A theoretical study on the electronically excited-state spectroscopic properties of phosphorus nitride

ABSTRACT The potential energy curves of 103 Ω states generated from the 39 Λ–S states of PN have been calculated using the internally contracted multireference configuration interaction method with the Davidson modification. Core-valence correlation and scalar relativistic corrections, as well as the basis-set extrapolation to the complete basis set limit are considered. The spin–orbit coupling is computed using the state interaction approach with the Breit-Pauli Hamiltonian. The spectroscopic parameters and molecular constants of bound and quasibound Λ–S and Ω states are evaluated. Our calculated spectroscopic results agree quite well with the available experimental data. The interactions among different electronic states in curve crossing regions have been discussed with the help of computed spin–orbit coupling matrix elements. The perturbations and predissociation phenomena of the A1 , b3 , D1Δ, E1Σ+, and 21 states and so on have been revealed. GRAPHICAL ABSTRACT


Introduction
Phosphorus plays an important role in biochemistry, being expected to be one of the candidates in star and meteorite evolution to provide the necessary life building material. Yet it is not a particularly prevalent element, with a cosmic abundance relative to hydrogen of P/H ∼ 2.8×10 −7 [1], less than that of many other third-row elements like Si  been identified in interstellar medium and atmospheres [2][3][4]. Hence, spectral information of gas phase PN has attracted intense research interest over the past several decades.
Spectroscopic evidence for the gaseous PN molecule was first observed in laboratory by Curry et al. [5,6] in the early 1930s. From that time on, a number of experimental studies were performed for this molecule before 1979. Measurements of the radiofrequency [7], microwave [8,9], fluorescence [10], photoelectron [11,12], and infrared spectrum [13] were recorded. The dissociation energy of the ground state was reported a couple of times [5,[14][15][16]. Huber and Herzberg [17] summarised the spectroscopic properties of PN as of that time. Since then, the PN molecule has been the object of numerous experimental studies. Most spectroscopy experiments concentrated on the A 1 -X 1 + [18][19][20][21][22], and E 1 + -X 1 + [23][24][25] band systems, the X 1 + [26][27][28][29], and several valence and Rydberg states [30][31][32]. Concerning the triplet states, the only transition observation was provided by Henshaw et al. [21], in which they tentatively identified the observed transition as the 3 + −X 1 + transition. In contrast to the scarcity of information on most electronic states, the X 1 + and A 1 states remain well characterised experimentally. Only little spectroscopic data of a couple of triplet states, e 3 − , d 3 , b 3 , have been approximately acquired by analysis of the perturbations of the vibrational levels of the A 1 state [20,22].
In theory, numerous ab-initio calculations have been performed on the electronic structure, spectroscopic parameters, and potential energy function of PN in the past several decades [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. Of these theoretical studies, the early theoretical study was reported by Gottscho et al. [33] in 1978. Eight -S electronic states were investigated using the self-consistent-field (SCF) method. Five years later, Grein and Kapur [34] calculated the potential energy curves (PECs) and spectroscopic parameters for the ground and eleven excited-states using the multireference single and double excitation configuration interaction (MRD-CI) method with Gaussian basis set. In the 1990s, a series of theoretical studies were performed by de Brouckere et al. [35][36][37], who calculated the PECs of the X 1 + , A 1 , and D 1 states using the same theoretical method as that of Grein and Kapur [34]. The rotational, rovibrational and transition properties of the A 1 -X 1 + and D 1 -A 1 band systems were determined. Recently, Abbiche et al. [38] studied the spectroscopic properties of the 27 -S states of PN using the variety of theoretical methods, including the explicitly correlated coupled cluster, complete active space self-consistent field, and multireference configuration interaction methods in combination with large correlation-consistent atomic orbitals basis sets AVQZ, AV5Z, and AV(5 + d)Z. The spectroscopic parameters of several states were determined, and the perturbations of the vibrational levels in the A 1 and E 1 + states were analyzed qualitatively. More recently, Qin et al. [39,40] performed a higher level calculation to investigate the transition properties of PN, in which the PECs of thirteen states were computed using the internally contracted multireference configuration interaction approach with the Davidson modification, core-valence and scalar relativistic corrections, as well as the basis-set extrapolation. The transition properties of fifteen band systems are predicted. Semenov et al. [41] made the PEC calculations for nine electronic states of PN using the icMRCI + Q/ECP10MWB method. The pure rotational ground state and rovibronic spectroscopy of A 1 -X 1 + band system were evaluated.
Despite a lot of studies have been reported during the past several decades, our knowledge on PN is far from enough. Firstly, only the X 1 + and A 1 states remain the best characterised electronic states. Some of the published data for this molecule are contradictory or questionable. In particularly, the dissociation energy is still uncertain due to the lack of further measurements. Secondly, little is known about most of excited-states regardless of numerous spectroscopic studies. Thirdly, the complete picture of spin-orbit coupling in the electronic excited-states of PN is still lack, even if the spin-orbit interaction plays an important role in spectroscopy and dynamics of the light molecules that contain atoms of the first and second rows of the periodic table. No intact molecular property investigations can be found in the literature for the states of PN. In addition, more information about the perturbations and predissociation mechanisms for this molecule is needed eagerly, which would help to determine the properties of the perturbing states accurately. So the spectroscopic properties of PN are worthy of further investigation theoretically.
In the present work, high-level ab-initio calculations including spin-orbit coupling effects have been made. The PECs of 39 -S states and 103 states of PN will be present. Based on the computed PECs, the electronically excited-state properties and perturbations and predissociation mechanisms have been studied. The spectroscopic parameters of bound and quasibound states will be reported. Various curve crossings and perturbations will be revealed, and some predissociation pathways will be predicted with the help of the computed spin-orbit coupling matrix elements.

Computational methods
The present calculations are performed with the MOL-PRO 2010 quantum chemistry package [52]. At a given internuclear distance, the molecule orbitals (MOs) of the ground state are calculated using the Hartree-Fock self-consistent field (HF) method at first. Then, the state-averaged complete active space self-consistent field (CASSCF) calculations are carried out using the HF orbitals as the starting guess values [53,54]. Next, utilising the CASSCF wavefunctions as reference, the energies of each -S state are computed using the internally contracted multireference configuration interaction (icM-RCI) approach [55,56] with the Davidson modification (+Q). In the CASSCF and subsequent icMRCI calculations, the C 2v subgroup of the C ∞v point group is employed. Eight valence molecular orbitals (MOs), including 4a 1 , 2b 1 , and 2b 2 symmetry MOs, are selected as the active space, which correspond to the atomic orbitals 3s3p of P and 2s2p of N. The rest of the inner electrons are kept frozen and not correlated. That is, the active space consists of ten electrons in eight molecular orbitals (referred to as CAS (10,8)).
The entirely uncontracted augmented correlation consistent polarised quintuple (aug-cc-pV5Z) and sextuple zeta (aug-cc-pV6Z) basis sets [57][58][59] are chosen to acquire the molecular energies. The energy extrapolation technique has been usually employed in high-level ab-initio calculations [60,61]. In our calculations, a twopoint energy extrapolation scheme given by Muller et al. [62] is used. For two successive correlation-consistent basis sets, the energy extrapolation formula is written as, Here, E total,∞ is the total energy extrapolated to the complete basis set (CBS) limit. And E total,n and E total,n+1 are the total energies obtained by the basis sets, aug-cc-pVnZ and aug-cc-pV(n+1)Z, respectively. Using the formula (1), the PEC of each state involved is extrapolated to the CBS limit using the aug-cc-pV5Z and aug-cc-pV6Z energies. Scalar relativistic effect is taken into account through the third-order Douglas-Kroll-Hess (DKH) one-electron integrals [63,64] combined with the relativistic contracted basis set cc-pV5Z [65]. The difference between the icMRCI energy with and without DKH approximation forms the scalar relativistic contribution (denoted as DK). The core-valence correlation corrections are included in the molecular energies. At the icMRCI/augcc-pCV5Z level of theory [58], the difference between the core-valence correlation and the frozen-core approximation energies is calculated to estimate the core-valence contribution (denoted as CV). Only the two electrons in the 1s inner shell of P atom are left uncorrected when we make the CV calculations for the PN molecule. The spin-orbit coupling effects are introduced using the state interaction approach with the full Breit-Pauli Hamiltonian [66] at the level of icMRCI theory in combination with all-electron aug-cc-pCVTZ (ACVTZ) [58] basis set.
The nuclear Schrödinger equation is solved using Le Roy's Level program [67] to determine the spectroscopic parameters and molecular constants. Table 1 gives the dissociation relationships for all -S electronic states arising from the lowest six dissociation asymptotes of the PN molecule. Altogether, 39 -S electronic states of PN are investigated using the aforementioned methods, including four -S states correlating with the first dissociation channel P( 4 S u ) + N( 4 S u ), namely, one 1 + , one 3 + , one 5 + , and one 7 + states; six -S states correlating with the second dissociation channel P( 2 D u ) + N( 4 S u ), namely, one 3 , one 3 + , one 3 , one 5 + , one 5 , and one 5 states; four -S states correlating with the third dissociationchannel P( 2 P u ) + N( 4 S u ), namely, one 3 − , one 3 , one 5 − , and one 5 states; six -S states correlating with the fourth dissociation channel P( 4 S u ) + N( 2 D u ), namely, one 3 , one 3 + , one 3 , one 5 + , one 5 , and one 5 states; four -S states correlating with the fifth dissociation channel P( 4 S u ) + N( 2 P u ), namely, one 3 − , one 3 , one 5 − , and one 5 states; and fifteen singlet -S states correlating with the sixth dissociation channel P( 2 D u ) + N( 2 D u ), namely, three 1 + , two 1 − , four 1 , three 1 , one 1 1 , and two 1 states.

PECs and spectroscopic properties of the -S states
The calculated energy separations between each higher dissociation limit and the lowest one are summarised in Table 1  Summarising the characters of 39 -S states of PN, we can obtain the following results. (1) The 1 5 + , 2 5 + , 2 5 − , and 2 5 states are bound or quasibound, and the other quintet and septet states are all repulsive in nature. Although the 4 5 state has a shallow potential well induced by two avoided crossings with the neighbouring 5 states, it could not support any vibration levels.
(2) All singlet and triplet -S excited states are bound or quasibound states. (3) Some avoided crossings exist between the states of the same spin and spatial multiply, which make the PECs of several states unsmooth  and several barriers formed, such as the 1 1 and 2 1 , the 2 3 + and 3 3 + states and so on. Based on the computed PECs, the ro-vibrational energy levels of bound and quasibound states are determined. It should be pointed out that the dissociation energies refer to the energy separation between the bottom of the potential well and its dissociation limit, whereas in the case of a quasibound state caused by the avoided crossing, refer to the energy separation between the bottom of the potential well and the top of the barrier.
The main electronic configurations of all bound and quasibound states are abstracted from the MRCI/AV6Z  wavefunctions near the equilibrium distance of each state, which are listed in Table S1 of the Supplementary materials for convenient discussion.

Spectroscopic parameters of the X 1 + and A 1 states
The computed spectroscopic parameters of the X 1 + and A 1 states of PN are summarised in Table 2 along with the available literature results for comparison. The ground X 1 + state of PN is mainly characterised by the 5σ 2 6σ 2 7σ 2 2π 4 3π 0 8σ 0 configuration at equilibrium distance as shown in Table S1, which holds 61 vibrational levels. Our computed R e , ω e , ω e x e , B e , and α e values are only 0.0008 Å, 2.97, 0.0759, 0.0005, and 0.0005322 cm −1 deviated from the experimental results [29], respectively. Our D e value of 6.3247 eV (D 0 = 6.2416 eV) for the ground state is close to the early experimental value of 6.3209 eV (D 0 = 6.30 eV) [17] and the theoretical values of 6.3589 [29], 6.3367 [45], and 6.3795 eV [39], while slightly smaller than the recent suggested value of 6.4396 eV by Cazzoli et al. [29]. As a whole, our calculations are superior to the other theoretical calculations [29,34,[38][39][40][41][42][43][44][45] when compared with the available experimental results.
The A 1 state shows a remarkable multiconfigurational character. It mainly forms from one-electron excitation 7σ 1 -3π 1 . Our computed T e for the A 1 state is the most accurate theoretical ones to this day. The largest deviation of the present T e from all the measurements is only 153.4 cm −1 (0.39%) [18]. While, the largest deviations of the T e values predicted by Qin et al. are 226.70 cm −1 (0.56%) [39] and 837.80 (2.08%) [40] from the experimental one [18], respectively. The computed R e , ω e , ω e x e , B e , and D e values of A 1 are 1.5393 Å, 1104.29 cm −1 , 7.2471 cm −1 , 0.7320 cm −1 , and 5.2242 eV, respectively, which match well with the experimental results [17,18,20]. Their relatively errors are within 0.20%, 0.11%, 0.35%, 0.18%, and 2.44%, respectively. Our computed D e value is only 0.028 eV larger than the experimental value [17]. While, the D e value of  [39] is 0.1134 eV small than the experimental value [17]. The present T e and D e values for the A 1 state are more accurate than those of previous theoretical studies. Tables S2 and S3 of the Supplementary materials show the vibrational energy levels, rotational and centrifugal distortion constants of the X 1 + and A 1 states, respectively, as well as the available experimental and theoretical values [18,20,22,28,47] for comparison. It should be pointed out that the experimental energy levels [28] are computed using the spectroscopic parameters [28] and the zero-point energy [69]. Our molecular constants are in reasonable agreement with the available experimentally determined ones if existing. For example, for the X 1 + state, the largest errors of G(ν), B ν and D ν are 18 (ν = 11), 0.00142 (ν = 6), and 0.2227 (ν = 11) cm −1 deviated from the experimental values [18], while, for the A 1 state, the largest deviations of G(ν), B ν and D ν values are not more than 3.8, 0.0025, and 0.6 cm −1 from the experimental values [20,22], respectively. We note that, our calculations indicate that the A 1 has a deep potential well with a depth of 42136.00 cm −1 holding 55 vibrational levels. However, only the A 1 ν ≤ 11 vibrational levels have been recorded in experiment to date. In the energy range from 37000 to 80000 cm −1 , the A 1 PEC crosses with the PECs of many other excited states, and some correlate with different dissociation limits and have different spin multiplicities. In order to explore why the higher vibrational levels of A 1 cannot be observed in experiment, the couplings among the A 1 and other states will be discussed in detail in Section 3.3.
In general, a detailed comparison of the spectroscopic parameters and molecular constants of the X 1 + and A 1 states indicates that good agreement exists between the available experimental data and our calculations. It validates that our calculated results are of high quality. So, it is expected that we could present a more accurate prediction on the spectroscopic properties for higher excited states and some perturbations and predissociation phenomena of PN.

Spectroscopic parameters of the states with ,
, and symmetries The spectroscopic parameters of nineteen -S states with the , and symmetries are listed in Table 3 along with available literature results for comparison.
The first excited state of PN belongs to 3 + symmetry. This state is mainly described by the electronic configuration 5σ 2 6σ 2 7σ 2 2π 3 3π 1 8σ 0 , arising from oneelectron excitation 2π → 3π. Another configuration namely, 5σ 2 6σ 2 7σ 2 2π 1 3π 3 8σ 0 contributes to a small extent. It is a typical bound state with a well depth of 3.1049 eV. As shown in Table 3, our computed T e , ω e , and ω e x e values are in good agreenent with the approximately experimental values [21], closer to the recent theoretical results [39], and prior to the theoretical results [38].
Similarly, the d 3 , e 3 − , C 1 − , D 1 , and 2 1 states can be basically represented by one dominant configuration 5σ 2 6σ 2 7σ 2 2π 3 3π 1 8σ 0 mixed with another secondary configuration. The d 3 state is 33077.02 cm −1 above the ground state. The e 3 − state is 5145.36 cm −1  [20], but it is worth noting that our calculated spectroscopic parameters of d 3 agree well with those predicted by Grein and Kapur [34], by Abbiche et al. [38], and by Qin et al. [39]. The C 1 − and D 1 states are also strong bound states holding 87 and 66 vibrational levels with the well depths of 5.4806 and 5.1078 eV, respectively. The D 1 state is 1153.12 cm −1 above the A 1 state, which is in accord with the value of 1300 cm −1 given by Grein and Kapur [34], and slightly larger than the value of 300 cm −1 predicted by de Brouckère et al [37]. For both D 1 and C 1 − states, our computed T e and R e values are slightly smaller, while, D e and ω e values are slightly larger than those of the MRCI method [38,39]. The 2 1 state has a deep potential well with depth of 2.1508 eV holding 50 vibrational levels.
In addition, the 2 3 + state holds one main configuration 5σ 2 6σ 2 7σ 1 2π 4 3π 0 8σ 1 at the equilibrium internuclear distance. The 3 3 + state can be represented by three main valence configurations, 5σ 2 6σ 2 7σ 2 2π 2 3π 2 8σ 0 (51.46%), 5σ 2 6σ 2 7σ 1 2π 3 3π 1 8σ 1 (17.82%), and 5σ 2 6σ 2 7σ 2 2π 3 3π 1 8σ 0 (15.50%) at the equilibrium internuclear distance. As shown in Figure 3, the 2 3 + state has a short bond length of 1.5288 Å and a shallow potential well holding 8 vibrational levels with a depth of 5706.28 cm −1 in the range of R < 2.0000 Å, and at larger internuclear distances it is repulsive induced by an avoided crossing with the 3 3 + state. Our spectroscopic parameters for the 2 3 + state are close to the recent theoretical results [38,39]. Similarly, an avoided crossing with the adjacent higher 3 + state makes the 3 3 + state possess a long equilibrium bond length of 1.9213 Å and a shallow potential well with the depth of 6652.28 cm −1 holding 7 vibrantional levels in the range of R < 2.1000 Å, and at larger internuclear distances it is also repulsive.
Regretfully, no spectroscopic data for the 2 5 + , 2 1 , 2 1 − , 4 1 + , 3 1 , and 1 1 states can be compared with either experimentally or theoretically directly. In addition, there are 15 triplet states due to P( 2 D u ) + N( 2 D u ), but they are not computed and discussed in this work. While, this asymptotic limit is very close to P( 4 S u ) + N( 2 P u ), therefore, the present results for the 3 3 + state are not entirely reliable.

Spectroscopic parameters of the states with the and symmetries
We summarise the spectroscopic parameters of ten states with the and symmetries in Table 4 along with available literature results for comparison.
Our calculated values of ω e = 653.55 cm −1 , B e = 0.7355 cm −1 , and T e = 8.10 eV are consistent with the derived values [30] and the calculated results [38], but prior to the  [39]. However, any judgment should be withheld until the experimental values have been more firmly established. The spectroscopic parameters of the 1 5 and 2 3 states are also in accordance with the theoretical results [38,39]. However, for the 4 3 state, our R e , ω e and D e values are somewhat larger than the corresponding computed results [38].
The rest five states, 3 3 , 1 1 , 3 1 , 4 1 , and 2 1 , lie 64592.04, 68307.97, 78275.19, 80019.13, 85150.23 cm −1 above the ground state, and possess 11, 27, 11, 10, and 7 levels, respectively. These states exhibit strong multiconfigurational characters due to avoided crossings with the adjacent states of the same symmetry and spin multiplicity. In detail, the avoided crossing between the 1 1 and 2 1 form a small barrier of 1297.16 cm −1 higher than the dissociation asymptote and a potential well with depth of 14761.64 cm −1 for the 1 1 state, and produce a large barrier of 9104.36 cm −1 higher than the dissociation asymptote and a shallow well with a depth of 5670.08 cm −1 for the 2 1 state. Similarly, the 3 1 and 4 1 states have flat potential wells with depths of 3599.38 and 5068.77 cm −1 , respectively due to avoided crossings with the neighbouring higher 1 state. The barriers of the 3 1 and 4 1 PECs locate nearby R = 2.7200 and 2.4200 Å, and are 276.55 and 3380.43 cm −1 higher than the dissociation asymptote, respectively. So the ν = 11 and 10 level appear to be the highest bound level supported by the 3 1 and 4 1 states, respectively.
Regretfully, to our knowledge, spectroscopic information in the literature is lack for the 1 1 , 3 1 , 4 1 , and 2 1 states either experimentally or theoretically. We expect our data reported here could be helpful for their definitive assignment, and may very well be corroborated by future experiments. Table 5 shows the dissociation relationships for the possible states and corresponding energy separations as well the experimental data [68].
Based on our calculations, the energy separations are 11381.62 ( 4 S 3/2 -2 D 3/2 of P), 10.12 ( 2 D 5/2 -2 D 3/2 of P), The spectroscopic parameters of bound and quasibound states and the dominant -S compositions near the equilibrium bond distance of each state are listed in Table 6.
The X 1 + PEC do not cross with the PECs of other states, and the crossing point between the a 3 + and b 3 PECs is located at R = 1.4016 Å, being away from the equilibrium internuclear distance of the a 3 + state. The PECs shape of the states arising from the X 1 + and a 3 + states would not change evidently. Consequently, the lowest bound states, X 1 + 0+ , a 3 + 0-, and a 3 + 1 , have almost the same spectroscopic properties as the corresponding pure -S states.
The b 3 state crosses with the d 3 state near the bottoms of their potential wells, and its crossing points with the e 3 − , 1 5 + , and 1 7 + states are far away from the equilibrium position of b 3 . These states have the same components as the b 3 and d 3 states. Due to the avoided crossing rule, the shapes of the = 1 and 2 components of the b 3 and d 3 states change greatly near their equilibrium distances, which has evident effects on their spectroscopic properties. For example, as shown in Table 6, for the = 1 and 2 components of the b 3 and d 3  Therefore, an avoided crossing between the = 3 components of the d 3 and 1 7 + states makes the dissociation energy of d 3 3 smaller than that of the corresponding -S state. The avoided crossings with the e 3 − 0+ , 1 5 + 0+ , and 1 7 + 0-states are far away from the bottom of the b 3 PEC, as a result of which the spectroscopic parameters of the b 3 0+ and b 3 0-states are similar to those of the pure -S states, but their D e values become smaller than that of the b 3 state. Our spin-orbit splittings of the b 3 and d 3 states have been calculated to be 462.22 and 44.68 cm −1 , respectively, being somewhat larger than the HF values [33] and the evaluated value from the perturbations on A 1 [20].
The e 3 − state splits into two components with the e 3 − 1 lying below e 3 − 0+ component. The computed fine structure splitting of the e 3 − state is only 4.61 cm −1 . Since the avoided crossings with the A 1 , b 3 , 1 5 + and 1 7 + states are far away from the R e of e 3 − , the R e and ω e values of two states are similar to those of the e 3 − state. And the dissociation energies of two states are smaller than that of the e 3 − state.
The spectroscopic parameters of the C 1 − 0-state are almost the same as the ones of the C 1 − state. While the avoided crossings with the = 0 − components of the 1 5 and 1 7 + states are far from the R e of the C 1 − state, which decrease its dissociation energy greatly. The PEC of A 1 crosses with those of the d 3 , e 3 − , 2 3 , 1 5 + , 1 5 , and 1 7 + states. The PEC of A 1 1 changes greatly due to the avoided crossing with the = 1 components of above mentioned states, which makes the equilibrium bond length become longer, frequency increase, and dissociation energy decrease.
Because the 1 5 + PEC crosses with the A 1 , D 1 , e 3 − , and d 3 PECs, the PEC shapes of = 0 + , 1, and 2 components of 1 5 + change greatly, which make their R e shorten, and ω e and D e values increase greatly.
The 1 5 state separates into six components with = -1, 0 − , 0 + , 1, 2, and 3 with the spin-orbit interactions. Since the 1 5 PEC crosses with the 1 5 + and 2 3 PECs far away from it equilibrium distance, the R e and ω e values of = -1, 0 − , 0 + , 1, 2, and the first well of = 3 components are quite close to those of the 1 5 state, while the D e values of = 0 + and 1 components are slightly larger than that of the corresponding -S state. The second potential well of the 1 5 3 state is located at R = 2.5334 Å, which is 3957.78 cm −1 higher than the first one with a depth of 5382.94 cm −1 . The 2 3 state splits into three components with a separation of 45.87 cm −1 in an inverted order. The 2 3 state crosses with the 1 5 state near its equilibrium internclear separation. Hence, the PECs of three components of 2 3 change greatly, which make the spectroscopic parameters of all three components change greatly. And the 2 3 2 state possesses two potential wells located at 1.8865 and 2.3563 Å with depths of 11049.79 and 2383.37 cm −1 , respectively.
The E 1 + 0+ PEC does not cross with the other PECs of the same symmetry near the equilibrium distance, which makes no influence on its T e , R e , and ω e . While, the avoided crossing with the 2 5 + 0+ state at long internuclear distance makes the D e value of E 1 + 0+ decrease.
The energy separations of the components of the b 3 , d 3 , 1 5 , 2 5 + , and 2 3 states are significant, hence, the spin-orbit coupling effects should not be neglected when analyzing the spectroscopic properties of these states.
In the energy range of 60000-90000 cm −1 , there are many curve crossings between the -S states, and when the spin-orbit coupling effects are considered, various avoided crossings would be come into being. The PECs of states in this region are much more complex than the lower energy regions due to the avoided crossing rule. And some states have two or more shallow potential wells. The depths of these wells are rather small, and some of them even cannot support one vibrational level. The spectra of PN in this energy range will be very diffuse and hard to be observed in experiment.

Curve crossings and perturbations
A larger number of excited states of PN lie in a small range as shown in Tables 2-4. Their PECs are therefore close together with numerous crossings occurring. We depict the amplified views of the curve crossing regions in Figure 6. This high density should favour their mutual interactions and some of them involve states of different spin multiplicities and symmetries, which makes the analysis of the observed rovrational resolved spectra quite difficult. Thus, an accurate ab-initio calculation is definitely helpful to explain the experimentally observed perturbations and identify the possibly predissociation phenomena. We have calculated the spin-orbit coupling matrix elements of interacting states in the crossing regions from the Cartesian components of the Breit-Pauli operator to determine the coupling strength. Figure S2 displays the evolution of the absolute values of spin-orbit matrix elements as functions of internuclear distances. The definitions of the schematic representation for the spin-orbit matrix elements used in these figures are given in Table S5 [18,20], Le Floch et al. [22] and de Brouckère et al. [37]. Nevertheless, Abbiche et al. [38] predicted that the A 1 ν' = 1 level is perturbed simultaneously by the e 3 − ν' = 9 level, the b 3 the ν' = 5 level, the d 3 the ν' = 10 level, and the C 1 − ν' = 4 level, which is significantly different from our conclusions. The absolute values of our computed spin-orbit matrix elements for (A 1 -d 3 ) and (A 1 -e 3 − ) are 14.59 and 13.98 cm −1 at the crossing points, respectively. On the one hand, these crossings lie in the potential wells of the d 3 and e 3 − states. On the other hand, their spin-orbit interactions are weak. Thus, the spin-orbit couplings between the A 1 state and the d 3 and e 3 − states could not induce certain predissociations, but enhance the perturbations on the lower vibrational spectrum of the A 1 state. In addition, the attractive sides of the A 1 and a 3 + PECs crosses at R = 1.2893Å, in between the A 1 ν' = 16 and 17 levels. The absolute value of the spin-orbit coupling matrix element for (A 1 -a 3 + ) is only 6.33 cm −1 at the crossing point, which is insignificant and could be negligible.
Base on the discussion above, we can conclude that the perturbations of the lower vibrational levels (υ = 0-3) of A 1 to be the result of the interaction with the adjacent singlet and triplet states, such as the d 3 , e 3 − , C 1 − , and D 1 states. The 2 3 , 2 3 , E 1 + , and 2 3 − states are identified here to be perturbing states of the higher ν' ≥ 14 vibrational levels of A 1 state. Meanwhile, the higher vibrational levels of the A 1 state is predissociative, mainly coming from the spin-orbit couplings with the 2 3 + , 3 3 + , 3 3 , and 4 3 states. For these reasons mentioned above, we could conclude that the ν' ≥ 14 vibrational levels of the A 1 state are hard to detect.
The D 1 state crosses with the 2 3 , 2 3 + , 3 3 + , 2 3 , 3 3 , 4 3 , 2 3 − , and E 1 + states at R = 2.0067, 2.3332, 2.5517, 2.3546, 2.4298, 3.2996, 2.3854, and 2.0931Å, respectively, in the region from 54000 to 72000 cm −1 . The D 1 -2 3 + , D 1 -3 3 + , and D 1 -2 3 − transitions are forbidden. The spin-orbit couplings between them are all equal to zero. The interactions between D 1 -3 3 and D 1 -4 3 are extremely weak and could be negligible. The absolute values of our computed spin-orbit matrix elements for (D 1 -2 3 ) and (D 1 -2 3 ) are 108.35 and 59.78 cm −1 at the crossing points, and the latter is in accord with the value of 58.95 cm −1 for (D 1 -2 3 ) reported by Abbiche et al. [38]. Because these crossing points are all in the potential wells of the 2 3 and 2 3 states, the D 1 (ν' ≥ 14) → 2 3 and D 1 (ν' ≥ 30) → 2 3 predissociation pathways would not be open. However, the spin-orbit interactions between the D 1 state and the 2 3 and 2 3 state are significant, the strong spin-orbit couplings will make the PECs and spectroscopic parameters of = 2 components of the D 1 , 2 3 , and 2 3 states change greatly as can be seen from Figure S1 and Table 6. The D 1 and E 1 + PECs have overlaps, thus, the ν' ≥ 14 vibrational levels of D 1 would be perturbed by those of the E 1 + state.
Besides crossing with the D 1 and A 1 states, the E 1 + state also crosses with the 2 3 , 2 3 , 3 3 , 4 3 , and 2 3 − states at R = 1.7801, 2.5873, 2.6224, 3.5837, and 3.0879 Å, with the 2 3 + state twice at R = 1.5085 and 2.4451 Å, and with the 3 3 + state at R = 2.7018 Å, respectively. The E 1 + ν' = 0 and 1 vibrational levels would be perturbed by the 2 3 ν' = 1 and 2 levels due to the vibronic coupling. The 2 3 state is 6098.52 cm −1 higher than the E 1 + state, and the potential well of the former is basically in that of the latter. Thus, the E 1 + ν' ≥ 10 vibrational levels could be perturbed by those of the 2 3 state, and the strong spin-orbit coupling between the E 1 + and 2 3 states would enhance this perturbation. This is essentially in agreement with the measurement reported by Coquart and Prudhomme [23], who observed twelve vibrational levels of the E 1 + state. Our calculations indicate the absolute value of our calculated spin-orbit matrix element for (E 1 + -3 3 ) is 19.89 cm −1 at the crossing point, which is in accord with the reported value by Abbiche et al. [38], and the spin-orbit interactions between the E 1 + state and the 4 3 and 2 3 − states are insignificant and could be negligible. The E 1 + -2 3 + and E 1 + -3 3 + transitions are forbidden. Their spin-orbit couplings are also equal to zero.
We find that the 2 1 state crosses with the 3 3 + , 2 3 − , and 4 3 states at R = 2.3519, 1.6943, 2.7053 Å, and crosses with the 2 3 + state twice at R = 1.6727 and 2.0335 Å, respectively. The absolute value of our computed spin-orbit coupling matrix element for (2 1 −3 3 + ) is equal to 54.80 cm −1 at the crossing point. Thus, the 2 1 ν' ≥ 16 level could predissociate via the 3 3 + state. While, the spin-orbit interactions between the 2 1 state and the 2 3 + , 2 3 − , and 4 3 states are extremely small and are insignificant. The 2 1 state crosses with the 3 1 + state twice at R = 1.5634 and 2.3673 Å, and with the 2 1 state at R = 1.7952 Å. The 2 1 ν' = 2 and 3 vibrational levels would be perturbed by the 3 1 + ν' = 5 and 6 levels, and the 2 1 ν' = 17 and 18 levels would be perturbed by those of the 3 1 + state. The 2 1 ν' = 0 vibrational level would be perturbed by the 2 1 ν' = 1 and 2 levels. In addition, the 2 1 state does not cross with the 1 1 and 3 3 states, while, the potential well of the 2 1 state is located in that of the 3 3 state, and the potential well of the 1 1 state lies in that of the 2 1 state. Therefore, the 2 1 ν' ≥ 5 vibrational levels would be perturbed by those of the 1 1 state due to the vibronic coupling. The 2 1 ν' ≤ 5 vibrational levels would be perturbed by the 3 3 1 ≤ ν' ≤ 14 levels due to the Coriolis coupling.

Conclusions
The PECs and spectroscopic parameters of 103 states generated from the 39 -S states of PN have been calculated employing the icMRCI + Q method in combination with the correlation -consistent basis sets. All the molecular energies are extrapolated to the CBS and corrected by the core-valence correlation and scalar relativistic effects. The spin-orbit coupling is computed using the state interaction approach with the Breit-Pauli Hamiltonian. The computed spectroscopic parameters and molecular constants of bound states match the available experimental data very well. Various curve crossings related to the A 1 , b 3 , D 1 , E 1 + , and 2 1 states, and so on are analyzed, and several possible predissociation pathways have been predicted with the help of the calculated spin-orbit coupling matrix elements. We propose that the d 3 , e 3 − , C 1 − , and D 1 states are responsible for the perturbations of the A 1 ν ≤ 3 vibrational levels due to the vibronic coupling. The 2 3 , 2 3 , E 1 + , and 2 3 − states are perturbing states of the A 1 ν' ≥ 14 vibrational levels due to vibronic and spin-orbit couplings. This induces the ν' ≥ 14 vibrational levels of the A 1 state are hard to detect. Meanwhile, several weak predissociation pathways could provide for the higher vibrational levels of the A 1 state via the 2 3 + , 3 3 + , 3 3 , and 4 3 states. The 2 3 + and 2 3 states are the perturbing states of the lower vibrational levels of the E 1 + state due to the vibronic coupling, and the higher vibrational levels of the E 1 + state would be perturbed due to the strong spin-orbit interaction with the 2 3 state. The 2 1 ν' ≤ 5 vibrational levels would be perturbed by the 3 1 + , 3 3 , 2 1 , and 1 1 states, and the 2 1 ν ≥ 16 vibrational levels may predissociate via the 3 3 + state. The b 3 state would be perturbed by the a 3 + , 2 3 , e 3 − , and 1 5 + states due to the vibronic coupling. The strong predissociations could occur for 2 3 via 1 5 , and for 1 5 via 1 7 + state. We hope that future improvements in experimental techniques will be able to verify these predictions.

Disclosure statement
No potential conflict of interest was reported by the author(s).