Stereodynamical Control of Nonadiabatic Quenching Dynamics of OH(A2S+) by H2: Full- Dimensional Quantum Dynamics


 The breakdown of the Born-Oppenheimer approximation is omnipresent in chemistry, but our detailed understanding of the nonadiabatic dynamics is still incomplete. In the present work, nonadiabatic quenching of electronically excited OH(A2S+) molecules by H2 molecules is investigated by a full-dimensional quantum dynamical method using a high quality diabatic potential energy matrix. Good agreement with experiment is found for the OH(X2P) ro-vibrational and L-doublet distributions. Furthermore, the nonadiabatic dynamics is shown to be controlled by stereodynamics, namely the orientation of the two reactants. The uncovering of a major (in)elastic channel, neglected in all previous analyses, resolves a long-standing experiment-theory disagreement concerning the branching ratio of the two electronic quenching channels.


Introduction
Although the Born-Oppenheimer (BO) approximation, 1 which assumes separability of nuclear and electronic motion, is widely accepted for characterizing reactions in their ground electronic states, there is general agreement that reaction dynamics can be signi cantly impacted by excited electronic states near an electronic degeneracy, where the motion of the electrons is strongly coupled with that of the nuclei.
While ultrafast nonadiabatic transitions near conical intersections (CIs) have been intensively studied in photochemistry, 2-8 non-BO effects have seldom been investigated in detail for collisions and bimolecular reactions. 9 Existing rst-principles theory of nonadiabatic reaction dynamics have mostly dealt with open-shell atoms, focusing on geometric phase effects [10][11][12] or spin-orbit excited electronic states. [13][14][15][16][17][18] In the present work, we extend the detailed rst-principles theory to the collision of electronically excited molecules, namely the hydroxyl radical in the OH(A 2 Σ + ) state, by studying the following processes in full dimensionality: OH(A 2 Σ + ) + H 2 → H + H 2 O (reactive quenching) (R1a) → OH(X 2 Π) + H 2 (non-reactive quenching) (R1b) → OH(A 2 Σ + ) + H 2 (elastic and inelastic scattering) (R1c) Here, both non-radiative quenching channels (R1a and R1b) necessarily require transitions from a higher to lower electronic state, due to nonadiabatic couplings near degeneracies such as conical intersections (CIs). 19,20 This system not only offers a prototype for understanding fundamental nonadiabatic dynamics in bimolecular collisions, but is also of great practical relevance to the laser-induced uorescence (LIF) monitoring of the omnipresent OH radicals in atmospheric chemistry and in combustion. 21,22 Pioneering experiments by Lester and coworkers identi ed the reactive quenching channel (R1a) 23 and measured the kinetic energy release of the hydrogen co-product. [24][25][26] These kinetic energy distributions were found to be bimodal, suggesting complex dynamics with at least two reaction pathways. These intrabeam measurements were con rmed by a crossed molecular beam experiment by Ortiz-Suárez et al. 27 Later experiments by the Lester group investigated the nonreactive quenching channel (R1b) with quantum state resolution. 25,[28][29][30] The OH(X 2 Π) product was found to be vibrationally cold but rotationally hot. A propensity for the A′ component of the OH(X 2 Π) Λ-doublet was also observed.
Furthermore, this group reported the branching ratio between the R1a and R1b channels, which favors the former. 25 The existence of quantum state-resolved experimental data makes this system a fertile proving ground for theoretical understanding of nonadiabatic dynamics in bimolecular collisions.
Several electronic states are involved in the nonadiabatic quenching channels. In the entrance channel, the OH(A 2 Σ + ) + H 2 asymptote correlates adiabatically with the 3 2 A state, while the doubly degenerate OH(X 2 Π) + H 2 asymptote correlates with the 1 2 A and 2 2 A states. In the reactive quenching channel, the 1 2 A state correlates adiabatically with the H 2 O + H asymptote. For C s geometries, the 1 2 A and 2 2 A states carry irreducible representations (irreps) A′′ and A′, while the higher 3 2 A state belongs to the A′ irrep. Early ab initio calculations identi ed a T-shaped (C 2v ) CI seam between the 3 2 A( 2 A 1 ) and 2 2 A( 2 B 2 ) states, with OH pointing its O-end to H 2 . 31 It was hypothesized that this CI was responsible for e cient quenching of OH(A 2 Σ + ), leading to both reactive and nonreactive quenching channels. Subsequent studies by Yarkony and coworkers revealed, however, that this CI seam actually spans the entire planar (C s ) geometry, extending from the C 2v to C ∞ v geometries, 32,33 where the irreps of the two states involved are A′/A′, B 2 /A 1 , and Π/Σ, respectively. In Fig. 1(a), this con uence of the CI seam is shown as a function of the H 2 -OH distance and the H 2 rotational angle, with O always pointing to H 2 . This planar CI seam is responsible for nonadiabatic transitions involved in R1a and R1b, and represents the focal point of the current dynamics investigation. Further studies by Dillon and Yarkony explored the non-planar portion of the con guration space. They identi ed an additional CI seam between the 1 2 A and 2 2 A states, facilitating a non-planar insertion pathway of HO into H 2 , 34,35 leading to H and H 2 O. This CI seam facilitates the population of the A′ and A′′ Λ-doublet states of the OH(X 2 Π) product, and presumably also for the bimodality of the H kinetic energy distribution in the R1a channel.
Based on this electronic structure understanding, several multi-dimensional coupled potential energy surfaces (PESs) have been reported. 29,[36][37][38][39][40] For nonadiabatic dynamics involving more than one electronic state, it is advantageous to express the Hamiltonian in a diabatic representation 5,41 to avoid singularities in the kinetic energy operator and cusps in the potential energy operator at the CI seams. While the resulting diabatic potential energy matrix (DPEM) is di cult to construct from adiabatic electronic structure calculations, once established, it greatly simpli es the dynamical calculations. To this end, reduced-and full-dimensional DPEMs considering two, 29, 37 three, 38,39 and four 40 electronic states have been reported. Based on a large number of multi-reference con guration interaction calculations, the latest global DPEM offers the highest delity in reproducing ab initio energies of the four lowest states and their couplings. 40 These high-quality DPEMs have opened the door for dynamical studies. [36][37][38]42 While the dynamics can only be accurately characterized quantum mechanically, such calculations are challenging because of the large energy release (> 4 eV), a large accessible phase space, and the complex multi-state dynamics. 9 Until now, quantum dynamics calculations have been restricted to planar geometries with two electronic states. 37,42 However, such a model is insu cient 38 since it neglects the 2 2 A state and important nonplanar dynamics. On the other hand, full-dimensional trajectory surface hopping (TSH) studies have been employed to gain insights into the quenching events, but the results had no quantum state resolution.
Interestingly, the TSH results favored the nonreactive quenching channel, in sharp contrast to the experiment. 25 The R1a/R1b branching ratio is a fundamental quantity in this nonadiabatic process, the experimental/theoretical discrepancy is therefore of considerable importance. It could be due to the semiclassical treatment of the nonadiabatic dynamics and/or inaccuracies in the DPEM. This issue is thus only resolvable by a full-dimensional quantum dynamics study on a globally accurate DPEM.
In this work, we report the rst full-dimensional investigation of the nonadiabatic collisional quenching of OH(A 2 Σ + ) by H 2 using a time-dependent quantum wave packet method on the recently developed DPEM. 40 We aim to resolve the aforementioned experiment-theory discrepancy, to validate the DPEM by comparing quantum state resolved product distributions with experiment, and to gain insight into the (stereo)dynamics of this prototypical nonadiabatic process.

Results
The calculated fraction for the nonreactive quenching channel (f R1b ) at the collision energy of 0.05 eV is 0.123, and that for reactive quenching R1a is f R1a = 0.100. The former is in excellent agreement with the experimental value (0.12 (5)). 25 However, our results suggest that nonreactive quenching is slightly favored over reactive quenching (R1a) at this energy. The preference to the R1b channel increases with the increasing collision energy, as the data in Table 1 show. This preference, consistent with the earlier TSH results based on a DPEM of Collins et al., 38 is in sharp contrast to the experimental report by Dempsey et al., 25 in which the reactive quenching channel dominates. As discussed in the next section, we attribute the experiment-theory discrepancy to the large yield of the R1c channel, which was neglected in the previous determination of the branching ratio. In the same Table, TSH results with zero impact parameter (b = 0) are also listed for comparison, and they provide a qualitatively but not quantitatively correct description of the process.
The internal state distributions of the OH(X 2 Π) product were calculated at the three collision energies. Figure 2 displays the OH(X 2 Π) ro-vibrational state-resolved probabilities for both the A′ and A′′ components of its Λ-doublet. The present calculations showed that the OH(X 2 Π) products are dominantly in the ground vibrational state with a broad rotational state distribution peaking at N OH =17, in good agreement with experimental observations, 28 shown in the top panels of the same gure. The peak of the distribution shifts towards higher N OH with increasing collision energy. Furthermore, Lester and coworkers reported that the OH(X 2 Π) products are mainly formed in the A′ component of this degenerate electronic state, 28 a trend reproduced by our calculated distributions.
The H 2 ro-vibrational state distribution has also been calculated, although there are currently no experimental data to compare. In Fig. 3, the nal H 2 vibrational and rotational state-resolved probabilities are displayed. The H 2 products are both vibrationally and rotationally hot, and only ortho-H 2 is formed.
The absence of the para-H 2 can be attributed to the permutation symmetry of the DPEM, as discussed in more detail in the next section. The H 2 vibrational state distribution extends to higher vibrational states when the collision energy is increased.

Discussion
Our wave packet calculations revealed that the fate of the collision between OH(A 2 Σ + ) and H 2 is strongly controlled by stereodynamics, which has important consequences in the quenching process. As recognized before, 31 the OH(A 2 Σ + ) + H 2 entrance channel on the excited 3 2 A state with the H-end of OH pointing towards H 2 () features barrierless access to a T-shaped van der Waals (vdW) well, which has a depth of 0.34 eV. The spectrum and lifetimes of predissociative states in this well have been a subject of extensive studies by Lester and coworkers. 31 However, this approach has no easy access to the CI seam because of a large barrier separating the vdW well from the upper cone of the CI, which corresponds to changing from to , as illustrated in Fig. 1(b) and in more detail in Figure S1 in Supporting Information (SI). As a result, the wave packet is largely re ected back to the OH(A 2 Σ + ) + H 2 channel by the repulsive wall. Noting that the O-H and H-H bond lengths in the vdW well (1.975 a.u. and 1.461 a.u.) are essentially the same as those of the free molecules, there is thus little vibrational excitation. The T-shape also dictates that rotational excitation is minimal because no torque results from momentum transfer. These features are shown in Figure S2 in SI. Importantly, this particular steric approach leads to the large R1c yield shown in Table 1, although the dominance of this (in)elastic channel is weakened at higher energies.
On the other hand, the upper cone of the 3 2 A-2 2 A CI can be accessed, also without a barrier as shown in Figs. 1(a) and 1(b), when OH(A 2 Σ + ) approaches H 2 with its O-end (). This CI seam has been identi ed in previous work, 32, 33 but most discussion on its effect has focused on the seam in the C 2v symmetry. 29,31 Interestingly, our quantum dynamics calculation indicates that the wave packets emerging on the 1 2 A and 2 2 A states are largely located near the H-H-O-H collinear geometry ( and ), as shown in Fig. 4 (and Movie S1 and Figure S3 in SI). This observation underscores the dominance of the collinear reaction pathway over the C 2v pathway proposed in previous work. 29,31 A closer examination of the CI seam revealed the origin of this behavior: The asymptote 3 2 A PES has a large anisotropy with respect to the H 2 rotational angle (), as shown in Fig. 1(a) (and Figure S4 in SI), which guides the incoming wave packet to the linear geometry. Indeed, the minimum energy crossing (MEX) with C ∞ v symmetry is ~ 0.631 eV lower than that at C 2v symmetry, the C s symmetry seam in between, as shown in Fig. 1(a) and Table S1. 0 ∘ Steric effects, controlled by PES anisotropy, have long been recognized in adiabatic collisions. [43][44][45] In an activated reaction, the access of the reactive transition state is often facilitated by a cone of acceptance, 46 which in uences not only the reactivity, but also product state distributions 47 and sometimes product branching. 48 It follows that stereodynamics could also have a signi cant impact on nonadiabatic dynamics, although such examples in collision processes are few and far between. 49,50 The nonadiabatic quenching process discussed here serves as an excellent example for nonadiabatic barrierless scattering between molecular reactants. Based on the stereodynamics described above, both the experimental and theoretical results can be rationalized. To begin with, we reconcile the aforementioned controversy concerning the branching ratio between the two quenching channels. In the work of Dempsey et al., 25 the yield of the nonreactive channel (f R1b ) was measured directly; however, the yield for the reactive quenching channel (f R1a ) was derived with the assumption that the OH(A 2 Σ + ) radiative quenching is near completion, supported by the fast uorescence decay time of OH(A 2 Σ + ) (165 ns) and the corresponding small yield. 29 While this assumption is reasonable, our quantum dynamics calculations revealed that the (in)elastic (R1c) channel is quite prominent due to the unique stereodynamics discussed above. Unfortunately, the existence of this channel was not considered in the branching ratio model of Dempsey et al. 25 Since the calculated absolute R1b fraction (f R1b ) is in good agreement with experiment, which further supports the accuracy of the DPEM, we speculate that the aforementioned assumption of neglecting the (in)elastic scattering channel might be awed. Indeed, if we subtract the fraction of the R1c channel (f R1c =0.772) from the experimental value for total nonreactive events (0.877), the fraction of the R1a channel (f R1a ) would be 0.105, which is very close to our calculated value of 0.100. Based on this argument, we propose a reinterpretation of the experimental results by including the (in)elastic yield, which leads to the conclusion that the reactive quenching is no longer the dominant channel. This conclusion is consistent with the earlier full-dimensional TSH results of Collins et al. 38 using their own DPEM.
An important caveat concerning the aforementioned branching ratio obtained from the quantum calculations is the lack of contributions of higher partial waves. While N tot > 0 quantum scattering calculations are beyond the scope of this work, we have investigated this question using TSH with all accessible impact parameters (b max = 5.5 Å or N max = 35). The calculated fractions for the three channels at E c =0.05 eV are as follows: 0.009 (R1a), 0.055 (R1b) and 0.93 (R1c). The increased dominance of the (in)elastic channel at large impact parameters is readily understood as the H 2 -OH approach to the CI region is partially blocked by a centrifugal barrier while the (in)elastic channel is hardly affected. The inclusion of the higher partial waves in the entrance channel does not qualitatively change the conclusion.
The validity of the DPEM is further con rmed by the product state distribution in the nonreactive quenching channel. The OH rotational excitation can be attributed to the large anisotropy on the 1 2 A and 2 2 A state PESs, which exerts a strong torque on the departing OH fragment. This driving force exists not only in C 2v , as shown in Figure S5 and discussed in previous work, 29 but also in C ∞ v . This is illustrated in Fig. 6 where the potential along the coordinate is displayed at the collinear CI seam. The A′ PES has larger anisotropy than the A′′ PES. On the other hand, the anisotropy along the angle is relatively small, leading to less rotational excitation of the H 2 product. The weak OH vibrational excitation stems from the fact that the equilibrium bond lengths of OH(A 2 Σ + ) (1.901 a.u.) and OH(X 2 Π) (1.825 a.u.) are similar. They are also quite close to that at the C ∞ v MEX (1.820 a.u.). On the other hand, the H 2 vibrational excitation can be attributed to the stretched H-H bond at the C ∞ v MEX. As shown in Table S1, the H-H distance at the MEX(C ∞ v ) is 1.776 a.u., which is signi cantly larger than its equilibrium value (1.406 a.u.). As shown in Movie S1, the wave functions on the 1 2 A and 2 2 A surfaces rst shrink to a small H-H distance right after the nonadiabatic transitions and bounce by the potential wall at the small H-H value. Subsequently, the wave packet moves in a zigzag path in the exit channel, indicating vibrational excitation. Finally, the reproduction of the observed Λ-doublet propensity underscores the importance of out-of-plane dynamics and the CI seam between the two lowest electronic states leading to the OH(X 2 Π) + H 2 products.
The exclusive population of the ortho-H 2 rotational states requires some additional explanation. The complete nuclear permutation and inversion (CNPI) group of H 3 O is the D 3h . For discussion of the permutation symmetry of H 2 in the absence of H exchange with the OH group, the G 4 CNPI subgroup isomorphic to the C 2v point group is su cient. In this subgroup, the electronic state correlating to the OH(A 2 Σ + ) reactant transforms as A 1 , and states correlating to the OH(X 2 Π) products transform as B 1 and B 2 . The character for the A 1 irrep under the permutation is + 1 (even), and that for the B 1 and B 2 irreps are − 1 (odd). Since the total symmetry (electronic + nuclear) is conserved, a para-H 2 (j = 0, even under permutation) reactant is expected to be transformed exclusively to ortho (odd j, odd under permutation) ones after nonadiabatic transitions.

Conclusions
The collisional quenching of OH(A 2 Σ + ) by H 2 has served as an important prototype for understanding nonadiabatic dynamics in bimolecular collisions. Although the underlying conical intersections have been identi ed for some time, the full-dimensional quantum characterization of the nonadiabatic dynamics has not been achieved until now. Using a highly accurate diabatic potential energy matrix that includes four lowest-lying electronic states, we report here for the rst time full-dimensional quantum dynamics study of this prototypical nonadiabatic process involving four atoms and six coordinates. Our results revealed that the fate of OH(A 2 Σ + ) + H 2 collision is largely determined by stereodynamics, namely the relative orientation between the two collisional partners. The quenching is made possible with the H 2 -OH approach as H 2 -HO collisions are ineffective in accessing the CI seam. Furthermore, nonadiabatic transitions in the former orientation occur mostly near the collinear C ∞ v CI seam, rather than the C 2v CI seam proposed in previous work. 29,31 Most interestingly, the existence of a major (in)elastic channel, not included in the prior experimental analysis, suggests a reinterpretation of the experimental results that is consistent with the theoretical ndings. This resolves a long-standing experiment-theory discrepancy concerning the reactive/nonreactive branching ratio. Finally, the OH(X 2 Π) and H 2 product state distributions were calculated and compared well with available experimental results. These results validate the accuracy of the recently developed diabatic potential energy matrix and shed valuable light on the complex nonadiabatic dynamics involved in the OH(A 2 Σ + ) quenching.

Methods
We employed a full-dimensional quantum wave packet method to study the collisional quenching of OH(A 2 Σ + ) by molecular hydrogen for zero total nuclear angular momentum (N tot =0) on the recently constructed four-state DPEM. 40 As the details of the discretization, propagation, and product state projection are given in SI, we only provide a skeletal description here. The scattering calculations were performed using an exact kinetic energy operator in the diatom-diatom Jacobi coordinates (Fig. 1(c)). The uxes to all three channels of R1 were computed from the wave packet initiated from the OH(A 2 Σ + ) + H 2 channel with both molecules in the ground ro-vibrational states and propagated in time using the splitoperator method. 51 The OH and H 2 internal state distributions in both the ground and excited states were obtained in the asymptotic region, while the internal state of the H 2 O product was not resolved. Absorbing potentials were placed in all three channels to enforce outgoing boundary conditions.
To validate the DPEM, scattering dynamics on the ground adiabatic state PES obtained from this DPEM was rst carried out. The reaction probability agrees well with the most accurate ground-state PES of Chen et al. 52 (Figure S6 in SI). This DPEM is further validated by the nonadiabatic dynamics discussed above in comparison with available experimental results.
In addition to the quantum dynamics calculations, we have also carried out TSH calculations in the adiabatic representation using ANT, 53 in order to support mechanistic insights and to calculate the branching ratios. The details of the calculation are given in SI.

Declarations
Data Availability: All the data corresponding to the ndings of this study are available from the corresponding author upon reasonable request. Table   Table I. Branching fractions for the three channels of R1 at three different collision energies.
The values in parenthesis were obtained from TSH calculations with zero impact parameters.