Photodissociation dynamics of N$_{3}^{+}$

The photodissociation dynamics of N$_3^+$ excited from its $^3 \Sigma_{\rm g}^{-}$ ground to the first excited singlet and triplet states is investigated. Three dimensional potential energy surfaces for the $^1$A$'$, $^1$A$''$, and $^3$A$'$ electronic states, correlating with the $^1 \Delta_{\rm g}$ and $^3 \Pi_{\rm u}$ states in linear geometry, for N$_3^+$ are constructed using high level electronic structure calculations and represented as reproducing kernels. The reference {\it ab initio} energies are calculated at the MRCI+Q/aug-cc-pVTZ level of theory. For following the photodissociation dynamics in the excited states, rotational and vibrational distributions $P(v')$ and $P(j')$ for the N$_2$ product are determined from vertically excited ground state distributions. Due to the different shapes of the ground state $^3$A$^{''}$ PES and the excited states, appreciable angular momentum $j' \sim 60$ is generated in the diatomic fragments. The lifetimes in the excited states extend to at least 50 ps. Notably, results from sampling initial conditions from a thermal ensemble and from the Wigner distribution of the ground state wavefunction are comparable.


I. INTRODUCTION
Many important processes occurring in the earth's atmosphere involve nitrogen-containing species as it is the most abundant element in the medium. Among these are charge transfer processes such as N( 4 S) + N + 2 (X 2 Σ + g ) ↔ N + ( 3 P) + N 2 (X 1 Σ + g ) that proceed via the formation of the N + 3 radical cation. An early study utilized the complete active space self-consistent field (CASSCF) approach and multireference configuration interaction method (MRCI) to calculate vertical excitation energies and specify the collinear dissociation paths of the electronically excited states of the N + 3 ion. 1 Given its importance in a variety of atmospheric reactions, the photodissociation dynamics of the N + 3 ion warrants a detailed investigation.
Reactions involving N + 3 are relevant in air plasmas at elevated temperatures. Under such conditions, N + 3 can serve as an N + donor to species such as NO, O 2 , SO 2 , N 2 O, CO 2 , or CO. [2][3][4][5] Thus, decomposition of N + 3 into N + + N 2 is of particular relevance in view of the general reaction scheme N + 3 + M → NM + + N 2 with subsequent decomposition of NM + . Species of particular importance that are formed through N + transfer reactions are NCO + (N + 3 + CO → NCO + + N 2 ) and NCO + 2 (N + 3 + CO 2 → NCO + 2 + N 2 ). 2 Furthermore, N + 3 together with N + 4 have been suggested to be the most abundant nitrogen-containing ions in the lower atmosphere of Titan where the N + 3 ion drives several reactions involving CH 4 , C 2 H 2 or C 2 H 4 through N + atom transfer. 6 Photodissociation reactions are induced by the absorption of one or more photons by a chemical species which can then disintegrate to form products. In a typical photolytic reaction, a reactant ABC with internal energy E int absorbs a photon with energy hν to form a transient activated complex [ABC]* which can undergo several types of primary photochemical processes followed by secondary processes. One possible outcome is the sub- A dynamics study for a chemical reaction usually begins with the generation of an accurate potential energy surface (PES) for the species. For the system of interest, 1-and 2-dimensional sections through the 3-dimensional PESs for the ground and electronically excited states have been previously determined at different levels of theory. 1,7,8 In addition, one-and 3-dimensional PESs for cyclic-N + 3 were determined, [9][10][11] and the vibrational levels up to ∼ 2 eV were computed. 10,11 Finally, ab initio MD simulations for the vibrational spectroscopy of linear and cyclic N + 3 have been carried out. 12 These studies were concerned with the electronic spectroscopy and the lower vibrational states on the ground state PES.
However, a full-dimensional PES for the electronic ground state has only become available recently. 13 The construction of high-dimensional PESs is a challenging problem for which machine learning-based approaches like neural networks or kernel-based methods have found wide applicability in recent times. [14][15][16][17] One such method, rooted in the theory of reproducing kernel Hilbert spaces (RKHSs), 18 is suitable for obtaining reliable PESs. For this, a training set of data is generated using electronic structure calculations which is then used to "train" the algorithm to produce a continuous surface by interpolating smoothly between the data points. Here this 'training set' constitutes the total electronic energy evaluated at various configurations using high-level electronic structure calculations. This method was utilized in previous work for the construction of an accurate PES for the ground state of N + 3 at the multi reference configuration interaction (MRCI) level of theory. 13 A similar methodology has been followed for the construction of high quality PESs for other triatomic species such as the [CNO] system, 19 N − 3 , 20 and NO 2 . 21 In the current work the focus is on those excited states of the N + 3 cation which are expected to be accessible during photoexcitation. Photodissociation reactions of the N + 3 ion involve transitions between the 3 Σ − g ( 3 A in bent geometry) ground state and energetically accessible excited states. The 1 A , 1 A and 3 A excited states of N + 3 are expected to play a role in dynamical processes owing to their energetic proximity to the ground state ( 3 Σ − g ).
The present work is structured as follows. First, the methods are presented. Next, the quality of the PESs is discussed and the results from the dynamics simulations for pho-toexcitation to the 3 A and the two singlet states are reported. Next, simulations starting from sampling the Wigner distribution of the ground state wavefunction are compared with the more conventional thermal initial conditions. Finally, the results are discussed and conclusions are drawn.

II. METHODS
First, the methods for generating and representing the potential energy surfaces are summarized. Following this, the QCT and quantum simulations for studying N + 3 →N 2 +N + photodissociation are described.

A. The N + 3 Potential Energy Surfaces
The 3-dimensional PESs for the 1 A , 1 A , and 3 A states of N + 3 were computed at the MRCISD 22,23 /aug-cc-pVTZ 24 level of theory with the Davidson quadruples correction 25 (MRCISD+Q) based on a CASSCF 26-29 reference wave function. Also, the 2d-PES at r NN = 2.25 a 0 (see below) for the 2 3 A state was determined. All electronic structure calculations were carried out using the Molpro 2019.1 program. 30,31 The active space for CASSCF included the full valence space. State-averaged (SA) calculations were carried out using 8 states in total, including the two lowest states of the 3 A (ground state), 1 A , 1 A , and 3 A symmetries, respectively.
For the electronic structure calculations, the grid was defined in Jacobi coordinates (R, r, θ) whereby r is the separation between nitrogen N1 and N2, R is the distance between N3 and the center of mass of N1 and N2, and θ is the angle between r and R; see Figure 2.
The grid includes distances r ∈ [1.55, 4.00] a 0 , R ∈ [1.5, 10.0] a 0 , and θ ∈ [0, 180 • ] from a 7-point Legendre quadrature. The products of the photodissociation reaction of N + 3 are N 2 and N + . Thus, the energy of the system for E(R → ∞, r) is the sum of the energies of the dissociation products. In the following, the "zero" of energy is set to this value according For the dynamics simulations the energies on the grid are most conveniently represented in a way that allows evaluation of energies and analytical gradients for arbitrary conformations.
Here, a reproducing kernel Hilbert space (RKHS) representation is employed. 14,18 According to the representer theorem 32 , a function f (x) for which values f (x i ) are given for arguments x, can always be approximated as a linear combination Here, K(x, x ) is a kernel function and α i are coefficients to be determined from matrix inversion. If the inner product φ(x), φ(x ) can be written as K(x, x ), the function is called a reproducing kernel of a Hilbert space H. 33 Here, reciprocal power decay kernel polynomials are used for the radial coordinates. For the R−coordinate kernel functions (k [n,m] ) with smoothness n = 2 and asymptotic decay m = 4 are employed, while n = 2 and m = 6 is used for the r dimension k [2,6] (x, x ) = 1 14 In both expressions, x > and x < are the larger and smaller values of x and x , respectively.
Such a kernel smoothly decays to zero giving the correct long-range behavior for atom-diatom type interactions. For the angle θ a Taylor spline kernel is used: Here, z > and z < are the larger and smaller values of z and z , respectively, and z is defined with z ∈ [0, 1]. Combination of the three 1-dimensional kernels leads to K(x, x ) = k [2,4] (R, R )k [2,6] (r, r )k [2] (z, z ), where, x, x are (R, r, z) and (R , r , z ), respectively. The coefficients α i and the RKHS representation of the PES are evaluated by using a computationally efficient toolkit. 14 The global reactive PES for each excited electronic state is constructed by smoothly connecting the three PESs for the three symmetry-equivalent reaction channels using a switching function, where the switching function ω j has an exponential form, Such a mixing using normalized weights is akin to that used in multi surface reactive MD. 34,35 The mixing parameters ρ j for each channel are obtained using least squares fitting.
For the 3 A ground state of N + 3 , the switching parameters are ρ = (0.65, 0.65, 0.65) a 0 . 13 The root mean squared error between the two sets of data is ∼ 0.

B. Quasi-Classical Trajectory Simulations
The QCT simulations used in the present work have been extensively described in the literature. 19,[36][37][38] Here, Hamilton's equations of motion are solved using a fourth-order Runge-Kutta numerical method. The time step was ∆t = 0.05 fs which guarantees conservation of the total energy and angular momentum.
For the photodissociation simulations, structures in the vicinity of the 3 A ground state PES and velocities were generated by drawing from a Maxwell-Boltzmann distribution at temperatures between 500 K and 3000 K. Each of the 500000 initial conditions was propagated for 50 ps on the ground state PES and the final positions and velocities were saved.
For a view of the ensemble of structures; see Figure S1. Following this, the entire population is projected vertically to the excited state PESs. Trajectories on the excited states are run until dissociation into products occurs or for a maximum of 50 ps. Configurations initially located around the ground state minima land in the vicinity of a potential well when projected onto the 3 A PES. Subsequent trajectories are initially confined in the region around the minima before dissociating into products. Examples for photodissociating trajectories from initial velocities generated at 1000 K on the 3 A PES are shown in Figure S2.
The product ro-vibrational states are determined following semiclassical quantization. Since the ro-vibrational states of the product diatom are continuous numbers, the states need to be assigned to integer quantum numbers for which a Gaussian binning (GB) scheme was used. For this, Gaussian weights centered around the integer values with a full width at half maximum of 0.1 were used. [38][39][40] It is noted that using histogram binning (HB) was found to give comparable results for a similar system.
and bending coordinates N1N2N3 using a Gaussian kernel-based interpolation method. Here, the triatomic is denoted as N1-N2-N3. This transformation allows the approximate assignment of the wavefunctions using quantum numbers obtained by counting the nodal planes along each coordinate.

A. The Potential Energy Surfaces of the Excited States
In the following the quality and topology of the excited state PESs determined in the present work is described. The 1 3 A ground state PES, correlating with the 3 Σ − g state in linear geometry and dissociating to N + ( 3 P)+N 2 (X 1 Σ + g ), was already discussed in earlier work. 13 Next up in energy are the 1 A and 1 A states (see left panel in Figure 1), which are degenerate for θ = 0, correlate with the 1 ∆ g state and dissociate to N + ( 1 D)+N 2 (X 1 Σ + g ). Upon bending away from the linear structure the degeneracy is lifted and two states -1 A and 1 A -emerge as is shown in Figure 1 middle and right panels. The overall shapes of the two PESs for the 1 A states follow that of the 3 A ground state PES.
At yet higher energy and most relevant for the photodissociation dynamics considered later, are the 3 A and 2 3 A states which correlate with the 3 Π u state in linear geometry. Both states dissociate to N + ( 3 P)+N 2 (X 1 Σ + g ), i.e. they have the same asymptote as the 1 3 A ( 3 Σ − g ) ground state. The 3 A state has a double minimum PES with a first minimum at R = 3.365 a 0 which is higher in energy than the corresponding N 2 ( 1 Σ + g )+N + ( 3 P) asymptote by 0.328 eV and separated from it by a barrier of 2.351 eV in the linear configuration; see The vertical excitation energies for the 1 ∆ g ← 3 Σ − g and the 3 Π u ← 3 Σ − g transitions have been determined or estimated from experiments. 42 For the 1 ∆ g ← 3 Σ − g transition the excitation energy estimated from photoelectron spectroscopy is 1.13 eV, and for the 3 Π u ← 3 Σ − g transition a laser spectroscopy study yielded an excitation energy of 4.54 eV. 43   with the 3 Π u state. In all panels the N1-N2 diatomic separation is fixed at r = 2.25 a 0 . The asymptote for the triplet states is E N 2 + E N + ( 3 P) and for the singlet states it is All energies are reported in eV. The bottom right panel shows the Jacobi coordinates (R, r, θ) used for describing the 3-dimensional PESs for N + 3 . The vector r is the N1-N2 separation, R is the separation between N3 and the center of mass of N1-N2, and θ is the angle between the two distance vectors.   their representation as a RKHS is given in Figure 3.  The crossing seams between the two singlet electronic states, which are degenerate in the linear configuration, are reported in Figure S3. The geometries at which the two states cross were stored whenever the energy difference between the 1 A and 1 A states was smaller than 10 −5 eV for a given geometry. Crossing seams are shown for three different values of the angle θ. Thus, for the excited vibrational states, there is a possibility of nonadiabatic transitions between the 1 A and 1 A states.
The photodissociation dynamics on the excited 3 A state is studied by evaluating the vibrational and rotational state distributions of the products. The product final state distributions from initial conditions generated at the different temperatures are shown in Figure   4. Vertical photoexcitation of a thermalized ensemble of molecules on the ground electronic state results in their promotion to the lowest energy vibrational level of the excited state.
As temperature increases higher v values become populated gradually with v max = 4. In Trajectories initialized from θ = 90 • to 180 • follow similar dynamical paths on account of the symmetry of the PES. The distribution of lifetimes P (τ ) of the N + 3 complex in Figure  4 indicates that a large fraction of the trajectories leads to photodissociation within 5 ps after excitation. However, there is also a number of trajectories with considerably longer lifetimes. Figure S4 shows one such trajectory at 1000 K with a lifetime of 17 ps.
Up to this point the entire ground state population was projected onto the excited state PES and the dynamics was followed. In other words, it was assumed that the resonance condition, hν = E photon = E es − E gs = ∆V , is always fulfilled. Here, hν is the energy of the incoming photon and E gs and E es are the energies of the molecule in the ground and excited states, respectively. However, experimentally typically only a fraction of the population is promoted from the lower to the upper state. Such processes constitute a subset of the trajectories discussed so far and are discussed next. An examination of the PESs corresponding to θ ≈ 0 • in Figure 1  of the ensemble (green) leads to a higher probability for longer lifetimes but the shape of the distribution is similar to that for exciting the entire ground state population (blue). For the linear geometry the 1 A and 1 A PESs are degenerate. As the middle row of Figure   2 shows, the most notable difference between the two singlet states is the presence of a potential well near θ = 90 • for the 1 A state. Thus, a trajectory starting with θ ≈ 0 • or 180 • on the 1 A state remains confined in the neighbourhood of the potential wells until sufficient energy has accumulated along the dissociative coordinate(s) to decompose whereas a trajectory with similar initial conditions but propagating on the 1 A state PES may potentially travel towards θ = 90 • on account of the presence of a minimum at this position.
However, the barrier to reach this minimum is ∼ 2.75 eV (see Table I) which is too high to be overcome at the conditions studied here. Nevertheless, away from the global minimum the shapes of the two singlet PESs differ somewhat.
The final state vibrational distributions from dynamics on the 1 A PES are dominated by

IV. DISCUSSION AND CONCLUSION
So far, the photodissociation dynamics of the N + 3 ion in the lowest singlet and triplet excited states was followed based on QCT simulations. Consistent with the shape differences between the 3 Σ − g ground state PES and the three electronically excited PESs high rotational excitations in j after photodissociation are found. To further corroborate this finding, simulations starting from sampling the Wigner distribution 44,45 of the 3 Σ − g ground state wavefunction were also carried out. The Wigner function f W (Γ) related to a three dimensional wavefunction in Jacobi coordinates Ψ(R, r, θ) is with Γ = (R, r, θ, P, p, P θ ). Initial conditions for photodissociation are generated by sampling the probability distribution f W (Γ) in Eq.10 using Metropolis Monte Carlo importance sampling. Here, Ψ(R, r, θ) is the ground state wavefunction for the 3 A PES. The collection of initial conditions generated on the ground state PES in this manner represent the quantum wavepacket which is then projected onto the excited state as before and the dynamics is followed from QCT simulations.
The resulting final state distributions are shown in Figure 6. The quantum ground state is 1421 cm −1 above the minimum energy of the PES, corresponding to 2000 K. Final state distributions from classical trajectories sampled from the Wigner distribution follow closely those from classical simulations with initial conditions at low temperature (500 K). Since "temperature" is not a meaningful physical quantity for such small systems, 46 T is rather more a label to distinguish how initial conditions were generated for ensembles which are increasingly energized. Given this, it is encouraging to see that final state distributions from classical trajectories starting from two very different strategies to generate initial conditions are consistent with one another.
To corroborate the pronounced coupling of the intermolecular modes it is also useful to consider the lower quantum bound states for the different PESs. For this, the 3-dimensional time-independent Schrödinger equation was solved for the lowest bound vibrational states on the the two lowest singlet electronically excited PESs, neglecting Renner-Teller coupling.  Table   II. It must be noted that the assignments are rather approximate due to the anharmonicities and strong couplings prevalent among the vibrational levels of the excited states. This can also be seen for the representative wavefunctions for lower energy levels of the 1 A and 1 A states shown in Figures S5 and Figure S6 respectively. Similar calculations for the triplet state 3 A were not performed as it predissociates and requires full scattering calculations, which are outside the scope of the present work.
The current work investigates the photodissociation dynamics of the N + 3 ion for the three lowest electronically excited states. For this, an ensemble of initial conditions is generated on the ground state PES and projected onto each of the excited states. For the 3 A PES, it is The labels are only approximate due to strong coupling. The angular momentum quantum number

SUPPORTING INFORMATION
The supporting information reports the initial conditions on the ground state PES, individual photodissociating trajectories, the crossing seams between the singlet PES, and wavefunctions for the fundamentals on the two singlet PESs.

DATA AVAILABILITY STATEMENT
All information necessary to construct the potential energy surfaces is available at https://github.com/MMunibas/N3p_PESs.    The states are indicated by the labels (ν 1 ν l 2 ν 3 ).