. In addition, the atmospheres of the recently characterized hot rocky planets, or lava planets, are likely to contain a whole range of unusual small molecules (Tennyson & Yurchenko 2017). In this work, performed as part of the ExoMol project (Tennyson & Yurchenko 2012), we concentrate on providing comprehensive line lists for the two open shell diatomic species: PO and PS. Oxygen and sulphur belong to the same group in the periodic table and, as a result, PO and PS have similar electronic structures (e.g. both have a X2Π ground electronic state) and their spectra show many analogies. After a number of failed attempts (Matthews, Feldman & Bernath 1987; Dimur et al. 2001; MacKay & Charnley 2001), phosphorus monoxide, PO, has been detected in a number of locations in space. The original detection, by Tenenbaum et al. (2007), was in the oxygen-rich, red Supergiant Star VY Canis Majoris and used microwave emissions near 240 and 284 GHz (7.2 and 8.5 cm−1). Subsequently PO has been observed in the wind of the oxygen-rich AGB star IK Tauri (De Beck et al. 2013) and in star-forming regions (Lefloch et al. 2016; Rivilla et al. 2016). In a number of these locations, PO appears to occur with similar abundance to the closed shell molecule PN for which an ExoMol line list has already been constructed (Yorke et al. 2014). As of yet there are no observations of phosphorus monosulphide, PS, in space (De Beck et al. 2013). A systematic attempt at its astronomical detection was performed by Ohishi et al. (1988) using the 45m telescope of the Nobeyama Radio Observatory and six distinct objects. Local thermodynamic equilibrium calculations by Tsuji (1973) indicate that PS should be the major P-bearing molecule in oxygen-rich circumstellar envelopes for temperatures below 2000 K. Available line lists for both PO and PS appear to be extremely limited. Long-wavelength transition frequencies are available for both species in the JPL data base (Pickett et al. 1998), but the transition intensities are all based on assumed or estimated values for the permanent dipole moments. A more up-to-date long-wavelength line list for PO is given by the Cologne Database for Molecular Spectroscopy (CDMS) (Müller et al. 2005). There have been several studies that aimed to obtain line frequencies and spectroscopic constants of PO from experimental and theoretical analyses of its spectrum (Ghosh & Verma 1978; Rao, Reddy & Rao 1981; Butler, Kawaguchi & Hirota 1983; Kanata, Yamamoto & Saito 1988; Qian 1995; de Brouckere 1999; Spielfiedel & Handy 1999; Bailleux et al. 2002; Metropoulos, Papakondylis & Mavridis 2003; Moussaoui, Ouamerali & De Mare 2003; Sun, Wang & Shi 2012; Liu et al. 2013). A review of the experimental work on PO prior to 1999 is given by de Brouckere (1999), while Liu et al. (2013) provide a more recent summary of ab initio studies. In addition to work on the spectrum of PO in its X2Π ground electronic state, there have also been extensive experimental studies of its excited electronic states. Early work on observed transitions is summarized by Huber & Herzberg (1979); Huber and Herzberg list in their compilation for PO 13 electronic states up to about 56 000 cm−1 (six 2Σ+ states, one 2Σ− state, three 2Π states, two 2Δ state and one 4Σ− state). Additional excited electronic states, including more quartet and sextet states, have been considered theoretically (Kanata et al. 1988; de Brouckere 1999, 2000; Spielfiedel & Handy 1999; Metropoulos et al. 2003; Moussaoui et al. 2003; Sun et al. 2012; Liu et al. 2013). PS was observed in the laboratory for the first time by Dressler & Miescher (1955) who detected two band systems corresponding to the C2Σ → X2Π and B2Π → X2Π electronic transitions, with wavelength ranges 2700–3100 and 4200–6000 Å, respectively. Since then a limited set of experiments on PS have followed (Narasimham & Subramanian 1969; Narasimham & Balasubramanian 1971; Jenouvrier & Pascat 1978; Balasubramanian, Dixit & Narasimham 1979; Lin, Balling & Wright 1987; Kawaguchi et al. 1988; Ohishi et al. 1988; Wang et al. 1993) with the most recent being the study of a submillimetre-wave rotational spectrum by Klein, Klisch & Winnewisser (1999). Several of these studies are considered further below. There have also been various theoretical studies conducted on the ground and electronic states of PS (Bruna & Grein 1987; Karna, Bruna & Grein 1988; Karna & Grein 1992; Moussaoui, Ouamerali & De Mare 1998; Kalcher 2002; Ben Yaghlane, Francisco & Hochlaf 2012), including a Multi-Reference Configuration Interaction (MRCI) study of the lowest 16 molecular terms by Ben Yaghlane et al. (2012). The aim of this work is to produce molecular line lists for 31P16O and 31P32S applicable for a large range of temperatures. 2 METHOD In the theoretical approach adopted by our group (Lodi & Tennyson 2010) the computation of a line list for the molecule of interest requires constructing potential energy curves (PECs), dipole moment curves (DMCs), spin–orbit couplings (SOCs) and, if necessary, other couplings such as angular momentum, spin–spin and spin–rotation (Tennyson et al. 2016b). These are then used to solve the relative nuclear motion Schrödinger equation, thus producing frequencies and intensities for the transitions of interest. 2.1 Ab initio electronic structure data 2.1.1 PO While there have been several ab initio studies of PO's many electronic states which yielded total energies and spectroscopic constants, none of them supply the PECs, SOCs and DMCs required for the construction of a line list. Ab initio curves were therefore computed using molpro (Werner et al. 2012). The chosen technique was MRCI+Q−r (internally contracted multireference configuration interaction with renormalized Davidson correction; the ‘relaxed reference’ energy was used) with the aug-cc-wCV5Z basis set. The calculation also included a relativistic correction curve computed as the expectation value of the mass–velocity plus one-electron Darwin operator (MVD1). The ab initio PEC was computed up to a nuclear separation of 4.5 Å and can be seen in Fig. 1. Figure 1. Open in new tabDownload slide Comparison of ab initio (MRCI+Q−r/aug-cc-pwCV5Z) and refined PECs of PO. Figure 1. Open in new tabDownload slide Comparison of ab initio (MRCI+Q−r/aug-cc-pwCV5Z) and refined PECs of PO. The ab initio SOC was obtained using the same level of theory. Fig. 2 shows the ab initio SO curve, as well as the refined curve (see below). The equilibrium ab initio SO value (106.4 cm−1) is in reasonable agreement with the empirical SO constant (ASO/2 = 112.1 cm−1) from Butler et al. (1983) and with the ab initio value 112.6 cm−1 of Liu et al. (2013). Our refined equilibrium SO value is 112.1 cm−1. Figure 2. Open in new tabDownload slide Comparison of PO spin–orbit curves. Solid line: ab initio curve; red dashed line: empirical curve produced using the morphing approach. Figure 2. Open in new tabDownload slide Comparison of PO spin–orbit curves. Solid line: ab initio curve; red dashed line: empirical curve produced using the morphing approach. Our value of the equilibrium bond length re = 1.482 Å obtained from the PEC of the X2Π state is in reasonable agreement with the most recent experimentally determined value re = 1.475 637 355(10) Å (Bailleux et al. 2002). The dissociation value De = 48 997.9 cm−1 also compares well with the value 48 980 cm−1 (6.073 eV) estimated by Rao et al. (1981). This ab initio PEC thus provides a suitable starting point for empirical refinement of the X2Π state. At the start of this work there was no DMC for the X2Π state of PO available in the literature with instead the majority of ab initio studies focusing on the myriad of low-lying PECs of PO. Moussaoui et al. (2003) computed a value for the dipole at equilibrium and recently Andreazza, de Almeida & Borin (2016) computed curves as part of their study on the formation of PO by radiative association. The experimental data are limited to the μ0 value by Rao et al. (1981). Therefore, it was decided to use the highest level of ab initio theory in this work to produce a suitable DMC for the line list calculation. Similar to the final ab initio PEC, the ab initio DMC was calculated using the MRCI+Q−r/aug-cc-pwCV5Z; it was calculated as the derivative of the MRCI+Q−r energy with respect to an external electric field along the internuclear axis for vanishing field strength (Lodi & Tennyson 2010). The dipole moment generated for the X2Π state of PO is shown in Fig. 3. Its value at the equilibrium bond length is 1.998 D, which is in reasonable agreement with 1.88±0.07 D (Kanata et al. 1988) (also adopted by CDMS; Müller et al. 2005) and therefore provides an adequate choice in the final calculation of the line list. These values are much larger than the value of 1.0 D assumed in the JPL line list (Pickett et al. 1998). Figure 3. Open in new tabDownload slide Ab initio DMC for the ground state of PO. Figure 3. Open in new tabDownload slide Ab initio DMC for the ground state of PO. 2.1.2 PS Ab initio curves for the ground (X2Π) and three excited (a4Π, B2Π and 4Σ−) electronic states of PS were generated using molpro and the MRCI+Q−r method with the aug-cc-pwCV5Z basis sets used for the ground state and aug-cc-pV5Z basis sets used for the excited states. The ab initio PEC of the a4Π state and the refined PECs for the X2Π and B2Π states considered in this paper are shown in Fig. 4. The dissociation limit obtained by our calculation for the X2Π electronic state appears to be close to the estimated De = 36 600 cm−1 (438±10 kJ mol−1) of Drowart et al. (1973), although we did not perform calculation at large enough bond lengths to quote an accurate value. Our equilibrium dipole moment of the X state is 0.523 D, which can be compared to the complete basis set extrapolation value by Muller & Woon (2013) of 0.565 D. Figure 4. Open in new tabDownload slide Refined PECs for the X2Π, B2Π electronic states and the ab initio PEC for the a4Π electronic state of PS. Figure 4. Open in new tabDownload slide Refined PECs for the X2Π, B2Π electronic states and the ab initio PEC for the a4Π electronic state of PS. The ab initio DMCs for the ground X2Π electronic state, excited a4Π electronic state and for the X2Π–B2Π electronic transition considered in this work for PS are shown in Fig. 5 . In order to reduce the numerical noise when computing the line-strengths using the duo program, we followed the recommendation of Medvedev et al. (2016) and represented analytically the ab initio DMCs (denoted by μ(r)). The following expansion with a damped-coordinate was employed: \begin{equation} \mu (r) = (1-\xi ) \sum _{n \ge 0} d_n z^n + d_{\infty } \, \xi , \end{equation} (1) where ξ is the Šurkus variable (Šurkus, Rakauskas & Bolotin 1984) \begin{equation} \xi =\frac{r^{p}-r^{p}_{\mathrm{ref}}}{r^{p}+r^{p}_{\rm ref }} \end{equation} (2) and z is given by \begin{equation*} z = (r-r_{\rm ref})\, \text{e}^{-\beta _2 (r-r_{\rm ref})^2-\beta _4 (r - r_{\rm ref})^4}. \end{equation*} Here p is an empirical parameter, rref is a reference position equal to re by default, dn are the expansion parameters, d∞ is the value of the dipole at r → ∞, and β2 and β4 are damping factors. These parameters defining the dipole moment expansion for the three ab initio DMCs considered in this work for PS are given in the supplementary material as a duo input file, while the functional form is now a part of the duo program. Figure 5. Open in new tabDownload slide Ab initio DMCs of PS which were represented analytically. Figure 5. Open in new tabDownload slide Ab initio DMCs of PS which were represented analytically. The SOCs were computed using the aug-cc-pVDZ basis set (valence only calculations) to speed up their calculation [SOCs are not expected to be very sensitive to the level of theory used, see Patrascu et al. (2014)] and are shown in Fig. 6. Our equilibrium SO ab initio value is 141.4 cm−1 and after refinement is 160.9 cm−1. The analogous experimental effective SO constant ASO/2 was determined to be 161.0 cm−1 by Jenouvrier & Pascat (1978). Figure 6. Open in new tabDownload slide Spin–orbit curves of PS, ab initio (solid) and refined (dashed). Figure 6. Open in new tabDownload slide Spin–orbit curves of PS, ab initio (solid) and refined (dashed). 2.2 Nuclear motion calculations The nuclear motion calculations were performed using the code duo (Yurchenko et al. 2016), which provides a variational solution to the nuclear motion problem and can account for virtually any coupling between the PECs of the molecule under study. A review of the theory on which duo is based can be found in Tennyson et al. (2016b). The necessary curves can be computed using ab initio electronic methods or by fitting to experimental data. Our general strategy (Tennyson 2012; Tennyson & Yurchenko 2017), which is followed here, is to use spectroscopically determined PECs and couplings, since these cannot be computed with sufficient accuracy by ab initio methods (McKemmish, Yurchenko & Tennyson 2016). Conversely, experience has shown that ab initio DMCs give results that are very reliable (Tennyson 2014) and can give intensities that are competitive in accuracy with the most precise laboratory measurements (Polyansky et al. 2015). In general, the nuclear motion problem can be solved with sufficient accuracy that the quality of the underlying ab initio curves is the main source of error in the calculations. 2.3 Refinement using experimental data 2.3.1 PO Table 1 lists the sources used in refining the PO X2Π PEC. Most of the 241 lines used correspond to pure rotational transitions observed in microwave and far-infrared studies, including hot bands Δv = 0 and v΄ ≤ 7. The only source of IR data is the laser spectroscopy study by Qian (1995) where 50 fundamental transitions (J ≤ 21.5) were reported. Butler et al. (1983) and Bailleux et al. (2002) resolved the hyperfine structure within each observed rotational transition. As hyperfine splitting is beyond the scope of this work, frequencies resulting from such transitions were averaged over for each rotational transition. Table 1. Experimental sources used in the empirical refinement of the PO PEC. Source Method Transitions lines Range (cm−1) Uncertainty (cm−1) Bailleux et al. (2002) Millimetre wave Δv = 0, v = 0 : 7, J ≤ 10.5 167 4.98–15.39 0.01 Qian (1995) Microwave, IR v = 1 ← 0, J ≤ 21.5, 46 1188.12–1245.12 0.0005 Butler et al. (1983) Mid-IR Δv = 1, v = 0 − 1, J ≤ 25.5, 28 2.1226–1254 0.0005 Verma & Singhal (1975) UV Tv for v = 0 : 11 Ev = 0–12 700.05 0.02 Source Method Transitions lines Range (cm−1) Uncertainty (cm−1) Bailleux et al. (2002) Millimetre wave Δv = 0, v = 0 : 7, J ≤ 10.5 167 4.98–15.39 0.01 Qian (1995) Microwave, IR v = 1 ← 0, J ≤ 21.5, 46 1188.12–1245.12 0.0005 Butler et al. (1983) Mid-IR Δv = 1, v = 0 − 1, J ≤ 25.5, 28 2.1226–1254 0.0005 Verma & Singhal (1975) UV Tv for v = 0 : 11 Ev = 0–12 700.05 0.02 Open in new tab Table 1. Experimental sources used in the empirical refinement of the PO PEC. Source Method Transitions lines Range (cm−1) Uncertainty (cm−1) Bailleux et al. (2002) Millimetre wave Δv = 0, v = 0 : 7, J ≤ 10.5 167 4.98–15.39 0.01 Qian (1995) Microwave, IR v = 1 ← 0, J ≤ 21.5, 46 1188.12–1245.12 0.0005 Butler et al. (1983) Mid-IR Δv = 1, v = 0 − 1, J ≤ 25.5, 28 2.1226–1254 0.0005 Verma & Singhal (1975) UV Tv for v = 0 : 11 Ev = 0–12 700.05 0.02 Source Method Transitions lines Range (cm−1) Uncertainty (cm−1) Bailleux et al. (2002) Millimetre wave Δv = 0, v = 0 : 7, J ≤ 10.5 167 4.98–15.39 0.01 Qian (1995) Microwave, IR v = 1 ← 0, J ≤ 21.5, 46 1188.12–1245.12 0.0005 Butler et al. (1983) Mid-IR Δv = 1, v = 0 − 1, J ≤ 25.5, 28 2.1226–1254 0.0005 Verma & Singhal (1975) UV Tv for v = 0 : 11 Ev = 0–12 700.05 0.02 Open in new tab We also included lower accuracy experimentally derived vibrational energies by Verma & Singhal (1975) with Tv with v ≤ 11. The availability of these vibrational terms values provided an important constraint for extrapolation of vibrational levels higher than v = 1. It is easier to start the duo refinement by fitting to energies, not transition frequencies. To this end, an ‘experimentally derived’ set of energy levels of PO was produced using the program PGOPHER (Western 2017) in conjunction with the experimentally derived spectroscopic constants for the ground (v = 0) and first excited (v = 1) vibrational states, as determined in Qian (1995). These energies (up to J = 26.5) were combined with the vibrational energies by Verma & Singhal (1975). 2.3.2 PS There is little laboratory data available on the X2Π state of PS. Dressler & Miescher (1955), Narasimham & Subramanian (1969), Narasimham & Balasubramanian (1971) and Balasubramanian et al. (1979) reported vibronic heads only and did not provide any information that can be used to refine this state. Kawaguchi et al. (1988), Ohishi et al. (1988) and Klein et al. (1999) give millimetre wave spectra that provide information on the rotational levels and spin–orbit splitting between the states. Ohishi et al. (1988) and Klein et al. (1999) provide hyperfine-resolved transition frequencies that were unresolved by averaging the frequencies of matching transitions with the same e/f, J΄, J″ and Ω values, where Ω is the projection of the total angular momentum. The e/f parity was not provided for |$\Omega =\frac{3}{2}$| in Ohishi et al. (1988) and Kawaguchi et al. (1988); it was assumed to be unresolved and were duplicated for later use. Kawaguchi et al. (1988) provides data on the vibrational fundamental. In addition, we used the program pgopher to derive a set of PS energy levels from the spectroscopic constants reported in the experimental paper by Klein et al. (1999). This gave a total of 316 energies for J ranging from 0.5 to 39.5 split between the v = 0 and v = 1 state. For the vibrationally excited states, we added the vibrational energies reconstructed from the vibrational Δv = 1 separations reported by Jenouvrier & Pascat (1978) from analysis of the B2Π–X2Π system. The energies coincided with consecutive vibrational transitions from v = 0 to 9. To make all data internally consistent, the experimental e/f parities for the lower state were converted into the +/− parities (Brown et al. 1975) using the standard relations: \begin{equation} \mathrm{e:\, }(-1)^{\mathrm{\text{}J-\frac{1}{2}}} \end{equation} (3) and \begin{equation} \mathrm{f:\, }(-1)^{\mathrm{\text{}J+\frac{1}{2}}}. \end{equation} (4) The selection rule +↔− was used to determine the parity for the upper state. The frequency and quantum numbers were repeated for both + and − whenever the parities were unavailable in the experimental data (all in the case of the |$^{2}\Pi _{\frac{3}{2}}$| sub-state). Σ΄ and Σ″ values (projections of the electronic spin on the molecular axis) were derived from |Ω| usually provided in experimental literature as |Ω| = |Λ ± Σ΄| and by matching the corresponding parity. Here Λ = ±1 is the projection of the electronic angular momentum and Ω is the projection of the total angular momentum on the molecular axis. 2.4 Fitting with duo duo offers a range of analytical functions for modelling PECs. Owing to its previous success in producing accurate PECs, it was decided to use the Extended Morse Oscillator (EMO) function (Lee et al. 1999) to obtain the final X2Π PECs for PO and PS denoted by V(r). The function is written \begin{equation} V(r)=D_{\rm e}\left[1-\exp \left(-{\sum _{k=0}^{N}}\beta _{k}\xi ^{k}(r-r_{\rm e})\right)\right]^{2}, \end{equation} (5) where ξ is the Šurkus variable, see equation (2). Here βk is an empirical parameter whose value (along with the parameter p) can be derived through refinement to experimental data. It is important to note that V(+∞) = De, as long as both p and ∑kβk are set greater than zero. duo also allows the SOC to be refined simultaneously with the PEC. An ab initio curve Fai(r) can be scaled using the morphing approach so that the empirical curve, F(r), is given by \begin{equation} F(r)=H(r)F_{\rm ai}(r) \end{equation} (6) with \begin{equation} H(r)=\left[(\mathrm{1}-\xi )\, {\sum _{k=0}^{N}}\, \beta _{k}\, \xi ^{k}+\xi \, t_{\infty }\right]\!, \end{equation} (7) where H(r) is the morphing function in terms of the Šurkus variable (ξ), t∞ is the value of the morphing function as r → ∞ and βk is the morphing expansion coefficient (Yurchenko et al. 2016). The morphing approach was used to refine the ab initio SOCs. As there were no available ab initio curves for spin–rotation (SR) and Λ-doubling effects, the functional form H(r) was applied to both couplings directly. The final duo input files for both PO and PS are given as part of the supplementary material. These files contain the various curves as well as the parameters used to run duo. 2.4.1 PO Experimental values of PO for dissociation De (Rao et al. 1981) and equilibrium bond length re (Bailleux et al. 2002) were held fixed until the final stages of the refinement process. The first four βk expansion coefficients from equation (5) were varied by fitting to experimentally derived energies of PO obtained using pgopher until a satisfactory fit was achieved. At this point the empirical energies were replaced by the actual, measured frequencies for comparison and further refinement. As the reference frequencies only include pure rotational and fundamental absorption lines, to increase constraints on the higher vibrational transitions, the differences between the higher vibrational energies (Verma & Singhal 1975) relative to the ground level (v = 0) were retained. At the final stage of refinement, two additional terms were introduced to account for spin-rotation and any additional Λ-doubling effects to further minimize Obs. − Calc. The resulting empirical PEC and SOC are shown in Figs 1 and 2, respectively: these are plotted with the respective ab initio curves for comparison. The final parameters as well as the corresponding curves are given as part of the duo input files in supplementary data. The accuracy of the fit ranges from 0.001 cm−1 for purely rotational transitions up to 0.05 cm−1 for vibrational transitions, producing a root mean square (RMS) error of 0.014 cm−1. The residuals are illustrated in Fig. 7, where the vibronic bands are indicated. Table 2 illustrates the Obs. − Calc. residuals for the vibrational excitations of PO and PS. In both cases the empirical term values are of limited accuracy (Verma & Singhal 1975; Jenouvrier & Pascat 1978), especially for the PS values from Jenouvrier & Pascat (1978). While the target accuracy of the PEC was achieved for low vibrational levels, having access to more experimental data for higher levels would help to verify the accuracy of extrapolated energy levels in the line list. Figure 7. Open in new tabDownload slide Observed minus Calculated (cm−1) PO and PS energy term levels and line positions after refinement. The discontinuity in the PS comparison is probably because the CDMS data switch from three to two hyperfine components at J = 36.5, which we average to one hyperfine-free value. Figure 7. Open in new tabDownload slide Observed minus Calculated (cm−1) PO and PS energy term levels and line positions after refinement. The discontinuity in the PS comparison is probably because the CDMS data switch from three to two hyperfine components at J = 36.5, which we average to one hyperfine-free value. Table 2. Obs. − Calc. residuals for PO and PS vibrational term values in cm−1. The experimentally derived energies of PO and PS are from Verma & Singhal (1975) and Jenouvrier & Pascat (1978), respectively. PO PS v Obs. Calc. Obs. − Calc Obs. Calc. Obs. − Calc 1 1220.161 1220.216 −0.055 1461.7 1461.4 0.3 2 2427.31 2427.291 0.019 2183.4 2183.2 0.2 3 3621.29 3621.242 0.04 2899.0 2899.0 0.0 4 4802.12 4802.078 0.04 3608.9 3608.8 0.1 5 5969.84 5969.799 0.04 4312.3 4312.6 −0.3 6 7124.41 7124.398 0.01 5010.2 5010.2 0.0 7 8265.85 8265.865 −0.01 5701.8 5701.7 0.1 8 9394.14 9394.184 −0.04 9 10509.30 10509.33 −0.03 10 11611.30 11611.29 0.01 11 12700.05 12700.03 0.02 PO PS v Obs. Calc. Obs. − Calc Obs. Calc. Obs. − Calc 1 1220.161 1220.216 −0.055 1461.7 1461.4 0.3 2 2427.31 2427.291 0.019 2183.4 2183.2 0.2 3 3621.29 3621.242 0.04 2899.0 2899.0 0.0 4 4802.12 4802.078 0.04 3608.9 3608.8 0.1 5 5969.84 5969.799 0.04 4312.3 4312.6 −0.3 6 7124.41 7124.398 0.01 5010.2 5010.2 0.0 7 8265.85 8265.865 −0.01 5701.8 5701.7 0.1 8 9394.14 9394.184 −0.04 9 10509.30 10509.33 −0.03 10 11611.30 11611.29 0.01 11 12700.05 12700.03 0.02 Open in new tab Table 2. Obs. − Calc. residuals for PO and PS vibrational term values in cm−1. The experimentally derived energies of PO and PS are from Verma & Singhal (1975) and Jenouvrier & Pascat (1978), respectively. PO PS v Obs. Calc. Obs. − Calc Obs. Calc. Obs. − Calc 1 1220.161 1220.216 −0.055 1461.7 1461.4 0.3 2 2427.31 2427.291 0.019 2183.4 2183.2 0.2 3 3621.29 3621.242 0.04 2899.0 2899.0 0.0 4 4802.12 4802.078 0.04 3608.9 3608.8 0.1 5 5969.84 5969.799 0.04 4312.3 4312.6 −0.3 6 7124.41 7124.398 0.01 5010.2 5010.2 0.0 7 8265.85 8265.865 −0.01 5701.8 5701.7 0.1 8 9394.14 9394.184 −0.04 9 10509.30 10509.33 −0.03 10 11611.30 11611.29 0.01 11 12700.05 12700.03 0.02 PO PS v Obs. Calc. Obs. − Calc Obs. Calc. Obs. − Calc 1 1220.161 1220.216 −0.055 1461.7 1461.4 0.3 2 2427.31 2427.291 0.019 2183.4 2183.2 0.2 3 3621.29 3621.242 0.04 2899.0 2899.0 0.0 4 4802.12 4802.078 0.04 3608.9 3608.8 0.1 5 5969.84 5969.799 0.04 4312.3 4312.6 −0.3 6 7124.41 7124.398 0.01 5010.2 5010.2 0.0 7 8265.85 8265.865 −0.01 5701.8 5701.7 0.1 8 9394.14 9394.184 −0.04 9 10509.30 10509.33 −0.03 10 11611.30 11611.29 0.01 11 12700.05 12700.03 0.02 Open in new tab 2.4.2 PS Again the PEC was fitted using an EMO. Values of re and De were kept fixed to their spectroscopic values, re = 1.89775 Å and De = 37004 cm−1. The ab initio PEC and SOC curves for the ground state of PS were used as the starting point in the fits. The experimental set for the fits comprised of the v = 0–9 energies from Jenouvrier & Pascat (1978), and the reconstructed energies from pgopher for J up to 39.5. Although the experimental data was limited, duo is able to extrapolate energies to higher values for the v and J quantum numbers. The empirical PEC is compared with the ab initio one in Fig. 1. Morphing was used to refine the SOC. The experimental value for re was used as the reference expansion point. t∞ represents the asymptote for the morphing function (r → ∞), and it equals unity in this case (Patrascu et al. 2014). Two morphing expansion parameters, A0 and A1, were included in the fit because the Obs. − Calc. was reduced even further; however, no significant changes were made when fitting to the expansion parameters for n > 1. Residues of the fit (Obs. − Calc.) for both molecules are plotted in Fig. 7 for all the data used in the fitting except for the less accurate vibrational energies of Verma & Singhal (1975) and Jenouvrier & Pascat (1978), which are collected in Table 2. The residuals build distinct vibronic v, Ω patterns and diverge somewhat at higher J indicating a deficiency in our model. One of the possible sources of error is the Λ-factor for Π states, which, when properly modelled, should originate from the electronic angular momentum coupling to Σ states (Brown & Merer 1979). The majority of the points are situated near the zero line. The RMS error of the fit was 0.026 cm−1 for all 316 experimental energies and 0.006 cm−1 excluding the lower accuracy vibrational data of Jenouvrier & Pascat (1978). For PS we have decided to include the two lowest excited electronic states a4Π and B2Π by using the corresponding ab initio MRCI+Q−r/aug-cc-pV5Z curves. In order not to destroy the accuracy of the refined model of the ground electronic state of PS, we omitted the couplings between these two states with the X state. The B2Π state PEC was refined by fitting to J = 0.5 and J = 1.5 rovibronic energies of this state, which we derived using parameters (Tv, re and Bv) of Jenouvrier & Pascat (1978) (RMS error is 0.41 cm−1). 3 LINE LIST CALCULATIONS 3.1 PO (X) and PS (X, B, a) line lists Line lists generated using duo are comprised of a states file and a transitions file, with extensions .states and .trans, respectively (Tennyson et al. 2016c). The .states file includes the running number n, energy term values (cm−1), total statistical weight, lifetime (Tennyson et al. 2016a), g-Landé factors (Semenov, Yurchenko & Tennyson 2017) and corresponding quantum numbers. The .trans file contains running numbers for the upper and lower levels, as well as the Einstein-A coefficients (Yurchenko et al. 2016). The maximum vibrational and rotational quantum numbers, vmax and Jmax are identified using the dissociation energy, D0. These numbers are given for both PO and PS in Table 3 . The duo integration range was chosen as r = [0.7, 4.0] Å for PO and r = [1.2, 4.0] Å for PS, and the respective grids comprised 501 points in conjunction with the Discrete Variable Representation (DVR) sinc method. The PO line list includes the electronic state X2Π, while the PS line list consists of transitions between the lowest three electronic states of X2Π, B2Π and a4Π. To reduce the size of the line list, for PS the lower state energy threshold is reduced to 25 000 cm−1, which should cover all thermal populations much higher than 5000 K. Table 3. Statistics for line lists for PO and PS. PO PS States (vmax) X(69) X(82), B(35), a(58) Jmax 234.5 320.5 νmax (cm−1) 12 000 37 000 |$E^{\prime }_{\rm max}$| (cm−1) 49 000 37 000 |$E^{\prime \prime }_{\rm max}$| (cm−1) 37 000 25 000 Number of energies 43 148 225 997 Number of lines 2096 289 30 394 544 PO PS States (vmax) X(69) X(82), B(35), a(58) Jmax 234.5 320.5 νmax (cm−1) 12 000 37 000 |$E^{\prime }_{\rm max}$| (cm−1) 49 000 37 000 |$E^{\prime \prime }_{\rm max}$| (cm−1) 37 000 25 000 Number of energies 43 148 225 997 Number of lines 2096 289 30 394 544 Open in new tab Table 3. Statistics for line lists for PO and PS. PO PS States (vmax) X(69) X(82), B(35), a(58) Jmax 234.5 320.5 νmax (cm−1) 12 000 37 000 |$E^{\prime }_{\rm max}$| (cm−1) 49 000 37 000 |$E^{\prime \prime }_{\rm max}$| (cm−1) 37 000 25 000 Number of energies 43 148 225 997 Number of lines 2096 289 30 394 544 PO PS States (vmax) X(69) X(82), B(35), a(58) Jmax 234.5 320.5 νmax (cm−1) 12 000 37 000 |$E^{\prime }_{\rm max}$| (cm−1) 49 000 37 000 |$E^{\prime \prime }_{\rm max}$| (cm−1) 37 000 25 000 Number of energies 43 148 225 997 Number of lines 2096 289 30 394 544 Open in new tab As an additional safeguard against enhanced intensities for high overtones resulting from numerical noise, we follow the procedure used by Wong et al. (2017) and use a dipole moment threshold of 10−8 D. Lifetimes for PS molecule were computed using an extended line list covering all transitions with the lower/upper state energies below 37 000 cm−1. They are shown in Fig. 8. Figure 8. Open in new tabDownload slide Lifetimes of the three lower electronic states of PS computed using the PS line list. Figure 8. Open in new tabDownload slide Lifetimes of the three lower electronic states of PS computed using the PS line list. Table 3 summarizes the statistics for our PO and PS line lists. Extracts from the .states and .trans files for PS are given in Tables 4 and 5 , respectively. Full tables for both PO and PS are available at http://ftp://cdsarc.u-strasbg.fr/pub/cats/J/MNRAS/xxx/yy or http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/MNRAS//xxx/yy, as well as the ExoMol website, www.exomol.com. Note that the states files include lifetime for each individual state (Tennyson et al. 2016a); these can be useful for a variety of issues including determining critical densities. Table 4. Extract from the .states file for 31P32S. n |$\tilde{E}$| gtot J τ g-Landé +/ − e/f State v Λ Σ Ω 1 0.000 000 4 0.5 inf −0.000767 + e X2Pi 0 1 −0.5 0.5 2 733.657 367 4 0.5 1.2003E+00 −0.000767 + e X2Pi 1 1 −0.5 0.5 3 1461.408 539 4 0.5 6.0697E−01 −0.000767 + e X2Pi 2 1 −0.5 0.5 4 2183.215 742 4 0.5 4.0934E−01 −0.000767 + e X2Pi 3 1 −0.5 0.5 5 2899.040 733 4 0.5 3.1065E−01 −0.000767 + e X2Pi 4 1 −0.5 0.5 6 3608.844 801 4 0.5 2.5151E−01 −0.000767 + e X2Pi 5 1 −0.5 0.5 7 4312.588 793 4 0.5 2.1216E−01 −0.000767 + e X2Pi 6 1 −0.5 0.5 8 5010.233 137 4 0.5 1.8413E−01 −0.000767 + e X2Pi 7 1 −0.5 0.5 9 5701.737 858 4 0.5 1.6315E−01 −0.000767 + e X2Pi 8 1 −0.5 0.5 10 6387.062 584 4 0.5 1.4689E−01 −0.000767 + e X2Pi 9 1 −0.5 0.5 11 7066.166 547 4 0.5 1.3393E−01 −0.000767 + e X2Pi 10 1 −0.5 0.5 12 7739.008 581 4 0.5 1.2337E−01 −0.000767 + e X2Pi 11 1 −0.5 0.5 13 8405.547 119 4 0.5 1.1461E−01 −0.000767 + e X2Pi 12 1 −0.5 0.5 14 9065.740 185 4 0.5 1.0724E−01 −0.000767 + e X2Pi 13 1 −0.5 0.5 15 9719.545 397 4 0.5 1.0095E−01 −0.000767 + e X2Pi 14 1 −0.5 0.5 16 10 366.919 955 4 0.5 9.5535E−02 −0.000767 + e X2Pi 15 1 −0.5 0.5 Column Notation 1 n Energy level reference number (row) 2 |$\tilde{E}$| Term value (in cm−1) 3 gtot Total degeneracy = gnsJ(J + 1) with gns = 2 4 J Rotational quantum number 5 τ Radiative lifetime (s) 6 g Landé factors 7 +/ − Total parity 8 e/f Rotationless parity 9 State Electronic state 10 v State vibrational quantum number 11 Λ Projection of the electronic angular momentum 12 Σ Projection of the electronic spin 13 Ω Ω = Λ + Σ (projection of the total angular momentum) n |$\tilde{E}$| gtot J τ g-Landé +/ − e/f State v Λ Σ Ω 1 0.000 000 4 0.5 inf −0.000767 + e X2Pi 0 1 −0.5 0.5 2 733.657 367 4 0.5 1.2003E+00 −0.000767 + e X2Pi 1 1 −0.5 0.5 3 1461.408 539 4 0.5 6.0697E−01 −0.000767 + e X2Pi 2 1 −0.5 0.5 4 2183.215 742 4 0.5 4.0934E−01 −0.000767 + e X2Pi 3 1 −0.5 0.5 5 2899.040 733 4 0.5 3.1065E−01 −0.000767 + e X2Pi 4 1 −0.5 0.5 6 3608.844 801 4 0.5 2.5151E−01 −0.000767 + e X2Pi 5 1 −0.5 0.5 7 4312.588 793 4 0.5 2.1216E−01 −0.000767 + e X2Pi 6 1 −0.5 0.5 8 5010.233 137 4 0.5 1.8413E−01 −0.000767 + e X2Pi 7 1 −0.5 0.5 9 5701.737 858 4 0.5 1.6315E−01 −0.000767 + e X2Pi 8 1 −0.5 0.5 10 6387.062 584 4 0.5 1.4689E−01 −0.000767 + e X2Pi 9 1 −0.5 0.5 11 7066.166 547 4 0.5 1.3393E−01 −0.000767 + e X2Pi 10 1 −0.5 0.5 12 7739.008 581 4 0.5 1.2337E−01 −0.000767 + e X2Pi 11 1 −0.5 0.5 13 8405.547 119 4 0.5 1.1461E−01 −0.000767 + e X2Pi 12 1 −0.5 0.5 14 9065.740 185 4 0.5 1.0724E−01 −0.000767 + e X2Pi 13 1 −0.5 0.5 15 9719.545 397 4 0.5 1.0095E−01 −0.000767 + e X2Pi 14 1 −0.5 0.5 16 10 366.919 955 4 0.5 9.5535E−02 −0.000767 + e X2Pi 15 1 −0.5 0.5 Column Notation 1 n Energy level reference number (row) 2 |$\tilde{E}$| Term value (in cm−1) 3 gtot Total degeneracy = gnsJ(J + 1) with gns = 2 4 J Rotational quantum number 5 τ Radiative lifetime (s) 6 g Landé factors 7 +/ − Total parity 8 e/f Rotationless parity 9 State Electronic state 10 v State vibrational quantum number 11 Λ Projection of the electronic angular momentum 12 Σ Projection of the electronic spin 13 Ω Ω = Λ + Σ (projection of the total angular momentum) Open in new tab Table 4. Extract from the .states file for 31P32S. n |$\tilde{E}$| gtot J τ g-Landé +/ − e/f State v Λ Σ Ω 1 0.000 000 4 0.5 inf −0.000767 + e X2Pi 0 1 −0.5 0.5 2 733.657 367 4 0.5 1.2003E+00 −0.000767 + e X2Pi 1 1 −0.5 0.5 3 1461.408 539 4 0.5 6.0697E−01 −0.000767 + e X2Pi 2 1 −0.5 0.5 4 2183.215 742 4 0.5 4.0934E−01 −0.000767 + e X2Pi 3 1 −0.5 0.5 5 2899.040 733 4 0.5 3.1065E−01 −0.000767 + e X2Pi 4 1 −0.5 0.5 6 3608.844 801 4 0.5 2.5151E−01 −0.000767 + e X2Pi 5 1 −0.5 0.5 7 4312.588 793 4 0.5 2.1216E−01 −0.000767 + e X2Pi 6 1 −0.5 0.5 8 5010.233 137 4 0.5 1.8413E−01 −0.000767 + e X2Pi 7 1 −0.5 0.5 9 5701.737 858 4 0.5 1.6315E−01 −0.000767 + e X2Pi 8 1 −0.5 0.5 10 6387.062 584 4 0.5 1.4689E−01 −0.000767 + e X2Pi 9 1 −0.5 0.5 11 7066.166 547 4 0.5 1.3393E−01 −0.000767 + e X2Pi 10 1 −0.5 0.5 12 7739.008 581 4 0.5 1.2337E−01 −0.000767 + e X2Pi 11 1 −0.5 0.5 13 8405.547 119 4 0.5 1.1461E−01 −0.000767 + e X2Pi 12 1 −0.5 0.5 14 9065.740 185 4 0.5 1.0724E−01 −0.000767 + e X2Pi 13 1 −0.5 0.5 15 9719.545 397 4 0.5 1.0095E−01 −0.000767 + e X2Pi 14 1 −0.5 0.5 16 10 366.919 955 4 0.5 9.5535E−02 −0.000767 + e X2Pi 15 1 −0.5 0.5 Column Notation 1 n Energy level reference number (row) 2 |$\tilde{E}$| Term value (in cm−1) 3 gtot Total degeneracy = gnsJ(J + 1) with gns = 2 4 J Rotational quantum number 5 τ Radiative lifetime (s) 6 g Landé factors 7 +/ − Total parity 8 e/f Rotationless parity 9 State Electronic state 10 v State vibrational quantum number 11 Λ Projection of the electronic angular momentum 12 Σ Projection of the electronic spin 13 Ω Ω = Λ + Σ (projection of the total angular momentum) n |$\tilde{E}$| gtot J τ g-Landé +/ − e/f State v Λ Σ Ω 1 0.000 000 4 0.5 inf −0.000767 + e X2Pi 0 1 −0.5 0.5 2 733.657 367 4 0.5 1.2003E+00 −0.000767 + e X2Pi 1 1 −0.5 0.5 3 1461.408 539 4 0.5 6.0697E−01 −0.000767 + e X2Pi 2 1 −0.5 0.5 4 2183.215 742 4 0.5 4.0934E−01 −0.000767 + e X2Pi 3 1 −0.5 0.5 5 2899.040 733 4 0.5 3.1065E−01 −0.000767 + e X2Pi 4 1 −0.5 0.5 6 3608.844 801 4 0.5 2.5151E−01 −0.000767 + e X2Pi 5 1 −0.5 0.5 7 4312.588 793 4 0.5 2.1216E−01 −0.000767 + e X2Pi 6 1 −0.5 0.5 8 5010.233 137 4 0.5 1.8413E−01 −0.000767 + e X2Pi 7 1 −0.5 0.5 9 5701.737 858 4 0.5 1.6315E−01 −0.000767 + e X2Pi 8 1 −0.5 0.5 10 6387.062 584 4 0.5 1.4689E−01 −0.000767 + e X2Pi 9 1 −0.5 0.5 11 7066.166 547 4 0.5 1.3393E−01 −0.000767 + e X2Pi 10 1 −0.5 0.5 12 7739.008 581 4 0.5 1.2337E−01 −0.000767 + e X2Pi 11 1 −0.5 0.5 13 8405.547 119 4 0.5 1.1461E−01 −0.000767 + e X2Pi 12 1 −0.5 0.5 14 9065.740 185 4 0.5 1.0724E−01 −0.000767 + e X2Pi 13 1 −0.5 0.5 15 9719.545 397 4 0.5 1.0095E−01 −0.000767 + e X2Pi 14 1 −0.5 0.5 16 10 366.919 955 4 0.5 9.5535E−02 −0.000767 + e X2Pi 15 1 −0.5 0.5 Column Notation 1 n Energy level reference number (row) 2 |$\tilde{E}$| Term value (in cm−1) 3 gtot Total degeneracy = gnsJ(J + 1) with gns = 2 4 J Rotational quantum number 5 τ Radiative lifetime (s) 6 g Landé factors 7 +/ − Total parity 8 e/f Rotationless parity 9 State Electronic state 10 v State vibrational quantum number 11 Λ Projection of the electronic angular momentum 12 Σ Projection of the electronic spin 13 Ω Ω = Λ + Σ (projection of the total angular momentum) Open in new tab Table 5. Extract of the first 15 lines from the PS .trans file. It contains the identification numbers f and i for upper (final) and lower (initial) levels, respectively, Einstein-A coefficients denoted by A (s−1) and transition frequencies ν (cm−1). i f A ν 98 398 98 831 5.1886E−17 0.012 535 97 958 97 524 4.6732E−17 0.012 571 101 886 102 318 7.6872E−17 0.012 588 103 188 102 758 8.6033E−17 0.012 611 100 140 100 573 6.2520E−17 0.012 617 99 701 99 268 5.6651E−17 0.012 627 101 445 101 012 6.8893E−17 0.012 630 98 396 98 829 5.1177E−17 0.012 655 101 884 102 316 7.5904E−17 0.012 672 103 186 102 756 8.4859E−17 0.012 678 97 956 97 522 4.6028E−17 0.012 698 100 138 100 571 6.1750E−17 0.012 726 101 443 101 010 6.8055E−17 0.012 730 i f A ν 98 398 98 831 5.1886E−17 0.012 535 97 958 97 524 4.6732E−17 0.012 571 101 886 102 318 7.6872E−17 0.012 588 103 188 102 758 8.6033E−17 0.012 611 100 140 100 573 6.2520E−17 0.012 617 99 701 99 268 5.6651E−17 0.012 627 101 445 101 012 6.8893E−17 0.012 630 98 396 98 829 5.1177E−17 0.012 655 101 884 102 316 7.5904E−17 0.012 672 103 186 102 756 8.4859E−17 0.012 678 97 956 97 522 4.6028E−17 0.012 698 100 138 100 571 6.1750E−17 0.012 726 101 443 101 010 6.8055E−17 0.012 730 Open in new tab Table 5. Extract of the first 15 lines from the PS .trans file. It contains the identification numbers f and i for upper (final) and lower (initial) levels, respectively, Einstein-A coefficients denoted by A (s−1) and transition frequencies ν (cm−1). i f A ν 98 398 98 831 5.1886E−17 0.012 535 97 958 97 524 4.6732E−17 0.012 571 101 886 102 318 7.6872E−17 0.012 588 103 188 102 758 8.6033E−17 0.012 611 100 140 100 573 6.2520E−17 0.012 617 99 701 99 268 5.6651E−17 0.012 627 101 445 101 012 6.8893E−17 0.012 630 98 396 98 829 5.1177E−17 0.012 655 101 884 102 316 7.5904E−17 0.012 672 103 186 102 756 8.4859E−17 0.012 678 97 956 97 522 4.6028E−17 0.012 698 100 138 100 571 6.1750E−17 0.012 726 101 443 101 010 6.8055E−17 0.012 730 i f A ν 98 398 98 831 5.1886E−17 0.012 535 97 958 97 524 4.6732E−17 0.012 571 101 886 102 318 7.6872E−17 0.012 588 103 188 102 758 8.6033E−17 0.012 611 100 140 100 573 6.2520E−17 0.012 617 99 701 99 268 5.6651E−17 0.012 627 101 445 101 012 6.8893E−17 0.012 630 98 396 98 829 5.1177E−17 0.012 655 101 884 102 316 7.5904E−17 0.012 672 103 186 102 756 8.4859E−17 0.012 678 97 956 97 522 4.6028E−17 0.012 698 100 138 100 571 6.1750E−17 0.012 726 101 443 101 010 6.8055E−17 0.012 730 Open in new tab 3.2 Partition functions Partition functions, Q(T), for 31P16O and 31P32S were computed using the program exocross (Yurchenko et al. 2017), which was also used to calculate absorption and emission cross-sections. The nuclear statistical weight of both species is gns = 2. The maximum temperature was set to 5000 K and partition functions were determined in increments of 1 K. Table 6 gives partition function values at selected temperatures and a comparison with various studies. The full tabulation is given in the supplementary material. Table 6. Comparison of calculated partition function Q(T) with those of Sauval & Tatum (1984), Irwin (1981), Barklem & Collet (2016)a and JPL (Pickett et al. 1998). T This work Irwin Sauval & Tatum Barklem & Collet JPL PO 300 1544.45 2094.29 2303.22 1539.8440 1000 7986.52 7987.18 9182.62 9269.00 2000 24 522.37 24 517.88 26 386.24 26 532.2 3000 50 383.83 50 326.68 52 435.51 53 203.8 4000 86 046.26 85 818.70 88 045.63 89 775.2 5000 132 015.51 131 377.58 133 966.53 136 893.4 PS 300 3525.47 3381.35 3532.94 3522.0557 1000 23 627.9 23 710.1 23 602.8 2000 83 950.66 82 351.29 83 383.6 3000 183 804.33 177 908.26 181 066.8 4000 329 835.22 313 112.07 318 202 5000 538 812.55 490 668.67 497 144 T This work Irwin Sauval & Tatum Barklem & Collet JPL PO 300 1544.45 2094.29 2303.22 1539.8440 1000 7986.52 7987.18 9182.62 9269.00 2000 24 522.37 24 517.88 26 386.24 26 532.2 3000 50 383.83 50 326.68 52 435.51 53 203.8 4000 86 046.26 85 818.70 88 045.63 89 775.2 5000 132 015.51 131 377.58 133 966.53 136 893.4 PS 300 3525.47 3381.35 3532.94 3522.0557 1000 23 627.9 23 710.1 23 602.8 2000 83 950.66 82 351.29 83 383.6 3000 183 804.33 177 908.26 181 066.8 4000 329 835.22 313 112.07 318 202 5000 538 812.55 490 668.67 497 144 aPartition function values from Sauval & Tatum (1984), Irwin (1981) and Barklem & Collet (2016) are doubled to allow for nuclear spin degeneracy and to bring them into line with the convention used here and by the other cited sources. Open in new tab Table 6. Comparison of calculated partition function Q(T) with those of Sauval & Tatum (1984), Irwin (1981), Barklem & Collet (2016)a and JPL (Pickett et al. 1998). T This work Irwin Sauval & Tatum Barklem & Collet JPL PO 300 1544.45 2094.29 2303.22 1539.8440 1000 7986.52 7987.18 9182.62 9269.00 2000 24 522.37 24 517.88 26 386.24 26 532.2 3000 50 383.83 50 326.68 52 435.51 53 203.8 4000 86 046.26 85 818.70 88 045.63 89 775.2 5000 132 015.51 131 377.58 133 966.53 136 893.4 PS 300 3525.47 3381.35 3532.94 3522.0557 1000 23 627.9 23 710.1 23 602.8 2000 83 950.66 82 351.29 83 383.6 3000 183 804.33 177 908.26 181 066.8 4000 329 835.22 313 112.07 318 202 5000 538 812.55 490 668.67 497 144 T This work Irwin Sauval & Tatum Barklem & Collet JPL PO 300 1544.45 2094.29 2303.22 1539.8440 1000 7986.52 7987.18 9182.62 9269.00 2000 24 522.37 24 517.88 26 386.24 26 532.2 3000 50 383.83 50 326.68 52 435.51 53 203.8 4000 86 046.26 85 818.70 88 045.63 89 775.2 5000 132 015.51 131 377.58 133 966.53 136 893.4 PS 300 3525.47 3381.35 3532.94 3522.0557 1000 23 627.9 23 710.1 23 602.8 2000 83 950.66 82 351.29 83 383.6 3000 183 804.33 177 908.26 181 066.8 4000 329 835.22 313 112.07 318 202 5000 538 812.55 490 668.67 497 144 aPartition function values from Sauval & Tatum (1984), Irwin (1981) and Barklem & Collet (2016) are doubled to allow for nuclear spin degeneracy and to bring them into line with the convention used here and by the other cited sources. Open in new tab For PO, our partition function is in good agreement with that of Irwin (1981) but agrees less well with Sauval & Tatum (1984) or the recent one of Barklem & Collet (2016), whose values appear to be too high. For PS, the agreement with Barklem & Collet (2016) is altogether more satisfactory, although their values become a little too low at the higher energies. The partition functions from this study were represented by the series expansion following the recommendation of Vidler & Tennyson (2000): \begin{equation} {\log _{10}(Q)=\sum _{i=0}^{8} a_{i}(\log _{10}(T))^{i}.} \end{equation} (8) The nine expansion coefficients denoted by ai are collected as part of the supplementary material. The fits are valid for temperatures up to 5000 K. 4 RESULTS 4.1 PO X2Π state Fig. 9 gives a comparison of our pure rotational spectrum with that given by CDMS (Müller et al. 2005) at T = 298 K. Hyperfine lines given in CDMS have been convolved for the comparison. The two spectra are in excellent agreement apart from a small difference in line intensities. This slight discrepancy arises from different values for the dipole moment: CMDS used the experimental equilibrium value 1.88 D by Kanata et al. (1988), while our ab initio dipole is 1.998 D at re. Figure 9. Open in new tabDownload slide Comparison of pure PO (left) and PS (right) rotational lines (T = 298 K) with those given in CDMS (Müller et al. 2005) and JPL (Pickett et al. 1998) data bases, respectively. Figure 9. Open in new tabDownload slide Comparison of pure PO (left) and PS (right) rotational lines (T = 298 K) with those given in CDMS (Müller et al. 2005) and JPL (Pickett et al. 1998) data bases, respectively. Fig. 10 shows our predicted, temperature-dependent absorption spectrum for PO. Figure 10. Open in new tabDownload slide PO absorption spectrum at T = 300 (bottom), 1000, 2000 and 3000 (top) K, presented with cross-sections on a logarithmic scale. A Gaussian profile with a Half Width at Half Maximum (HWHM) = 10 cm−1 was used. Figure 10. Open in new tabDownload slide PO absorption spectrum at T = 300 (bottom), 1000, 2000 and 3000 (top) K, presented with cross-sections on a logarithmic scale. A Gaussian profile with a Half Width at Half Maximum (HWHM) = 10 cm−1 was used. 4.2 PS X2Π state Fig. 9 shows a comparison of the rotational spectrum (T = 298 K) of PS simulated using our line list with that from the JPL data base (Pickett et al. 1998). The latter is also a simulated spectrum where a generic dipole value of 2 D was used; therefore, the JPL's intensity scale is arbitrary. We recommend that JPL's intensities are rescaled using value for the dipole. The Q-branches from pure rotational and fundamental bands are illustrated in Fig. 11, where the absorption spectra of PS at T = 400 K are shown. These sharp features are important to astronomers because they are easily identifiable; however, the transitions themselves are relatively weak. Figure 11. Open in new tabDownload slide The Q-branches in the pure rotational and fundamental absorption bands of PS at T = 400 K. A Gaussian profile with HWHM of 0.02 cm−1 was used. Figure 11. Open in new tabDownload slide The Q-branches in the pure rotational and fundamental absorption bands of PS at T = 400 K. A Gaussian profile with HWHM of 0.02 cm−1 was used. Fig. 12 displays the temperature-dependent absorption spectrum of PS. The spectra are prominent in the infrared as expected for rovibrational transitions within the ground electronic state. The fundamental band (v = 1 ← 0) at about 13.6 μm corresponds to the highest peak in the spectra (∼1 × 10−19 cm2 molecule−1). The vibrational overtones at shorter wavelengths than the fundamental band are progressively weaker. Figure 12. Open in new tabDownload slide Absorption spectrum for the ground state of PS as a function of temperature (T = 300, 1000, 2000 and 3000 K). Gaussian profiles with HWHM = 1 cm−1 (left-hand panel) and 10 cm−1 (right-hand panel) were used. Figure 12. Open in new tabDownload slide Absorption spectrum for the ground state of PS as a function of temperature (T = 300, 1000, 2000 and 3000 K). Gaussian profiles with HWHM = 1 cm−1 (left-hand panel) and 10 cm−1 (right-hand panel) were used. Cross-sections were calculated for transitions corresponding to the allowed rovibrational transitions within the X2Π, a4Π and B2Π terms and for B2Π–X2Π transitions. The B2Π–X2Π band peaks in the UV region of the electromagnetic spectrum as expected. Our predicted temperature-dependent B2Π–X2Π absorption spectrum is shown in Fig. 12. At low temperatures it shows sharp features in the 3000–4000 Å region. These features gradually diminish at higher temperatures. Although the B2Π–X2Π band remains featureless at high temperatures, it could possibly be used as a tracer for PS in low-temperature astronomical environments. Fig. 13 shows a comparison of a synthetic (T = 2000 K) B–X emission spectrum with a chemiluminescence spectrum from the reaction Cs+PSCl3 observed by Lin et al. (1987). Figure 13. Open in new tabDownload slide Predicted B–X emission spectrum of PS at T = 2000 K (lower curve) with a Gaussian profile with HWHM = 50 cm−1, compared to PS chemiluminescence from the reaction Cs+PSCl3 by Lin et al. (1987) (upper curve). The cross-section scale refers to our computed values only. Figure 13. Open in new tabDownload slide Predicted B–X emission spectrum of PS at T = 2000 K (lower curve) with a Gaussian profile with HWHM = 50 cm−1, compared to PS chemiluminescence from the reaction Cs+PSCl3 by Lin et al. (1987) (upper curve). The cross-section scale refers to our computed values only. 5 DISCUSSION AND CONCLUSION The new line lists for PO (X2Π) and PS (X2Π, B2Π and a4Π) are the most comprehensive to date. The PO line list, covering the X2Π state, should be sufficient for modelling infrared spectra for this species at long wavelengths. The PS line list has a larger coverage, up to 37 000 cm−1 and considers two low-lying excited electronic states, B2Π and a4Π in addition to the ground X2Π state. This uses the refined curves for the X and B states, but with the addition of the unrefined ab initio curves for the a4Π electronic state as relevant experimental data was unavailable. This line list is designed to aid the identification of spectral features of PS at short wavelengths, although a more accurate line list will be necessary for a full spectral analysis. Our line list shows a strong banded structure for absorption at 300 K in the 3500 Å region. However, at higher temperatures the absorption becomes a very broad feature with little to no structure suggesting that observation of the B2Π–X2Π is unlikely to be useful for detecting PS in hot environments. The Q-branch transitions from the spin–orbit split components can be used as a diagnostic for PS in observational spectra due to the characteristically sharp peak in both absorption and emission. However, the transitions themselves are relatively weak, and so sensitive detectors would be required. It is difficult to provide specific uncertainties for theoretical calculations. However, our fits to the experimental transition frequencies give some insight into the accuracy of the line positions. For PO, this is 0.001 cm−1 for pure rotational transitions and 0.05 cm−1 for transitions involving vibrational excitation. These estimates cover the range were experimental data is available (v ≤ 11 and J ≤ 22.5); for values outside these ranges the uncertainty is expected to grow approximately linearly with v and quadratically with J. Comparison with the rather uncertain experimentally determined dipole suggests that our computed intensities for PO are unlikely to be accurate to better than 10 per cent, and are possibly worse than this. For PS, there is less high-quality experimental data for line positions available. For transitions within the X state, pure rotational transitions are accurate to about 0.006 cm−1 for J ≤ 39.5; the accuracy of the vibrational transition frequencies is limited by a lack of high-accuracy measurements and are unlikely to good to better than 0.2 cm−1. In the absence of any experimental determinations, we can only estimate the uncertainty of vibration–rotation transition intensities as being between 10 and 20  per cent. The vibronic line list for PS is considerably less accurate with transition frequencies unlikely to be better than 0.5 cm−1 and intensities only reliable to about ±50 per cent. These line lists, which we call the POPS line lists, can be downloaded from the http://cdsarc.u-strasbg.fr or from www.exomol.com. So far phosphorus mononitride (PN) (Yorke et al. 2014) and phosphine (PH3) (Sousa-Silva et al. 2015) are the only phosphorus-bearing molecules whose line lists have been computed as part of the ExoMol project. PN was the first P-bearing molecule to be astronomically observed by Turner & Bally (1987) and Ziurys (1987) in Orion KL, Sagittarius B2 and W51; PH3 was discovered in Jupiter and Saturn from Voyager data in 1975 (Bregman, Lester & Rank 1975; Ridgway, Wallace & Smith 1976), and recently detected around the C-rich AGB star, IRC+10216 (Agúndez et al. 2008). This paper presents line lists for PO and PS. A line list for PH is currently under construction while an empirical line list for CP, which can also be downloaded from the ExoMol website, was provided by Ram et al. (2014). Acknowledgements This work was supported by the ERC under the Advanced Investigator Project 267219. We also acknowledge the networking support by the COST Action CM1405 MOLIM. This work made extensive use of the Legion HPC at UCL. REFERENCES Agúndez M. , Cernicharo J. , Pardo J. R. , Guélin M. , Phillips T. G. , 2008 , A&A , 485 , L33 Crossref Search ADS Agundez M. , Cernicharo J. , Decin L. , Encrenaz P. , Teyssier D. , 2014 , ApJ , 790 , L27 Crossref Search ADS Andreazza C. M. , de Almeida A. A. , Borin A. C. , 2016 , MNRAS , 457 , 3096 Crossref Search ADS Bailleux S. , Bogey M. , Demuynck C. , Liu Y. , Walters A. , 2002 , J. Mol. Spectrosc. , 216 , 465 Crossref Search ADS Balasubramanian T. K. , Dixit M. N. , Narasimham N. A. , 1979 , Pramana , 12 , 707 Crossref Search ADS Barklem P. S. , Collet R. , 2016 , A&A , 588 , A96 Crossref Search ADS Ben Yaghlane S. Francisco J. S. Hochlaf M. , 2012 , J. Chem. Phys. , 136 Bregman J. D. , Lester D. F. , Rank D. M. , 1975 , ApJ , 202 , L55 Crossref Search ADS Brown J. M. , Merer A. J. , 1979 , J. Mol. Spectrosc. , 74 , 488 Crossref Search ADS Brown J. M. et al. , 1975 , J. Mol. Spectrosc. , 55 , 500 Crossref Search ADS Bruna P. J. , Grein F. , 1987 , J. Phys. B: At. Mol. Opt. Phys. , 20 , 5967 Crossref Search ADS Butler J. E. , Kawaguchi K. , Hirota E. , 1983 , J. Mol. Spectrosc. , 101 , 161 Crossref Search ADS De Beck E. , Kaminski T. , Patel N. A. , Young K. H. , Gottlieb C. A. , Menten K. M. , Decin L. , 2013 , A&A , 558 , 9 Crossref Search ADS de Brouckere G. , 1999 , J. Phys. B: At. Mol. Opt. Phys. , 32 , 5415 Crossref Search ADS de Brouckere G. , 2000 , Chem. Phys. , 262 , 211 Crossref Search ADS Dimur C. , Pauzat F. , Ellinger Y. , Berthier G. , 2001 , Spectra Chim. Acta A , 57 , 859 Crossref Search ADS Dressler K. , Miescher E. , 1955 , Proc. Nat. Acad. Sci. , 68 , 542 Drowart J. , Myers C. E. , Szwarc R. , Auwera-Mahieu A. V. , Uy O. M. , 1973 , High Temp. Sci. , 5 , 482 Ghosh S. N. , Verma R. D. , 1978 , J. Mol. Spectrosc. , 73 , 266 Crossref Search ADS Huber K. P. Herzberg G. , 1979 , Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules . Van Nostrand Reinhold Company , New York Google Scholar Crossref Search ADS Google Preview WorldCat COPAC Irwin A. W. , 1981 , ApJS , 45 , 621 Crossref Search ADS Jenouvrier A. , Pascat B. , 1978 , Can. J. Phys. , 56 , 1088 Crossref Search ADS Kalcher J. , 2002 , Phys. Chem. Chem. Phys. , 4 , 3311 Crossref Search ADS Kanata H. , Yamamoto S. , Saito S. , 1988 , J. Mol. Spectrosc. , 131 , 89 Crossref Search ADS Karna S. P. , Grein F. , 1992 , Mol. Phys. , 77 , 135 Crossref Search ADS Karna S. P. , Bruna P. J. , Grein F. , 1988 , J. Phys. B: At. Mol. Opt. Phys. , 21 , 1303 Crossref Search ADS Kawaguchi K. , Hirota E. , Ohishi M. , Suzuki H. , Takano S. , Yamamoto S. , Saito S. , 1988 , J. Mol. Spectrosc. , 130 , 81 Crossref Search ADS Klein H. , Klisch E. , Winnewisser G. , 1999 , Z. Naturforschung A , 54 , 137 Lee E. G. , Seto J. Y. , Hirao T. , Bernath P. F. , Le Roy R. J. , 1999 , J. Mol. Spectrosc. , 194 , 197 Crossref Search ADS PubMed Lefloch B. et al. , 2016 , MNRAS , 462 , 3937 Crossref Search ADS Li Y.-L. , Sun S. , Chan L. S. , 2013 , Ecol. Evol. , 3 , 115 Crossref Search ADS Lin K. K. , Balling L. C. , Wright J. J. , 1987 , Chem. Phys. Lett. , 138 , 168 Crossref Search ADS Liu H. , Shi D. H. , Sun J. F. , Zhu Z. L. , 2013 , J. Quant. Spectrosc. Radiat. Transf. , 121 , 9 Crossref Search ADS Lodi L. , Tennyson J. , 2010 , J. Phys. B: At. Mol. Opt. Phys. , 43 , 133001 Crossref Search ADS Maciá E. , Hernández M. V. , Oró J. , 1997 , Orig. Life Evol. Biosph. , 27 , 459 Crossref Search ADS PubMed MacKay D. , Charnley S. , 2001 , MNRAS , 325 , 545 Crossref Search ADS Matthews H. E. , Feldman P. A. , Bernath P. F. , 1987 , ApJ , 312 , 358 Crossref Search ADS McKemmish L. K. , Yurchenko S. N. , Tennyson J. , 2016 , Mol. Phys. , 114 , 3232 Crossref Search ADS Medvedev E. S. , Meshkov V. V. , Stolyarov A. V. , Ushakov V. G. , Gordon I. E. , 2016 , J. Mol. Spectrosc. , 330 , 36 Crossref Search ADS Metropoulos A. , Papakondylis A. , Mavridis A. , 2003 , J. Chem. Phys. , 119 , 5981 Crossref Search ADS Milam S. N. , Halfen D. T. , Tenenbaum E. D. , Apponi A. J. , Woolf N. J. , Ziurys L. M. , 2008 , ApJ , 684 , 618 Crossref Search ADS Moussaoui Y. , Ouamerali O. , De Mare G. R. , 1998 , J. Mol. Struct. (THEOCHEM) , 425 , 237 Crossref Search ADS Moussaoui Y. , Ouamerali O. , De Mare G. R. , 2003 , Int. Rev. Phys. Chem. , 22 , 641 Crossref Search ADS Muller H. S. P. , Woon D. E. , 2013 , J. Phys. Chem. A , 117 , 13868 Crossref Search ADS PubMed Müller H. S. P. , Schlöder F. , Stutzki J. , Winnewisser G. , 2005 , J. Mol. Struct. (THEOCHEM) , 742 , 215 Crossref Search ADS Narasimham N. A. , Subramanian T. K. B. , 1969 , J. Mol. Spectrosc. , 29 , 294 Crossref Search ADS Narasimham N. A. , Balasubramanian T. K. , 1971 , J. Mol. Spectrosc. , 37 , 371 Crossref Search ADS Ohishi M. et al. , 1988 , ApJ , 329 , 511 Crossref Search ADS Patrascu A. T. , Hill C. , Tennyson J. , Yurchenko S. N. , 2014 , J. Chem. Phys. , 141 , 144312 Crossref Search ADS PubMed Pickett H. M. , Poynter R. L. , Cohen E. A. , Delitsky M. L. , Pearson J. C. , Müller H. S. P. , 1998 , J. Quant. Spectrosc. Radiat. Transf. , 60 , 883 Crossref Search ADS Polyansky O. L. , Bielska K. , Ghysels M. , Lodi L. , Zobov N. F. , Hodges J. T. , Tennyson J. , 2015 , Phys. Rev. Lett. , 114 , 243001 Crossref Search ADS PubMed Qian H. B. , 1995 , J. Mol. Spectrosc. , 174 , 599 Crossref Search ADS Ram R. S. , Brooke J. S. A. , Western C. M. , Bernath P. F. , 2014 , J. Quant. Spectrosc. Radiat. Transf. , 138 , 107 Crossref Search ADS Rao T. V. R. , Reddy R. R. , Rao P. S. , 1981 , Physica B & C , 106 , 445 Crossref Search ADS Ridgway S. T. , Wallace L. , Smith G. R. , 1976 , ApJ , 207 , 1002 Crossref Search ADS Rivilla V. M. , Fontani F. , Beltrán M. T. , Vasyunin A. , Caselli P. , Martín-Pintado J. , Cesaroni R. , 2016 , ApJ , 826 , 161 Crossref Search ADS Sauval A. J. , Tatum J. B. , 1984 , ApJS , 56 , 193 Crossref Search ADS Semenov M. , Yurchenko S. N. , Tennyson J. , 2017 , J. Mol. Spectrosc. , 330 , 57 Crossref Search ADS Sousa-Silva C. , Al-Refaie A. F. , Tennyson J. , Yurchenko S. N. , 2015 , MNRAS , 446 , 2337 Crossref Search ADS Spielfiedel A. Handy N. C. , 1999 , Phys. Chem. Chem. Phys. , 1 , 2401 Crossref Search ADS Sun J. F. , Wang J. M. , Shi D. H. , 2012 , Int. J. Quantum Chem. , 112 , 672 Crossref Search ADS Šurkus A. A. , Rakauskas R. J. , Bolotin A. B. , 1984 , Chem. Phys. Lett. , 105 , 291 Crossref Search ADS Tenenbaum E. D. , Woolf N. J. , Ziurys L. M. , 2007 , ApJ , 666 , L29 Crossref Search ADS Tennyson J. , 2012 , WIREs Comput. Mol. Sci. , 2 , 698 Crossref Search ADS Tennyson J. , 2014 , J. Mol. Spectrosc. , 298 , 1 Crossref Search ADS Tennyson J. , Yurchenko S. N. , 2012 , MNRAS , 425 , 21 Crossref Search ADS Tennyson J. , Yurchenko S. N. , 2017 , Mol. Astrophys. , 8 , 1 Crossref Search ADS Tennyson J. , Hulme K. , Naim O. K. , Yurchenko S. N. , 2016a , J. Phys. B: At. Mol. Opt. Phys. , 49 , 044002 Crossref Search ADS Tennyson J. , Lodi L. , McKemmish L. K. , Yurchenko S. N. , 2016b , J. Phys. B: At. Mol. Opt. Phys. , 49 , 102001 Crossref Search ADS Tennyson J. et al. , 2016c , J. Mol. Spectrosc. , 327 , 73 Crossref Search ADS Tsuji T. , 1973 , A&A , 23 , 411 Turner B. E. , Bally J. , 1987 , ApJ , 321 , L75 Crossref Search ADS Verma R. D. , Singhal S. R. , 1975 , Can. J. Phys. , 53 , 411 Crossref Search ADS Vidler M. , Tennyson J. , 2000 , J. Chem. Phys. , 113 , 9766 Crossref Search ADS Visscher C. , Lodders K. , Fegley B. Jr , 2006 , ApJ , 648 , 1181 Crossref Search ADS Wang H. Q. , Gu Y. S. , Liu C. P. , Yin Y. J. , Cao D. Z. , Li S. T. , 1993 , Acta Chim. Sin. , 51 , 18 Werner H.-J. , Knowles P. J. , Knizia G. , Manby F. R. , Schütz M. , 2012 , WIREs Comput. Mol. Sci. , 2 , 242 Crossref Search ADS Western C. M. , 2017 , J. Quant. Spectrosc. Radiat. Transf. , 186 , 221 Crossref Search ADS Wong A. , Yurchenko S. N. , Bernath P. , Mueller H. S. P. , McConkey S. , Tennyson J. , 2017 , MNRAS , 470 , 882 Crossref Search ADS Yamaguchi T. et al. , 2011 , Publ. Astron. Soc. Jpn. , 63 , L37 Crossref Search ADS Yamaguchi T. et al. , 2012 , Publ. Astron. Soc. Jpn. , 64 , 105 Crossref Search ADS Yorke L. , Yurchenko S. N. , Lodi L. , Tennyson J. , 2014 , MNRAS , 445 , 1383 Crossref Search ADS Yurchenko S. N. , Lodi L. , Tennyson J. , Stolyarov A. V. , 2016 , Comput. Phys. Commun. , 202 , 262 Crossref Search ADS Yurchenko S. N. Amundsen D. S. Tennyson J. Waldmann I. P. , 2017 , A&A Ziurys L. M. , 1987 , ApJ , 321 , L81 Crossref Search ADS SUPPORTING INFORMATION Supplementary data are available at MNRAS online. Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article. © 2017 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society
TI - ExoMol molecular line lists – XXIII. Spectra of PO and PS
JF - Monthly Notices of the Royal Astronomical Society
DO - 10.1093/mnras/stx2229
DA - 2017-12-11
UR - https://www.deepdyve.com/lp/oxford-university-press/exomol-molecular-line-lists-xxiii-spectra-of-po-and-ps-yjd3MDRFiH
SP - 3648
VL - 472
IS - 3
DP - DeepDyve
ER -