Rovibronic spectroscopy of PN from first principles

We report an ab initio study on the rovibronic spectroscopy of the closed-shell diatomic molecule phosphorous mononitride, PN. The study considers the nine lowest electronic states, $X\,{}^{1}\Sigma^{+}$, $A\,{}^{1}\Pi$, $C\,{}^{1}\Sigma^{-}$, $D\,{}^{1}\Delta$, $E\,{}^{1}\Sigma^{-}$, $a\,{}^{3}\Sigma^{+}$, $b\,{}^{3}\Pi$, $d\,{}^{3}\Delta$ and $e\,{}^{3}\Sigma^{-}$ using high level electronic structure theory and accurate nuclear motion calculations. The ab initio data cover 9 potential energy, 14 spin-orbit coupling, 7 electronic angular momentum coupling, 9 electric dipole moment and 8 transition dipole moment curves. The Duo nuclear motion program is used to solve the coupled nuclear motion Schr\"{o}dinger equations for these nine electronic states and to simulate rovibronic absorption spectra of $^{31}$P$^{14}$N for different temperatures, which are compared to available spectroscopic studies. Lifetimes for all states are calculated and compared to previous results from the literature. The calculated lifetime of the $A\,{}^{1}\Pi$ state shows good agreement with an experimental value from the literature, which is an important quality indicator for the ab initio $A$-$X$ transition dipole moment.


Introduction
Phosphorus is considered to be one of the key elements as a source of life and replication on our planet, 1,2 with PN being one of the candidates in star and meteorite evolution to provide the necessary life building material. There have been multiple observations of PN in different media in space: hot dense molecular clouds, 3,4 energetic star forming regions, 5,6 cold cloud cores, 5,7 red giant stars [8][9][10] and protoplanetary nebula. 8 Regardless of high astrophysical and astrobilogical importance, phosphorous mononitride is one of the experimentally least-well studied diatomic molecules 0 a Department of Physics  of its isoelectronic group (P 2 , SiO, N 2 , CS).
The molecule was reported for the first time by Curry et al. 11 using a spectroscopic study. Many other subsequent spectroscopic studies have taken place including photoelectron, 12 fluorescence, 13 matrix infrared, 14 microwave [15][16][17] and Fourier Transform Infra-Red (FTIR). 18 Most of the high resolution spectroscopy experiments with PN concentrated on the electronic system A 1 Π -X 1 Σ + , 11,13,[19][20][21] with E 1 Σ + -X 1 Σ + being confirmed afterwards, [22][23][24] and several valence and Rydberg states have been studied as well. 25,26 Measured lifetimes can be important indicators of intensities and Einstein A coefficients. So far only one experimental work reports lifetime measurements for the A 1 Π state of PN molecule 27 using Hanle effect, with several ab initio works providing computed values for PN lifetimes. 28,29 The mass spectrometric experiment by Gingeric 30 reports PN's dissociation energy D 0 to be 6.35±0.22 eV, which is lower than experimental D 0 values by Huffman et al. 31 and Uy et al. 32 , 7.1±0.05 eV and 7.57±0.03 eV. In a combination of a high-level ab initio and microwave spectroscopy study, Cazzoli et al. 17 suggested a D 0 value of 6.27 eV. This is close both to the originally predicted value by Curry et al. 33 and to the experimental value of Gingeric 30 .
Several theoretical investigations of PN are available in the literature. The most recent was carried out by Qin et al. 29 who reported spectroscopic constants for the lowest five singlet (X 1 Σ + , e 3 Σ − ) and two quintet electronic states of PN. In that paper all the states were studied at the internally contracted multi-reference configuration interaction (icMRCI) level of theory with Davidson correction (+Q). Similarly, Abbiche et al. 34 reported an ab initio study of seven states of PN, In this work we present a comprehensive ab initio spectroscopic model for the nine lowest electronic states of phosphorous monon- consisting of potential energy curves (PECs), transition dipole moments curves (TDMCs), spin-orbit coupling curves (SOCs) and angular momentum coupling curves (AMCs) using the icMRCI+Q method and to calculate the rovibronic energies and transition probabilities as an ab initio line list for PN. Producing such line lists for molecules of astrophysical significance is one of the main objectives of the ExoMol project. 35 39 and aug-cc-pCV5Z 40 basis set for nitrogen. The initial complete active space self-consistent field (CASSCF) calculation over which the configuration interaction calculations were built was for the X 1 Σ + state only. In conjunction with ECP for phosphorus, the active space was selected to be (6,2,2,0) with (1,0,0,0) closed orbitals. The state averaging set contained 96 states: 24, 24, 24 and 24 states of the singlets, triplets, pentets and septets. This level of theory will be referenced to as icMRCI+Q/ECP10MWB.
The calculations of the spin-orbit couplings were too difficult to perform for the icMRCI+Q/ECP10MWB level of theory, taking too long to complete and producing wrong data. We therefore decided to use a non-ECP level of theory for our spin-orbit calculations. To this end, we selected the ab initio level of theory similar to that used by Qin et al. 29 with an active space of (9,3,3,0). In this case the state averaging set consisted of 11 singlet configurations (4 A 1 , 2 B 1 , 2 B 2 and 3 A 2 ). The Douglass-Kroll correction was taken into account with or without core-valence correlation. These levels of theory will be referenced to as icMRCI/aug-cc-pV5Z(-DK) and icMRCI/aug-cc-pWCV5Z(-DK), respectively, with or without DK.

DUO calculations
We use the program DUO 36,43 to solve the coupled Schrödinger The goal of this paper is to build a comprehensive ab initio spectroscopic model for this electronic system of PN based on the icM-RCI+Q/ECP10MWB and icMRCI/aug-cc-pV5Z-DK ab initio curves.
We therefore do not attempt a systematic refinement of the ab initio curves by fitting to the experiment, which will be the subject of future work. In order to facilitate the comparison with the experimental data, we, however, perform some shifts of the equilibrium energy T e and bond length r e values, as described in further detail below. We also smooth some of the ab initio curves as described in detail in the Appendix.
In DUO calculations, the coupled Schrödinger equations are solved on an equidistant grid of points, in our case 501, with bond lengths r i ranging from r = 0.85 to 5 Å using the sinc DVR method 48 . Our ab initio curves are represented by sparser and less extended grids (see above). For the bond length values r i overlapping with the ab initio ranges, the ab initio curves were projected onto the denser DUO grid using the cubic spline interpolation.
The following functional forms were used for extrapolation out- Table 1 Comparison of spectroscopic constants taken from previous works and calculated from our ab initio curves (icMRCI+Q/ECP10MWB): Dissociation energy D e in cm -1 (rounded to 3 s.f.), electronic equilibrium energy T e in cm -1 , equilibrium bond length r e in Å, harmonic constant ω in cm -1 , rotational constant B e in cm -1 .

State
for short range and

Results of ab initio calculations
The lowest 9 singlet and triplet PECs (  Table 1 presents spectroscopic constants for these states estimated using the corresponding (spin-orbit-free) PECs and compares to previous studies 17,19,29,33,34,41,42 . These values agree well for the states X 1 Σ + , The ab initio SOCs, EAMCs and TDMCs are shown in   Table. 2.

The
A 1 Π-X 1 Σ + band using different levels of ab initio theory As the A 1 Π-X 1 Σ + band is one of the most important spectroscopic systems of PN, here we compare the X 1 Σ + and A 1 Π PECs and the A 1 Π -X 1 Σ + TDMC computed using different levels of theory: icM-RCI+Q/ECP10MWB, icMRCI/aug-cc-pV5Z-DK and icMRCI/augcc-pwCV5Z-DK, see Figs. 5 and 6. As part of the spectroscopic model for PN, even a slight change of these curves can have a profound effect on the simulated spectra or lifetimes.
In Figure 6, we present a comparison of TDMCs of the A 1 Π and X 1 Σ + states calculated at different levels of theory and results previously calculated by Qin et al. 29 . The TDMC calculated with the ECP10MWB stands out the most with larger values of the dipole moment around the equilibrium (i.e. in the spectroscopically rel-   Comparison of transition dipole moment between A 1 Π and X 1 Σ + states of PN at the different levels of theory (ECP10MWB, icMRCI/augcc-pV5Z-DK, icMRCI/aug-cc-pWCV5Z-DK) and previously reported results 29 .
evant region) and the rest of the ECP-free methods giving curves which lie close to each other. We show below that the ECP10MWB DMC has improved the A-X lifetime of PN, the only known experimental evidence of the transition probability of this system.
The aug-cc-pWCV5Z-DK and previous results by Qin et al. 29 show smaller values of TDMC which will lead to weaker intensities and longer lifetimes.

Results of DUO calculations
In this study we work directly with the ab initio data in the grid representation without representing the curves analytically, with exception of the X 1 Σ + state for which we use an empirical potential energy function from Yorke et al. 49 . Using their PEC, we essentially reproduce the X 1 Σ + states ro-vibrational energies of the  YYLT line list.
For DUO calculations, we selected the following set of curves: the icMRCI+Q/ECP10MWB PECs shown in Fig. 1 for all but the X 1 Σ + state. The comparison between the X 1 Σ + ab initio PEC from this work and the empirical PEC from 49 can be seen in Fig. 7.
Similarly for (T)DMCs all curves are ab initio, see Fig. 4, apart from the ground state dipole moment X 1 Σ + |µ z |X 1 Σ + , which was also taken from Yorke et al. 49 . The comparison between two DMCs is shown in Fig. 8 to 88 000 cm -1 and J ≤ 270. In these calculations, the lower states were capped by the energy threshold 60 0000 cm -1 , which is close

Partition function
The partition function of 31 P 14 N computed using our ab initio line list is shown in Fig. 9, which is compared to that recently reported by Barklem

Spectral comparisons
Using the ab initio 31 Here we mainly concentrate on discussing these band systems, due to the lack of experimental detection for other bands calculated in this work.

X 1 Σ + band
The pure rotational X state transitions represent the main source of the PN observations in interstellar-medium and stellar spectra. [3][4][5][6][7][8][9][10] Apart from astrophysical observations, the band has been analysed in multiple lab experiments as well. [14][15][16]18 An overview of the calculated absorption band of PN at 2000 K is shown in Figure 11.

A 1 Π-X 1 Σ + band
The visible A-X band system was first observed by Curry et al. 33 and there have been several subsequent laboratory observations. 13,19,21,54 Below we compare our results to the experimental spectra; an overview spectrum for the A 1 Π-X 1 Σ + band system at Figure 11 Overview of the calculated X 1 Σ + state rotation-vibration absorption spectrum of PN at T = 2000 K with a Gaussian line profile of HWHM=1 cm -1 .

Figure 12
Calculated absorption spectra of the A 1 Π-X 1 Σ + system at 2000 K with a Gaussian profile of HWHM=1 cm -1 . Fig. 12. Figure 13 compares a DUO generated spectrum of the A 1 Π-X 1 Σ + (2,0) band to a part of the spectrum from recent experimental observations by Le Floch et al. 21 . Even at such a high resolution the main trends of the spectrum are reproduced. However, we have to note a shift of −2217 cm -1 and some differences in intensities. This will be corrected with a further improvement of our model by fitting the A 1 Π PEC to the experimental data.  PEC was adjusted to the experimental values previously reported by Ghosh et al. 19 to reproduce this spectrum, there is still a difference of 0.6 nm between ab initio spectrum and experiment, which we aim to resolve with further improvements of our spectroscopic model via empirical refinements. We attribute the relative difference in strength of peaks between theory and experiment to difficulty in reproducing the non-LTE behaviour of the experiment, due to the lack of necessary information in the original paper.

Lifetimes
In this work, lifetimes of PN are calculated using EXOCROSS 51 , based on the states and transitions files generated by DUO. Ex-oCross calculates lifetimes as follows: where τ i is radiative lifetime, A i j is Einsteins A coefficients, and i and j stand for upper and lower states, respectively. While the lifetimes reported in other works cited below are for vibrational levels and the above formula is for an individual state; however we have found no strong J-dependency for the PN lifetimes. This means that J = 0 does indeed give a good approximation for the vibrational state lifetime. The lack of this dependency can be seen in Fig. 16. The methodology used is described in detail by Ten-  nyson et al. 55 . Figure 16 shows structures for different vibrational levels, which speaks to the perturbed nature of the A 1 Π state.
The lifetimes of 31 P 14 N in the A 1 Π state for (v = 0) were measured by Moeller et al. 27 , and also were calculated by Qin et al. 29 and de Brouckere et al. 28 . While Moeller et al. 27 report a lifetime of 227 ± 70 ns for v = 0 of A 1 Π our value is only slightly higher at 341 ns, which is closer than previous theoretical calculations of 695.4 ns 29 and 742.4 ns 28 by a factor of 2. Table 3 provides a more detailed comparison between our calculated lifetimes and those of Qin et al. 29 . We assume that the major differences in the A 1 Π lifetimes and previous calculation to the difference in the transition dipole moment between X 1 Σ + and A 1 Π comparison of which can be seen in Fig. 6. In order to check that, we have extracted the A 1 Π TDMC data from the original source 29 and rerun the lifetime calculation, then getting a comparable result of 677 ns for v = 0. For the b 3 Π lifetimes seem to be in agreement with Qin et al. 29 for all but v=0, but same cannot be said for the D 1 ∆ and e 3 Σ − states. We attribute the bulk of differences in lifetimes for these states to the differences in TDMCs.
Lifetimes for other singlet and triplet states are reported in Table 4. From there we can see that the radiative lifetimes are in milliseconds for C 1 Σ − d 3 ∆ and a 3 Σ + , and nanoseconds for E 1 Σ + .
Long lifetimes for the C 1 Σ − , d 3 ∆ and a 3 Σ + states are an indication of low probability of transition from these states, leading us to believe that these states would be very difficult to observe in an experiment. This is confirmed with current experimental evidence for PN, as

Conclusion
In this work, a comprehensive ab initio spectroscopic model for the nine lowest electronic states of PN is presented. A full set of potential energy, (transition) dipole moment, spin-orbit coupling, and electronic angular momenta coupling curves for these 9 electronic states was produced ab initio using the icMRCI+Q/ECP10MWB and icMRCI/aug-cc-pV5Z(-DK) methods. These curves were then processed via the DUO program to solve the fully-coupled nuclear-