State-to-state dissociative photoionization of molecular nitrogen: the full story

N2 is a major constituent of Earth and planetary atmospheres. First, evidenced in 1952, the dissociative photoionization of molecular nitrogen, N2, plays an important role in the species abundance, out of equilibrium evolution, and chemical reactivity of diverse media including upper atmospheres (the so-called ionospheres) and plasma. Many scenarios were proposed for rationalizing the dissociative ionization mechanisms and exit channels, which are reviewed here, mainly involving the N2 + (C∑u , v) vibrational levels state-to-state dynamics on which we focus. We show, however, that previous studies are not comprehensive enough for fully shedding light on the complex undergoing processes. As a complementary global work, we used state-of-the-art quantum chemistry, time dependent and independent theoretical approaches associated to advanced experimental techniques to study the unimolecular decomposition of the N2 + ions forming the N + N products. In addition to the already suggested spin-orbit-induced predissociation of the cationic C∑u + state, we documented a new mechanism based on vibronic coupling and tunneling dissociation. Besides, the quantum processes highlighted here should be also in action in the dynamics of electronically excited larger molecular systems involved in physical and chemical phenomena in plasma and in various natural environments on Earth and beyond. ARTICLE HISTORY Received 20 May 2020 Accepted 29 September 2020

+ (C 2 ∑ u + , v + ) vibrational levels state-to-state dynamics on which we focus. We show, however, that previous studies are not comprehensive enough for fully shedding light on the complex undergoing processes. As a complementary global work, we used state-of-the-art quantum chemistry, time dependent and independent theoretical approaches associated to advanced experimental techniques to study the unimolecular decomposition of the N 2 + ions forming the N + + N products. In addition to the already suggested spin-orbit-induced predissociation of the cationic C 2 ∑ u + state, we documented a new mechanism based on vibronic coupling and tunneling dissociation. Besides, the quantum processes highlighted here should be also in action in the dynamics of electronically excited larger molecular systems involved in physical and chemical phenomena in plasma and in various natural environments on Earth and beyond.

I. Introduction
The dissociative photoionization of molecular nitrogen, N 2 , plays an important role in diverse media such as the upper Earth's atmosphere (ionospheres), in planetary atmospheres, in astrophysical media (interstellar medium) and in plasma. Indeed, atomic and molecular nitrogen and their ions are important species in planetary science and astrophysics because of the relatively large N 2 /H 2 universal abundance ratio of 1.6 × 10 −4 . In particular, molecular nitrogen is a major constituent of Earth's atmosphere [1] and of the upper atmospheres of Jupiter, Saturn and its largest moon Titan [2,3,4]. It protects the Earth from high-energy vacuum ultraviolet (VUV) and extreme ultraviolet (XUV) radiations and plays a major role in the chemistry of the thermosphere [5,6]. The photochemistry of N 2 in a variety of environments, e.g., the interstellar medium, extrasolar planets, and circumstellar envelops is believed to be responsible for the synthesis of N-containing molecules, including prebiotic molecules (e.g., HCN, CH 3 CN) [7][8][9], which are precursors of larger molecules such as the building blocks of life [10][11][12], and also triggers the formation of N-containing aerosols such as those found in Titan's atmosphere [13].
VUV and XUV radiations interacting with neutral N 2 lead to its ionization and its dissociative ionization to form N 2 + and N + N + , respectively. In addition, the neutral dissociation of N 2 by absorbing VUV light produces excited N radical atoms and even excited N + ions [14,15,16]. These processes take place, for instance, in Earth's upper atmosphere after absorption of the solar radiation in the VUV and XUV ranges [17]. They occur along the potential energy curves (PECs) of the electronic-excited states of the N 2 + cation, where both spin-orbit and nonadiabatic relaxation phenomena are in action. N 2 + is also formed in the bow shock wave found in front of heat shields during entry into the Earth's atmosphere. Moreover, it is suspected to be responsible for the escape of He + ions from the atmosphere after charge transfer collisions between N 2 + and neutral He [18]. For the realistic modeling of these phenomena, one needs accurate descriptions of the electronic states of this ion and in-depth understanding of its formation and destruction dynamics via N 2 photo -and dissociative ionizations.
The dissociative photoionization of molecular nitrogen was evidenced for the first time in 1952. Since then, several scenarios were proposed in order to explain the formation of the N + N + fragments following the predissociation of the cation. These works are reviewed in Section 2. However, we will show, that the various involved processes are complex and cannot be understood within a single picture. At present, we performed complementary experimental works on the single photon ionization of N 2 using synchrotron VUV light source coupled to advanced double imaging electron/ion coincidence detector. We investigated the dynamics of the N 2 + cation on a newly computed set of N 2 + PECs and their couplings using two well-established state-of-the-art theoretical methodologies. These up-to-date experimental and theoretical results complement those already available. The full examination of the processes that may occur upon formation of an N 2 + ion reveals that, in addition to the already suggested spin-orbit-induced predissociation, which are in action for the low vibrational levels of the N 2 + (C 2 ∑ u + ) state, the upper levels correspond to resonances, which dissociation is documented here for the first time. In summary, we shed light on the complex dynamics undertaken by the N 2 + ions forming the N + + N products, which remained not fully explained for more than 60 years. This work should help in understanding the state-to-state complex dynamics of electronically excited larger molecular systems relevant for astrophysics, planetary sciences and in plasma physics. Indeed, N 2 + C state represents a prototype molecular system, where the observed features and processes are easily found in large molecules.

II. Overview on the dissociative ionization of N 2 via the N 2 + (C 2 ∑ u + ) state
Back in 1952, Douglas [19] observed a strong enhancement of the N 2 + (C 2 ∑ u + , v + = 3) bands in a He + N 2 discharge. He showed that this is due to inverse predissociation, i.e. N( 4 S) + N + ( 3 P) → N 2 + (C 2 ∑ u + ), where the N + ions are produced via charge transfer reactions involving the He + ions. In 1959, Carroll [20] documented the direct predissociation of the C state to form N + + N, which occurs in the vicinity of the v + = 3 level. Since then, a multitude of experimental and theoretical works were devoted to the identification of the mechanism(s) responsible for this predissociation. Indeed, the production of N + ions was studied using mass spectrometry [21][22][23][24]. The N 2 + (C 2 ∑ u + ) ions are also characterized to be long-lived enough to fluoresce and hence are responsible for the second negative system of N 2 + (i.e. the C 2 ∑ u + -X 2 ∑ g + emission transition). In the early seventies, Wankenne and Momigny [21] determined a lifetime of 0.8 μs for the N 2 + (C 2 Σ u + ) ion before radiationless decomposition. However, a much shorter radiative lifetime (τ(v + ≤ 2) = 0.09 μs) was deduced by Govers and coworkers [24,25] using either electron or ion impact-induced N 2 + (C 2 ∑ u + ) predissociation. In 1976, Erman [26] rationalized both observations after direct measurement of the lifetimes of the N 2 + (C 2 ∑ u + , v + = 1-5) vibrational levels using high-resolution high-frequency deflection technique. He determined a radiative lifetime of 78.9 ±3.0 ns for v + = 2 level and lifetimes of 4-5 ns for v + ≥ 3 and attributed the sharp drop of these lifetimes to the already noticed predissociation of the v + ≥ 3 states populating the lowest dissociation limit, L1: N( 4 S) + N + ( 3 P). The corresponding predissociation rate was measured ~2 x 10 8 s −1 . In 1974, Asbrink and Fridh [27] observed the population of the upper dissociation limits by dissociative ionization of N 2 . Later on, the threshold photoelectron-photoion coincidence spectroscopic investigations of the valence and inner-valence states of N 2 + by Nicolas et al. [28] and by Aoto et al. [29] measured the following branching ratios (BRs): BRs = 100% to the L1 limit for C state v + levels located in the energy range between L1 (N + ( 3 P) + N( 4 S)), and L2 (N + ( 1 D) + N( 4 S)); for those located between L2 and L3 (N + ( 3 P) + N ( 2 D)), BRs ~ 60% to L1 and 40% to L2; and BRs ~ 100% to L3 for those located above L3. Recent experimental and theoretical investigation dealing with the dissociation dynamics of N 2 + induced by isolated attosecond XUV pulses in combination with few-optical-cycle near-infrared/visible (NIR/VIS) pulses showed that most of the N 2 + ions decay to populate L2 (BR = 77.1%) and L3 (BR = 14.0%) and to a lesser extent the following upper limit L4 (BR = 8.5%), whereas a 0% branching ratio is measured for L1 [30]. The use of higher energy XUV femtosecond light induces the population of the upper dissociation limits of N 2 + [31]. In sum, different BRs are measured depending on the specific vibrational or electronic states of N 2 + populated by ionizing N 2 (X 1 ∑ g + ). Moreover, Govers and co-workers showed the existence of a large isotope effect on these predissociations [25,[32][33][34][35][36]. All these findings are the signature of a complex dynamics undergone by the N 2 + ions. In the 24-28 eV energy domain above N 2 (X 1 ∑ g + , v" = 0), the formation of the N + + N pair corresponding to the dissociation limits L1, L2 and L3 is ascribed to the predissociation of the N 2 + (C 2 ∑ u + ) state. The question of which states are involved in such predissociation has been the subject of much discussions in the course of which several scenarios were suggested [26,28,[37][38][39][40][41][42][43][44]. They can be gathered into two main mechanisms known as the 'direct' and the 'accidental' mechanisms. The direct predissociation involves the conversion into the continuum of the N 2 + (B 2 ∑ u + ) state either due to finite vibrational interaction or after crossing a repulsive state, or a repulsive part of an attractive state. On the other hand, the accidental predissociation starts by the population of a bound intermediate state, which is then predissociated by a second repulsive state. In both cases, the implication of the N 2 + quartet states and their spin-orbit couplings with the neighboring doublets was invoked. Up-to-date, none of these mechanisms, however, fully explained the state-to-state dissociative photoionization of N 2 and the subsequent complex unimolecular decomposition dynamics undertaken by the N 2

III. On the potentials and couplings of the N 2 + ion a. Potential energy curves
The full interpretation of the state-to-state dissociative photoionization of N 2 requires the mapping of the PECs of all N 2 + cationic electronic states correlating to L1, L2 and L3 and those electronic states correlating to the upper asymptotes lying in energy below 28 eV with respect to N 2 (X 1 ∑ g + , v" = 0). We thus computed N 2 + states of doublet, quartet, and sextet spin multiplicities in both ungerade and gerade symmetries. The electronic calculations were performed using the MOLPRO 2015 program suite [45] in the D 2h point group symmetry. We used the complete active space selfconsistent field (CASSCF) [46,47] technique followed by the internally contracted multi-reference configuration interaction (MRCI) [48][49][50] approach. We described the nitrogen atom by a large basis set of aug-cc-pV5Z quality [51,52], which is augmented by 3s and 2p diffuse Gaussiantype orbitals (GTOs). This results in 254 contracted Gaussian functions, to be used. In CASSCF, the active space was composed of the valence molecular orbitals (MOs) of N 2 + to which we added one σ g and one π g MOs for better relaxation of the wavefunctions of the N 2 + electronic states whose configurations differ in their σ and π orbital occupations. We used an active space larger than the valence and a diffuse basis set for better description of the possible diffuse nature of the valence-Rydberg states of N 2 + as discussed for neutral N 2 [53,54]. For MRCI calculations, all configurations in the CI expansion of the CASSCF wavefunctions were taken as a reference and all valence electrons were correlated. Hence, we considered more than 4.8 × 10 8 uncontracted configuration state functions per symmetry. Using these highly correlated wavefunctions, we deduced the radial, the rotational and the spin-orbit couplings between the electronic states involved in the C 2 ∑ u + state predissociation. The spin-orbit integrals were evaluated, in Cartesian coordinates, over the MRCI wavefunctions, where the effective Breit-Pauli SO operator, H SO , as implemented in MOLPRO was used [55].
The N 2 + MRCI/aug-cc-pV5Z PECs are displayed in Figure 1. For the bound states, the respective potential energy curves were incorporated into nuclear motion treatments in order to deduce a set of spectroscopic constants for the N 2 + states. Thus, we used the method of Cooley [63] and the derivatives at the minimum energy distances and standard perturbation theory. The corresponding data are listed in Table 1. This table lists the spectroscopic parameters of the bound states including their equilibrium distances, vibrational and rotational constants, and excitation energies. The comparison to the previous experimental and previous theoretical data shows an overall good agreement, asserting the good quality of our PECs. The pattern of the vibrational levels of the C 2 ∑ u + state is also obtained using a variational treatment and time-dependent approach. Figure 1 shows the high density of electronic states lying in the vicinity of the N 2 + (C 2 ∑ u + ) state that may participate to its direct or accidental predissociation mechanisms. Both of them require the couplings with the N 2 + (C 2 ∑ u + ) state either by vibronic or spin-orbit couplings and the subsequent mixings of their wavefunctions. Nevertheless, standard selection rules reveal that only ungerade electronic states are in action in the dissociative single photon ionization of N 2 , ruling out the contribution of the gerade states in this energy domain. Moreover, only quartet electronic states correlate adiabatically to the second dissociation limit L2. Thus, their implication cannot be excluded for the formation of, at least, N + ( 1 D) + N products. Consequently, we present in Figures 2 and 3 the electronic states and their mutual couplings that are a priori participating to the formation of N + N + ions in the 24-28 eV energy domain. Figure 2 shows that the adiabatic PEC of the C 2 ∑ u + state exhibits two potential energy wells: the inner potential where we locate 14 vibrational levels and two quasi-bound shape resonances; and the outer potential that also accommodates several bound levels with a large interatomic equilibrium distance (see Section VI). Both potential wells are separated by a barrier due to the avoided crossing with the 3 2 ∑ u + state. At these large internuclear separations, the C 2 ∑ u + state interacts with the 3 2 ∑ u + state by radial coupling (Figure 3(a)) and with the lower B 2 ∑ u + state as discussed in Refs [43,44]. The shape of the radial coupling between the C 2 ∑ u + and 3 2 ∑ u + states ( Figure 3(a)) exhibits a maximum for NN distances between 3-3.5 bohr i.e. close to the maximum of the C 2 ∑ u + state potential barrier due to the avoided crossing between both states. This indicates thus that the excited vibrational states of the C 2 ∑ u + state are coupled with the 3 2 ∑ u + . Although the vibronic eigenstates have always components on both electronic states (cf. 6 T. AYARI ET AL. Section VI), it is especially the highly excited vibrational states of the C 2 ∑ u + state that are coupled with the 3 2 ∑ u + since an important part of these vibrational wavefunctions is located in the region where the radial coupling . The reference energy is taken as that of N 2 (X 1 ∑ g + , v" = 0). The adiabatic ionization energy of N 2 is taken from Ref [74]. The lowest dissociation limit (L1 @ 24.2884 eV) is located as given in Ref [56]. L2 (@ 26.1874 eV) and L3 (@ 26.6724 eV) were located using the excitation energies of N + and N, respectively. The thick line corresponds to N 2 + ground state PEC. Table 1. Spectroscopic constants of the electronic states of N 2 + investigated in the present work. We give the equilibrium distance (R e in Å), vibrational constants (ω e , ω e x e , ω e y e , in cm −1 ) and the rotational constants (B e , α e , in cm −1 ). T 0 (in eV) is the adiabatic excitation energy calculated as the difference between the energies of the minimum of the ground state and the minimum of the considered excited state including the zero point vibrational energy corrections. is large. The radial coupling between the C 2 ∑ u + and B 2 ∑ u + states is maximum around 2.46 bohr [43], close to the minimum of the C 2 ∑ u + state where the energy gap at the avoided crossing is larger than in the C 2 ∑ u + -3 2 ∑ u + interaction. It is noteworthy that the radial coupling overlap and that the two non-adiabatic interactions cannot be considered as separated two-bytwo systems. Indeed, the three 2 ∑ u + states strongly interact. The 1 2 ∑ u − and 2 4 ∑ u − states are crossing the C 2 ∑ u + state and may couple together by spin-orbit ( Figure 3(d)). The 4 π u states are also crossing the C 2 ∑ u + state, where spinorbit conversions may occur (Figure 3(d)). These states are also mutually coupled vibronically (Figure 3(b)).  [60]. i.MRCI/aug-cc-pCV5Z. Ref [61]. j.Molecular constants have been measured using fast-ion-beam laser spectroscopy. Ref [62].

IV. Radiative, predissociative, and natural lifetimes of the N 2
The calculated potentials of Figure 2, spin-orbit couplings and dipole transition moments displayed in Figure 3, were incorporated into the LEVEL [64] and BCONT programs of Le Roy [65] to deduce the radiative and the spin-orbit-induced predissociation lifetimes of the vibrational levels of N 2 The radiative lifetimes were deduced using the following formula where A v 0þ v 00þ corresponds to the spontaneous emission Einstein coefficient from an ionic v' + level to a v" + level. This needs the transition moments between both initial and final electronic states as well as their potential energy curves. The data are given in Table 2. The validity of our approach was already noticed, for instance, for the SH (A 2 ∑ + ) state through the direct comparison between our theoretical data and experimental measurements [66].
The C 2 ∑ u + vibrational levels may decay radiatively to populate either the X 2 ∑ g + or the D 2 Π g states ( Figure 1). The dipole moments of the C-X and C-D of these allowed transitions are given in Figure 3(c). Table 2 lists the radiative lifetimes of N 2 + (C 2 ∑ u + , 0 ≤v + ≤ 15) as computed using the LEVEL code [64]. For the C 2 ∑ u +_ X 2 ∑ g + radiative decays of the 0 ≤ v + ≤ 6 levels, the corresponding radiative lifetimes are in a good agreement with the results deducted using the + involved in the predissociation branching ratios calculations of the N 2 + (C 2 ∑ u + ) state. The reference energy is taken as that of N 2 (X 1 ∑ g + , v" = 0). For obtaining the diabatic PECs from the adiabatic ones, we used the B -C radial coupling from Ref [43]. in addition to the couplings presented in Figure 3 empirical RKR potentials [67]. Again, this confirms the good quality of our ab initio PECs and dipole moments. When compared to previous experimental radiative lifetimes, a good agreement is found with those measured by Govers and coworkers and by Erman for v + ≤ 2 (Table 2). Nevertheless, none of them gave an accurate estimation for v + ≥ 3 since it is hard to deduce these quantities from experiments. Table 2 shows that the C 2 ∑ u +_ D 2 π g radiative decays affect the lifetimes of the upper levels of the C (v + ≥ 7) states. It is worth noting that the lifetimes of the upper levels exhibit an irregular evolution, which may be related to the non-monotonic evolution of the corresponding transition dipole moments along the NN distance. Experimental measurements are welcomed for confirmation. Table 2 gives also the predissociative lifetime obtained by three approaches. First, the lifetime is computed using the BCONT code [65]. We considered the C state spin-orbit-induced predissociation by the 1 4 π u , 2 4 Figure 3. Radial couplings between the 2 ∑ u + states (in A) and those between the 4 π u states (in B). In C), we give the transition dipole moments between the C 2 ∑ u + state and some electronicexcited states of N 2 + . In D), we plot the spin-orbit couplings between the electronic states depicted in Figure 2  π u and 2 4 ∑ − u states, respectively. Mainly, the predissociative 0 ≤ v + ≤ 10 levels lifetimes are due to the 1 4 π u induced predissociation leading to L1. For the upper levels, the 2 4 π u and 2 4 ∑ − u states are in action populating thus L2 and L3. Nevertheless, the corresponding lifetimes are somehow too long to be a realistic, signature of another dynamics (cf. Section VI). Second, we extract a lifetime from the time evolution of the norm P(t) of the wave packet initially prepared in a vibrational level of the C state. By adding an absorbing complex potential in the asymptotic region in each electronic state, the decrease of the total norm is a signature of the dissociation through the different channels. Note that the decay is always nonexponential and the rate is estimated from the initial quasi linear part of ln(P(t)). Finally, we report the lifetimes computed by the Fermi Golden rule in Ref [43] by considering the interaction between the C and B states only. In summary, the first method focuses on the spin-orbit coupling and discards the strong non-adiabatic interaction while the third one discards both the spin-orbit and the important interaction with the 3 2 ∑ u + state, which has a strong influence on the dynamics. It is thus not surprising that the results may be rather different since the methods sometimes use a truncated electronic basis set. The complex dynamics in the complete basis set provides more oscillatory results, which could be explained by the possibility of trapping some components of the wave packet in the wells of the π states leading to a slower decay. Indeed, the expected behavior is more complex than a textbook coupling between a bound state and a continuum. Nevertheless, the comparison of these different results remains interesting since they may illustrate the relative role of the different processes. Still, the results given in Table 2 confirm the already established sudden drop of the natural lifetimes of the N 2 + (C 2 ∑ u + , 3 ≤ v + ≤ 15) levels because of the dominance of the predissociation processes, which will be our focus on the following parts of this manuscript.

V. Present state-to-state N 2 + ion unimolecular decomposition investigations
To complement previous work on the predissociation of N 2 + (C 2 ∑ u + , v + ), we performed a combined experimental and first-principles treatment to fully investigate the state-to-state predissociation of the N 2 + (C 2 ∑ u + , v + ) upon ejection of an electron from the neutral N 2 (X 1 ∑ g + , v" = 0). Thus, we recorded the state-to-state BRs for N 2 + (C 2 ∑ u + , v + ) → L1/L2/L3 using the mass-selected kinetically analyzed ions formed after N 2 ionization by the VUV light delivered by the DESIRS beamline at the French SOLEIL synchrotron facility. Theoretically, we computed the corresponding BRs using time-independent and time-dependent approaches where the nuclear motions were treated on a newly computed set of the N 2 + PECs possibly involved in the C state predissociation and of their radial and/or spin-orbit couplings. In order to highlight the possible-specific contribution of the electronic states lying in close to the C state, we performed several computations, where we examined all the possible processes, and derived their involvement in the C state predissociation. Our data together with those obtained previously are used to shed light on the specific processes undergone by N 2

a. Experimental determination of the BRs
Experiments were performed at the French SOLEIL synchrotron facility, more specifically at the molecular beam endstation SAPHIRS [68] located at the VUV undulator-based beamline DESIRS [69]. Nitrogen gas was sourced from Air Liquide (N60 grade) and used without further purification to create a molecular beam by expanding half a bar of N 2 through a 30 µm nozzle. The resulting supersonic beam traversed a 1 mm skimmer before crossing the synchrotron light at the center of the DELICIOUS3 double imaging photoelectron photoion coincidence spectrometer [70]. In the context of the present study of state-selected fragmentation, our imagingbased set-up offers the unique opportunity to collect 100% of the emitted electrons allowing a very efficient measurement of electron/ion coincidences. The photon energy was set at 27.2 eV with a 1.5 meV bandwidth, while the polarization axis was chosen so that it was contained in the detection plane. The electrons and ions produced were accelerated in opposite directions by a 90 V/cm electric field. The ions were analyzed and detected by a modified Wiley McLaren spectrometer to yield the three dimensional translational energy distribution with a resolution of 15%, while the electrons energy distribution was obtained from their velocity map image through an Abel inversion [71] with resolutions down to a few % for the fastest photoelectrons. The data is then organized through the coincidence scheme to yield kinetic energy correlation diagrams, from which dissociation limits can be observed and their relative branching ratios extracted. Note that the translational energy of the parent ion has been fitted to a Boltzmann energy distribution to obtain an estimate of the molecular beam temperature, 80 ± 10 K.
The results of the state-to-state dissociative single photon ionization of N 2 are given in Figure 4, which shows the intensity of the N + collected fragments with respect to the electron and ion kinetic energy. These intensities are shown in this figure from the vibrational level (v + =3) where the predissociation of the C state starts. From this image, obtained at the single photon energy of hv = 27.2 eV, the branching ratios leading to the three accessible dissociation limits can be obtained for each vibrational level. Figure 5 shows the kinetic energy distributions of the N + fragment for the vibrational levels 11 ≤ v + ≤ 15 obtained through vertical projections of the matrix shown in Figure 4 centered at the electron kinetic energy corresponding to the vibrational levels, and using a width of ± 50 meV to obtain sufficient signal-to-noise, and the branching ratios to the first three limits of dissociation, which are also reported in Table 3. Note that for 3 ≤ v + ≤ 10 levels, the only dissociation limit energetically available is L1, so that dissociation is 100% to this limit, as shown in Figure 6, while L2 and L3 open for v + ≥11 and v + ≥13, respectively.
The branching ratios have been extracted by fitting three Gaussian functions to the curves shown in Figure 5, one for each dissociation limit, and agree reasonably well with those deduced earlier by Nicolas et al. [28] using the TPEPICO technique. There are, however, some noticeable differences, especially for v + ≥ 14, where our data show some signal in L2 and L1 (see Figure 5) -albeit with relatively high error bars due to the low signal-to-noise -whereas Nicolas et al. found 100% fragmentation to L3. Also, for v + = 11, they reported more fragmentation to the L2, while we find 74% and 26% BRs to L1 and L2, respectively, in better accordance with the present theoretical calculations that predict mainly dissociation to L1. These differences could be explained by the different experimental techniques and/or treatment methods, since our BRs are extracted from a single photon energy within a 100 meV electron binding energy bandwidth and the translational energy of the fragment ion is measured in 3D, while Nicolas et al. only had access to the 1D translational energy distribution, which accuracy might depend on the KER angular asymmetry, and performed a photon energy scan with For v + =14 and v + =15, the curves are also shown after multiplication by an order of magnitude to accentuate the weak dissociation to L2 and L1. The branching ratios to the three dissociation limits are provided for the five vibrational levels in Table 3.   [72]. Nevertheless, our data show that dissociation to the L1 is gradually quenched by the opening of the L2 and L3 channels, but does not completely disappear, with L2 also dramatically decreasing with the opening of L3. Note however, that due to the experimental resolution the v + = 11 and perhaps v + =12 branching ratios might be affected by the presence of the 2 Π g progression, that could contribute up to 9% to these bands as reported by Baltzer et al. [73] Other progressions, reported only in threshold photoelectron spectra [36,74,75,76], could also contribute but since they were not seen in the high-resolution photoelectron spectrum by Baltzer et al., they are dismissed in this work, although they are shown in Figure 6 for reference.
We have also recorded the TPEPICO curves of N 2 + and N + in the photon energy region between 23.6 eV and 24.8 eV, i.e., below the N 2 + (C 2 ∑ u + ) v + = Figure 6. Number of dissociation events towards the three available limits, L1 (black), L2 (blue) and L3 (red), recorded at hv = 27.2 eV as a function of the electron kinetic energy. The relative intensity of the curves corresponds to the branching ratios to the three dissociation limits. The positions of the N 2 + (C 2 ∑ u + ) vibrational levels from Yoshii et al. [75] are marked in magenta, as well as four different progressions in gray S1(2 2 Π g ), S2( 2 ∑ u − ), S3( 2 Δ u ) and S4( 2 Π u ) from the same reference. Figure 7. These curves have also recently been acquired with a higher signal-to-noise over a larger energy range encompassing vibrational levels up to v + = 6, and discussed in great detail by Hrodmarsson et al. [36]. Therefore, they will not be further mentioned here, except to note that N 2 + does not fully dissociate above v + = 3. Hrodmarsson et al. reported survival rates for the parent ion 14 N 2 + of a few % for v + = 3, 4, while for the 15 N 2 + isotope the survival gradually went from 37% at v + = 3, to 5% at v + = 7.

b. Theoretical determinations of the BRs
The C 2 ∑ u + → 1 2 ∑ u − pathway can be ruled out since the respective spin-orbit integral is close to zero (Figure 3(d)). We considered the C 2 ∑ u + state interacting, via spin-orbit, with only the 1 4 Π u , 2 4 Π u and 3 4 Π u quartet states or after adding either the 2 4 ∑ − u , the 1 2 ∑ − u or the 2 2 Π u states. Note that the 1 4 Π u state correlates adiabatically to L1, the 2 4 Π u and 2 4 ∑ u − states to L2, and the 1 2 ∑ u − , 2 2 Π u and 3 4 Π u states to L3. We computed the state-to-state Figure 7. Threshold photoelectron photoion coincidence (TPEPICO) curves obtained for N + (red) and N 2 + (black) by selecting only the photoionization events that correlate with the emission of a threshold electron. The threshold selection is achieved using the slow photoelectron method outlined by Poully et al. [77] and the parameters are set to deliver a total resolution of 15 meV. The vertical-dashed line represents the first dissociation limit towards N + ( 3 P) + N( 4 S). Positions of the N 2 + (C 2 ∑ u + ) 0 ≤ v + ≤ 3 vibrational levels are from Yoshii et al. [75].
predissociation BRs by considering each set of electronic states and their couplings using both a time-dependent and time-independent techniques to solve the nuclear motions. Within the time dependent treatment, the evolution of the C 2 ∑ u + vibrational states is performed in the diabatic representation of the B 2 ∑ u + , 3 2 ∑ u + , 2 4 ∑ u − and 4 Π u states ( Figure 2) together with the corresponding spin-orbit and their diabatic potential couplings. This allows us to observe possible vibronic predissociation mechanisms involving either the 4 Π u states and/or the 2 ∑ u + states, in addition to the spin-orbit one.

Predissociation and time-independent dynamics of N 2 + (C 2 ∑ u + ) ions
The predissociation branching ratios of the C 2 ∑ u + state for the first 16 vibrational levels (0 ≤ v + ≤ 15) were calculated using the BCONT code [65]. BCONT allows the calculation of the isoenergetic predissociation from a discrete rovibrational level (v, J) of a bound potential energy curve to the continuum of a second dissociative state whose asymptote is situated below this level (v, J). These calculations consider the potential energy curves of the two electronic states and their spin-orbit couplings to deduce the predissociation lifetimes. For instance, the data introduced in this code are the potentials of the initial state C 2 ∑ u + and those crossing this state and which are coupled to it by spin-orbit coupling. For each vibrational level, the percentage of connection (i.e. the BRs) was calculated as follows: where x = 1, 2 or 3 and Lx = L1, L2 or L3 are the different dissociation limits and I Lx are the transition intensities assigned to each dissociation limit for each vibrational level. The data are given in Table 3.

Time-dependent wavepacket dynamics of N 2
Photodissociation dynamics is performed in the diabatic representation by taking as initial condition different vibrational eigenstates of the adiabatic C 2 ∑ u + potential energy curve with average internuclear distance centered in the inner-well. The adiabatic-to-diabatic DðRÞtransformation matrix is block diagonal with respect to ∑ and Π subspaces. For each symmetry, the corresponding block is obtained by solving the coupled equations @ R D ð�=�Þ ðRÞ ¼ F ð�=�Þ ðRÞD ð�=�Þ ðRÞ where F ð�=�Þ ðRÞ is the matrix of the radial nonadiabatic coupling elements of the first derivative @=@R in the ∑ or Π manifold. Then, the full adiabatic potential energy matrix V adia ðRÞ containing the off-diagonal spin-orbit couplings between the ∑ and Π states is transformed to the diabatic representation V dia ðRÞ ¼ D T ðRÞV adia ðRÞDðRÞ. The propagation is carried out by the split operator algorithm [78] extended to the nonadiabatic case [79]. The kinetic energy operator is applied in the impulsion and diabatic representation while the potential energy operator is expressed in the position and eigen basis set of the V adia ðRÞ matrix including the spin-orbit couplings. The computation uses a grid of 2 13 points from R = 2 to 15 bohr. The time step is 1 a.u. (0.0241 fs). An optical potential is added at the boundary for R > 12 bohr to avoid unphysical reflection on the grid limit. This induces a decrease of the wavepacket norm as soon as some component in a given electronic state reaches the asymptotic region. The propagation is continued until the residual norm becomes constant when only trapped components survive. The branching ratios are then estimated from the partial photodissociation cross sections that are obtained from the Fourier transform 1 ; tÞ in electronic state m evolving from the initial vibrational state v + in the asymptotic region (here R 1 = 10 bohr). E is here the energy above the minimum of the C 2 ∑ u + state.
The partial cross-section is given by hc is the fine structure constant, a 0 is the bohr radius, μ is the reduced mass and k m ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 2μðE À E m Þ p =h is the wave number in dissociative channel m with an asymptotic energy E m [80,81].
In order to illustrate that the results strongly depend on the completeness of the electronic basis set, we compare the cross sections corresponding to the Franck-Condon initial wave packet, which is a weighted coherent superposition of all the vibrational states of the C state. The initial wave packet is the ground vibrational state of the ground electronic state promoted in the adiabatic C state (assuming a constant transition dipolar moment in the Condon approximation). In Figure 8, we compare the results obtained with the full basis set including the three 2 ∑ u + , three 4 Π u and one 4 ∑ u − states (panels (a) and (b)) with those given in a truncated basis containing only the three 2 ∑ u + states (panels ((c) and (d)) or only the coupled B 2 ∑ u + -C 2 ∑ u + as in Ref [43] (panels (e) and (f)). This minimum basis set overestimates the role of the B state as can be seen by comparing panels (f) and (d). Moreover, in the Franck-Condon case, the spin-orbit channel plays a significant role as shown in panel (b). Note that this example is a kind of average and the relative contribution in the dissociation towards L1 may depend on the initial vibrational state. These computations also reveal that both vibronic couplings and spin-orbit couplings are involved in the predissociation. The three 2 ∑ u + states are strongly interacting by the radial couplings. Models truncating the basis to the couple B 2 ∑ u + -C 2 ∑ u + or C 2 ∑ u + -3 2 ∑ u + are not sufficient.

Results
The BRs are listed in Table 3. Although different BRs are obtained using these approaches, we should view them as complementary to fully understand the complex dynamics undertaken by the N 2 + ions in the C levels. In any case, both theoretical approaches show that the C levels (i.e. 3 ≤ v + ≤ 10), located in the energy range between L1 and L2, decay to populate L1, those between L2 and L3 (i.e. 11 ≤ v + ≤ 13) lead to L1 and L2, whereas mainly L3 is reached through the decomposition of N 2 + (C 2 ∑ u + , v + = 14, 15) ions. The 1 4 Π u and 2 4 Π u states are also crucial to explain the predissociation of the v + ≤ 13 levels to form L1 and L2 species. For the N 2 + (C 2 ∑ u + , v + = 14, 15) ions a supplementary channel leading specifically to L3 opens and leads to a different evolution partly due to shape resonances. This involves the electronic states correlating to this dissociation limit such as the 3 2 ∑ u + , 3 4 Π u , D 2 Π u and 2 4 ∑ u − states. The influence of the 3 4 Π u state or the D 2 Π u state is viewed however as negligible, and both are ruled out in the predissociation of the N 2 + (C 2 ∑ u + ) ions.

VI. The N 2 + (C 2 ∑ u + ) state full story
While the first principle computations are well-established accurate methodologies, Table 3 shows that none of these methods concord fully with the measured BRs for the C state predissociation. Nevertheless, we can consider them as complementary approaches to shed light on the complex and nontrivial mechanisms undergone by the N 2 + (C 2 ∑ u + , v + ) ions. For the v + ≤ 10 vibrational levels, the C 2 ∑ + u state relaxes either by pre-dissociation to the first dissociation limit L1 or by fluorescence to the electronic ground state of N 2 That was proved by the present and previous experimental studies [19][20][21][22][23][24][25][26][27][28][29][30] and the branching ratios calculated in this work show that this is due to predissociation leading to L1 by vibronic interaction with the B 2 ∑ u + and by spin-orbit coupling with 1 4 Π u . The BR values estimated from the final population in the Π states are in a good agreement with those measured presently and the Nicolas et al. [28] ones. For v + = 11 and 12, the predissociation of the C 2 ∑ + u state is distributed between the two first dissociation limits L1 and L2. Experimentally, we determined BRs of ~74% for L1 and ~26% for L2, while time-independent approach and time-dependent simulation without the B 2 ∑ + u state lead to ~100% for L1. For the v + = 12 level, both theoretical percentages are distributed between the L1 and L2 limits with a predominance of L1. This is attributed mainly to the predissociation by the 1 4 Π u and 2 4 Π u states. The measured BRs for this vibrational level are however of ~50% for each limit. For v + = 13, we measure 41.7% to L1, 54.7% to L2 and a small percentage (~3.6%) for the third dissociation limit L3. These values are in close agreement with those computed here by the time-independent method taking into account the spin-orbit predissociation of the C state by the 4 Π u and 2 4 ∑ u − states (Figure 2). For the v + = 14 and v + = 15 vibrational levels, experimental measurements show a dissociation predominantly to the third limit with small contributions to L2 and L1, in very good agreement with our both theoretical predictions. The computations show that the spin-orbit coupling is less efficient here and that predissociation due to vibronic coupling is instead in action. Indeed, the spin-orbit couplings were not enough to change the calculated branching ratios. The N 2 + (C 2 ∑ u + ) predissociative system is difficult to analyze by timedependent methods because the profile of the potential energy curves of ∑ and Π symmetry is favorable to trapping part of the wavepackets in the bound region leading to long lifetimes. Figure 9 gives the probability of occupying each adiabatic electronic state when the system is prepared in Figure 9. Time evolution of the wavepacket initially prepared in the v + = 0, 4, 10, 11, 12, 13, 14 and 15 vibrational levels of the C 2 ∑ u + state of N 2 + populating the C 2 ∑ u + , B 2 ∑ u + , 3 2 ∑ u + , 1 4 Π u , 2 4 Π u , 3 4 Π u and 2 4 ∑ u − states. These evolutions are confirmed by longer dynamics (not shown). For v + = 15, the populations are almost those at t = 12 ps. different vibrational states localized in the inner well of the C 2 ∑ u + . This reveals the early dynamics. The most remarkable observation is the very fast occupation of the three C 2 ∑ u + , B 2 ∑ u + and 3 2 ∑ u + states due to the vibronic coupling for any initial state. The three states are populated directly during the first femtoseconds with slightly different weights with respect to v + and Rabi oscillations of weak amplitudes persist during the whole process. This non-adiabatic interaction is responsible of a part of the leakage towards L1 or L3. Spin-orbit coupling compete with this mechanism and allow the evolution towards the three L1, L2 and L3 asymptotes. The populations in the different states is not sufficient to extract the branching ratio among the dissociation channels due to the trapped components, which can decay by fluorescence only. Insight on the photodissociation is given by the timedependent localization probability χ v þ m ðR 1 ; tÞ � � � � 2 at a large internuclear distance (R 1 = 10 bohr here) and more precisely by the integrated cross sections extracted from each component χ v þ m ðR 1 ; tÞ. The branching ratio (see Table 3) is obtained by summing the contribution coming from the correlated channels (B 2 ∑ u + and 1 4 Π u for L1, 2 4 Π u and 2 4 ∑ u − for L2, C 2 ∑ u + , 3 2 ∑ u + and 3 4 Π u for L3). The decay towards the B 2 ∑ u + by vibronic coupling confirms the contribution of this state in the predissociation via the L1 asymptote but it is not the only one since transition via the 1 4 Π u also occurs.
The second noteworthy point is the large branching ratio towards L3 and the fast decay of the v + = 14 and v + =15 states. For v + = 14, the full decay requires ~600 ps. For v + = 15 level, the decay is even shorter and it needs ~6 ps. The difference of behavior is due to the shape of the vibrational initial states. As shown in Figure 10, the vibrational states with v + < 14 form a progression of successive bound states in the inner well of the C state (with 2 ≤ <R NN > ≤ 4 bohr) while the v + = 14 and v + =15 states are shape resonances near the top of the barrier. To illustrate this, Figure 10(a) gives the initial average interatomic distances of the eigenstates of the adiabatic C state. The states of the continuum are discretized by the diagonalization in a finite basis set. We clearly see the eigenstates that correspond to the inner well of the C state (with 2 ≤ <R NN > ≤ 4 bohr) and those which correspond to the outer well (with 4 ≤ <R NN > ≤ 6 bohr). We can see also the quasi continuum eigenstates where the value of the average distance (<R NN > ≥ 8 bohr) is due to the finished grid used. Indeed, the fact that the average is large proves that these are discretized states of the continuum. Note that the parts located in the C state inner part potential are the only ones for which the Franck-Condon factors are large starting from N 2 (X 1 ∑ g + , v" = 0). Afterwards, the levels v + = 14 and v + = 15 are shape resonant states in the continuum of state C 2 ∑ u + (Figure 9(e,f)) continuing the series of bound states in the inner-well. For the v + = 14 level, we see that, although the wavefunction part trapped in the inner-well of the C state exhibits 14 nodes, noticeable oscillations in the dissociation continuum, i.e., at long internuclear distances, are observed ( Figure 10(e)). A similar discussion applies for the v + = 15 wavefunction (Figure 10(f)), where the area under these oscillations is even larger. These eigenvectors of the C adiabatic state are the only ones having a notable Franck-Condon factor above the L3 dissociation limit (note that Baltzer et al. [73] observe three more resonances above v + = 15). They are characterized by their localization inside the inner well and a weak amplitude beyond the barrier. All the neighboring eigenvectors have the opposite behavior with a large amplitude outside the barrier and a very weak amplitude in the well. This strong structuration of the eigenvectors is typical of a shape resonance due to tunneling. For instance, we clearly observe a strong variation of the shape of the eigenfunctions close to the barrier, which is typical of what is currently called a shape resonance due to a barrier. It is well-known that below and above the resonance energy the eigenvectors have a large amplitude in the outer region and a very weak amplitude inside the well while it is the contrary close to the resonance. As Figure 10. In A): Average position of the first 100 vibrational eigenstates of the C 2 ∑ u + state with a discretization of the dissociation continuum. In the blue region, we give those located mainly in the inner potential well of the C state (i.e. v + = 0-15). The v + quantum number does not correspond to the eigenstate number for v + > 10 but continues the series of eigenstates located in the inner well with an increasing number of nodes. The v + = 0-15 levels are marked by red crosses. In the red region, those located in the outer potential well. In the yellow region, those delocalised in the continum (but artificially localized at long distance due to the finite grid). In the inset, we give an enlargement for the 20 first eigenstates where we can distinguish the v + =0 -12 eigenstates located in the inner potential well of the C state (red region). In B-F), we plot the vibrational wavefunctions of the N 2 + (C 2 ∑ u + ) state for v + = 0, 4, 13, 14 and 15, respectively. The energies of these levels with respect to the vibrationless N 2 ground state are also given. See text for more details.
this is an adiabatic view, this is not sufficient to describe the dissociation. The prompt direct population of the C asymptotic region by tunneling is only one of the processes that compete with vibronic and spin-orbit transition towards other electronic states. This direct tunneling component explains the faster decay of these states.
In sum, the dynamics for the upper C state vibrational levels was never fully described before by enlightening the competing role of vibronic couplings among the three 2 ∑ u + states and spin-orbit interaction with the 2Π u and 2 4 ∑ u − states. It is the first time that a resonance-induced dissociation is documented for dissociative photoionization.

VII. Conclusion
In addition to previous experimental and theoretical findings dynamics, we have combined direct state-of-the-art experimental measurements and high ab initio calculations to highlight the roles of the excited states of N 2 + in the predissociation phenomena of the C 2 ∑ u + state. The experiments consisted on the detection of the N + + N fragments formed after single photon ionization into state-selected vibrational states of N 2 + and the deduction of the state-to-state branching ratios populating the three lowest dissociation limits of N 2 + via the decomposition of the N 2 + (C 2 ∑ u + ) ions. These branching ratios were also computed using both time-dependent and timeindependent complementary approaches where the nuclear motions were treated in a full new set of N 2 + PECs lying close to the C state. The comparison of experimental data and those derived from these computations exposes the competing processes between vibronic couplings allowing dissociation towards L1 via the B 2 ∑ u + state and the spin-orbit leading to the 4 Π u states for the low levels, whereas a new mechanism was identified for the upper ones. Indeed, the v + = 14, 15 levels are identified as resonances, whose dissociation proceeds via vibronic coupling to the 3 2 ∑ u + state and spin-orbit coupling with the D 2 Π u and 2 4 ∑ u − states forming directly and dominantly L3 products in good agreement with the experiments.
Although we treated the dissociative single photon ionization of a small (diatomic) molecule, the processes described here are complex. Our work showed that neither the time dependent nor the time independent theoretical approaches can give a complete vision on these complex dynamics undertaken after ionizing N 2 . In sum, we showed that one needs an interplay between diverse computational and experimental investigations to disentangle the complex dynamics of the predissociation of the N 2 + (C 2 ∑ u + ), which was pointed out more than 60 years ago. This should help in the correct modeling of these important phenomena in atmospheric, planetary and plasma media. Similar complex situations can be found for other important molecular systems. For instance, the present findings may help to fully understand the isotopic effects observed recently in the ultrafast dynamics of dissociative photoionization of N 2 and the unexpected branching ratios measured recently during the dissociation dynamics of N 2 + induced by isolated attosecond XUV pulses in combination with few-optical-cycle near-infrared/visible (NIR/VIS) [30]. Similar approaches can be applied also to explain the isotopic effects in the VUV photodissociation of the N 2 and CO molecules [16,82] and that of larger molecular systems relevant for astrophysical, planetary, and atmospheric media and having environmental impacts such as HNCO [83].