Born–Oppenheimer and Renner–Teller coupled-channel quantum reaction dynamics of O(³P) + H₂⁺(X²Σ_g⁺) collisions

Abstract

We present Born–Oppenheimer (BO) and Renner–Teller (RT) time dependent quantum dynamics studies of the reactions O(³P) + H₂⁺(X²Σ_g⁺) → OH⁺(X³Σ⁻) + H(²S) and OH(X²Π) + H⁺. We consider the OH₂⁺ X̃²A′′ and òA′ electronic states that correlate with a linear ²Π species. The electronic angular momenta operators L̂ and L̂² are considered in nonadiabatic coupled-channel calculations, where the associated RT effects are due to diagonal V^RT potentials that add up to the PESs and to off-diagonal C^RT couplings between the potential energy surfaces (PESs). Initial-state-resolved reaction probabilities P_I, integral cross sections σ_I, and rate constants k_I are obtained using recent ab initio PESs and couplings and the real wavepacket formalism. Because the PESs are strongly attractive, P_I have no threshold energy and are large, σ_I decrease with collision energy, and k_I depend little on the temperature. The X̃²A′′ PES is up to three times more reactive than the òA′ PES and H₂⁺ rotational effects (j₀ = 0, 1) are negligible. The diagonal V^RT potentials are strongly repulsive at the collinearity and nearly halve all low-energy observables with respect to the BO ones. The off-diagonal C^RT couplings are important at low partial waves, where they mix the X̃²A′′ and òA′ states up to ∼20%. However, V^RT effects predominate over the C^RT ones that change at most by ∼19% the BO values of σ_I and k_I. The reaction O(³P) + H₂⁺(X²Σ_g⁺) → OH⁺(X³Σ⁻) + H(²S) is probably one of the most reactive atom + diatom collisions because its RT rate constant at room temperature is equal to 2.26 × 10⁻¹⁰ cm³ s⁻¹. Within the BO approximation, the present results agree rather well with recent quasiclassical and centrifugal-sudden data using the same PESs.

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I. Introduction

The atom–molecular ion reactions

O(³P) + H₂⁺(X²Σ_g⁺) → OH⁺(X³Σ⁻) + H(²S), ΔD₀ = −2.44 eV proton transfer, H channel, (1.1)

→ OH(X²Π) + H⁺, ΔD₀ = −1.74 eV hydrogen transfer, H⁺ channel, (1.2)

can play a significant role in atmospheric elementary processes, where the OH₂⁺ reaction intermediate is produced by water ionization and was observed since 1974.^1–3 Although the water molecule and related ionic species (O⁺, OH⁺, H₂O⁺, H₃O⁺, etc.) are important in the astrochemical context, it should be noted that the importance of reactions (1.1) and (1.2) in astrochemistry is still unclear.⁴ Moreover, the O(³P) + H₂⁺(X²Σ_g⁺) system is related to other interesting ionic systems as, e.g., the X(¹S) + H₂⁺ → XH⁺ + H (X = He, Ne, Ar) and O⁺(⁴S) + H₂ → OH⁺ + H reactions.^5,6

The dynamics of the collisions (1.1) and (1.2) was investigated in an old experimental work reported by McClure et al.⁷ for the deuterated variant D₂⁺, and in some very recent theoretical contributions by Paniagua et al.,⁸ Martínez et al.,⁹ and Gannouni et al.¹⁰ We plot schematically in Fig. 1 the relevant electronic correlation diagram, simplifying those of ref. 8 and 11 and using the energies given in ref. 8. The two lowest potential energy surfaces (PESs) are barrierless and present strongly bound minima at −8.16 and −7.54 eV with respect to the reactants. The product channels (1.1) and (1.2) are uncoupled within the adiabatic Born–Oppenheimer (BO) approximation and occur, respectively, on the two lowest X̃²A′′ and òA′ PESs of OH₂⁺, which transform as ²B₁ and ²A₁, respectively, in C_2v symmetry.

Fig. 1 Correlation diagram of OH₂⁺ X̃²A′′ (²B₁, red) and òA′ (²A₁, blue).

The ground PES (X̃²A′′) includes the ground electronic state of the water cation, H₂O⁺(X̃²B₁), in the minimum energy path (MEP) connecting reactants and OH⁺ + H products; while the first excited PES (òA′) includes the first excited electronic state of the water cation, H₂O⁺(òA₁), in the MEP involving reactants and OH + H⁺ products. These two electronic states of H₂O⁺ correspond to the degenerate components of a ²Π_u species, when the molecule is in a linear arrangement, and the associated nonadiabatic (NA) Renner–Teller (RT) effects can remarkably influence the dynamics of these collisions, as they also occur for the X̃²B₁ − òA₁ optical spectrum¹² of H₂⁺O.

This is especially important here because the deep potential wells and ²Π configurations are geometrically and energetically close and most collisions proceed via OH₂⁺ intermediate collision-complexes.^8,9 When the RT interactions are large, we really have two coupled reactions (1.1) and (1.2) that mainly proceed on both interacting PESs. The contribution of the ²A₁ − ²B₂ conical-intersection to the dynamics has not been considered in this work, and its influence has been recently studied in the photodissociation of H₂O⁺ of ref. 13.

The X̃²A′′ and òA′ PESs are based on ab initio calculations⁸ at the internally-contracted multi-reference configuration-interaction level, employing the Davidson correction (icMRCI + Q) and the augmented correlation-consistent polarized-valence quadruple-zeta basis set (aug-cc-pvQZ). More than 2000 ab initio points of both PESs were then fitted to a many-body expansion. Paniagua et al.⁸ also obtained BO quasiclassical trajectories (QCTs), integral cross sections, and thermal rate constants of both reactions and of their deuterated variants (D₂⁺ and HD⁺) on both PESs. Following that work, Martínez et al.⁹ employed the quantum-mechanical real wavepacket (RWP) method to obtain BO centrifugal-sudden (CS) initial-state-selected reaction probabilities and cross sections. Large reaction probabilities and decreasing cross sections were obtained for both reaction channels. Channel (1.1) was found to be nearly three times more reactive than (1.2).

Extending the previous studies,^8,9 the present paper reports a RWP study of the dynamics of the (1.1) and (1.2) collisions, at the NA RT coupled-channel (CC) level. We employ the OH₂⁺ X̃²A′′ and òA′ PESs of ref. 8 and the NH₂ RT matrix elements of Zhou et al.,¹⁴ taking into account that these triatoms are isoelectronic and have similar rovibronic structures.¹² The RT matrix elements were already employed by Zhou et al.¹⁴ and by Lin et al.¹⁵ in NH₂ and N(²D) + H₂(X¹Σ_g⁺) spectroscopic and collisional studies, respectively. Obviously, the results obtained in this study can be useful for a first estimation of quantum dynamical properties taking into account the presence of nonadiabatic effects in the title reactions. Nevertheless, higher accuracy could be reached once the RT matrix elements for this reaction are explicitly calculated, although small differences between both systems are expected.

The paper is organized as follows. Section II presents some theoretical and computational details, enlightening the origin, formulae, and physical meanings of the adiabatic (diagonal) and nonadiabatic (off diagonal) RT terms. Section III reports initial-state-resolved reaction probabilities for the lowest rotational states of H₂⁺ in the ground vibrational state, and for the product reaction channels (1.1) and (1.2). We take into account many total angular momentum quantum numbers J, both overall parities p = ±, and the Coriolis couplings among all K ≥ 0 projections of J along the O–(H₂⁺) axis. We report initial-state-resolved integral cross sections and rate constants in Section IV, discussing the influence of the RT effects on both product channels and comparing our BO results with previous QCT^8,9 and RWP CS⁹ calculations and we, finally, present in Section V the main conclusions.

II. Theory and calculation details

Following our previous studies,^16–18 the triatomic system ABC is described by the reactant A + BC Jacobi coordinates (R, r, and γ), a body-fixed reference frame with the z axis along R, a spinless rovibronic Hamiltonian Ĥ that contains both the total Ĵ and the electronic L̂ angular momentum, and the C_s(M) or C_2v(M) permutation-inversion groups that are isomorphic with the C_s or C_2v point groups, respectively. As usual, we employ atomic units unless stated otherwise.

The largest RT effects are due to the matrix elements of L̂_R² or L̂_R, which are respectively diagonal or off-diagonal in an adiabatic electronic representation |σΛ〉 with parity σ = ± and orbital angular momentum quantum number Λ ≥ 0 at the linearity. Omitting the L̂_± minor contributions, Ĥ contains the L̂_R²/sin² γ, Ĵ_RL̂_R/sin² γ, and Ĵ_±L̂_R cot γ operators, and the RT effects are thus responsible for the coupling among three degrees of freedom: the electronic and total rotations around R and the angular motion. The OH₂⁺ electronic states are |−1〉 = X̃²A′′ and |+1〉 = òA′ (or X̃²B₁ and òA₁ in C_2v, respectively). Eqn (3) of ref. 18 shows that the electronic matrix elements of L̂_R² and L̂_R define two different contributions to the RT effect.

The former RT term is associated with L̂_R² and is defined by the diagonal potentials

where B and b are half the inverse inertial moments associated with R and r, respectively. These potentials add up to the usual BO PESs, thus generalizing the BO approximation, and present RT barriers that diverge at the linearity. Therefore, these RT terms inhibit the reactivity with respect to the BO one, provided the collision samples quasilinear geometries. This preferably occurs when the K = 0 component of the wavepacket predominates, because the Legendre polynomials |j0〉 are maximal at the linearity.

The latter RT term is due to L̂_R and to the real, symmetrical, and (J,K)-dependent off-diagonal couplings

where λ^±_JK = [J(J + 1) − K(K ± 1)]^1/2. This equation contains three terms: the first is K-diagonal and the other two involve the RT–Coriolis interaction. When K or K′ > 0, these terms are different from zero and account for both the breakdown of the BO approximation and the nonadiabatic interactions between the electronic states. These couplings also diverge at the linearity if K′ = K.

Considering H₂⁺ in the ground vibrational state, ν₀ = 0, we calculate BO and RT coupled-channel (CC) initial-state-resolved reaction probabilities , where e₀ and (j₀,K₀) label the initial OH₂⁺ electronic and H₂⁺ rotational state, respectively, and E_col is the collision energy. We here consider j₀ = 0 and 1, E_col from 0.05 to 0.4 eV, J ≤ 75, and the Coriolis couplings among all K projections. Some checks performed until J = 30 showed that the deep minima involved in this study make necessary the propagation of the full set of K projections. At higher J values a similar behavior is expected according to the general trend observed in insertion reactions occurring through a deep minimum along the MEP.^19,20

The reaction probabilities are obtained through the real wavepacket (RWP) formalism of Gray and Balint-Kurti,^21,22via a scaled and shifted Hamiltonian Ĥ_s, solving an arccos mapping of the equation of motion. We employ an initial complex Gaussian WP, |ψ₀〉 = |a₀〉 + i|b₀〉, with the R-dependent part given by

where μ_R is the reduced mass associated with R and E₀ is the initial collision energy. The main equations to solve are²¹

|a₁〉 = Ĥ_s|a₀〉 − (1 − Ĥ_s²)^1/2|b₀〉, first propagation (complex WP), (2.4)

|a_n+2〉 = 2Ĥ_s|a_n+1〉 − |a_n〉, other propagations (real WP). (2.5)

The calculations are performed using reactant Jacobi coordinates, the square root involving the Ĥ_s operator is evaluated via a Chebyshev expansion, and eqn (2.5) corresponds to a standard Chebyshev propagation of just a real WP, which is damped at large R and r values by the Gaussians of eqn (28) and (29) of ref. 21. At the end of the propagations, the reaction probabilities are obtained through a flux method.²² The RWP method shares common features with the time independent studies of Chen and Guo²³ and Kroes and Neuhauser,²⁴ and it is essentially coincident with the Chebyshev WP approach of Lin and Guo.²⁵

We propagate CC WPs at J ≤ 10, J = 30, 50, 60, 65, 70, and 75, according to the numerical parameters of Table 1, which correspond to 6 042 330 basis states for each electronic species at J = 0, converging the probabilities in the range E_col = 0.05–0.4 eV within 2% and 40 000 iterations. Owing to the high cost of RT CC calculations and the usual J shifts of the probability thresholds, we employ a J fitting-interpolation technique¹⁹ at any (j₀,K₀) pair for obtaining reaction probabilities at all non-propagated WP J values. The comparison with some CC data shows that this approach gives good results for probabilities and cross sections, and error bars of ∼2% for room-temperature rates, as Fig. 7 of ref. 20 shows for the conical-intersection collision OH(A²Σ⁺) + H(²S).

Table 1 Parameters of the quantum dynamics calculations. Values in atomic units, unless otherwise specified

a Due to the symmetry of the system this corresponds to 70 j values and γ points.
Initial Gaussian g0(R)
Width α, R0 center, and E0 translational energy	0.3, 30, and 0.4 eV
R range and number of grid points	0–37 and 417
r range and number of grid points	0.5–18.5 and 207
Associated Legendre functions |jK〉, jmax	140^a
R and r absorption start at	32 and 13.5
Absorption strength	0.0005
Flux analysis at r	12.5

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Cross sections σ_e0j0(E_col) and rate constants k_e0j0(T), in the temperature range T = 200–900 K, are obtained through the usual expressions

where f_el is the population of the initial electronic state e₀ and depends on the electronic partition functions, Because the doublet PESs were calculated without accounting for the O(³P_M) spin–orbit (SO) interaction, three different and approximate strategies can be employed for estimating the O(³P) partition function:

(1) without fine structure, in a fully consistent way with the PES calculation.

(2) , with the SO energy levels²⁶ε₂ = 0, ε₁ = 158, and ε₀ = 227 cm⁻¹ referred to the ground state O(³P₂), and with q_el,O(³P) increasing with T from 5 to 9. This is the usual procedure which is not, however, strictly consistent with the PES calculation.

(3) , with ε₂′ = −78, ε₁′ = 80, and ε₀ = 149 cm⁻¹ referred to the O(³P) term.⁸ This proposal^27,28 agrees with the energy zero at the O(³P) + H₂⁺ asymptote without SO and can give a better agreement with the experiment.²⁹ However, is not fully meaningful, because it evolves from a divergent value at low T up to 9 at high T, and it is not consistent with the PESs.

In conclusion, the first choice is both fully consistent and physical and we therefore propose using f_el = 1/9 through this study, although the other choices are also depicted once for comparison. The H and H⁺ channels correspond to the cross sections and in the BO approximation, respectively, but they are associated with and at the RT level, where the and cross sections are due to the nonadiabatic couplings C^RT of eqn (2.2). Analogous expressions can be considered for the rate constants.

Note that the parameters of Table 1 produce reliable results just for E_col ≥ 0.05 eV and that their use at lower energies can give wrong cross sections for the O + H₂⁺ ionic collision. In principle, this drawback can be removed using other initial WPs below 0.05 eV, with larger α, R₀, and radial grids, as we did for the H + HeH⁺ reaction,³⁰ where these modifications described were able to reproduce with high accuracy the close coupling time independent results³¹ also in the cold energy range. Nevertheless, considering the additional and intensive RT CC calculations that would be necessary, we follow here a simpler strategy. Guided by some time-independent quantum-dynamics calculations through the ABC code³² and by QCT⁸ results at E_col < 0.05 eV, in a similar procedure to that employed for the Ne + H₂⁺ reaction,³³ we fit here the WP cross sections at E_col ≥ 0.05 eV using the well known expression³⁴

σ(E_col) = C_sE^s_col, (2.9)

which is then employed for extrapolating σ down to E_col = 0.005 eV. Therefore, in Sections III and IV we present RWP reaction probabilities and cross sections in the range E_col = 0.05–0.4 eV, and rate constants are obtained from the above cross sections and from the extrapolated ones in the range 0.005–0.05 eV.

III. Reaction probabilities

For the reader's convenience, we remember that the product channels H, proton transfer, and H⁺, hydrogen-atom transfer, eqn (1.1) and (1.2), are separated in the BO approximation, where a given collision always evolves on a single PES (the X̃²A′′ or the òA′ PES). Moreover, the single-state dynamics is affected by the V^RT potential (2.1) and the C^RT couplings (2.2) vanish identically at J = 0, when the WP remains on the initial surface.

Fig. 2 reports BO and RT reaction probabilities P^J−_X00 and P^J+_A00 for both channels H and H⁺ and initial electronic states, at J = 0, 10, and 50, p = −(²A′′) or +(²A′), j₀ = 0, and K₀ = 0. At low J values, the BO probabilities (left panels) do not present any threshold and increase sharply up to high values, with mild oscillations around a value of ∼0.90. The resonance structure found suggests the existence of an indirect insertion mechanism, favored by the deep minima of both PESs. The formation of short-lived collision complexes that break down rather quickly towards the product channels has also been reported in QCT calculations on the same PESs.⁹ As usual, high J values strongly damp the intensity of the oscillations observed, owing to the averaging effects of the Coriolis couplings among many K values, suggesting an increasing contribution of direct collisions in the process for J ≥ 30, in agreement with QCT results.⁹

Fig. 2 BO (left) and RT (right) CC reaction probabilities. e₀ = X̃²A′′ (above) and òA′ (below). J = 0 (red), 10 (green), 50 (blue). p = − (above) and + (below). j₀ = 0, K₀ = 0, E_col ≥ 0.05 eV. Probabilities due to C^RT are shown as dashed lines.

The energy thresholds depend on the BJ(J + 1) centrifugal barriers in the entrance valley and on the J-dependent energy shifts found on the òA′ excited surface, which correlates with the excited OH + H⁺ products, are much larger than those found on the ground X̃²A′′ PES. At J = 50, for example, the X̃²A′′ and òA′ BO thresholds are equal to ∼0.05 and ∼0.20 eV, respectively. This is consistent with the òA′ D_∞h equilibrium geometry, where B ∼ R⁻² can achieve large values, with a nearly collinear OHH dynamics on this excited PES, and with the larger values of the X̃²A′′ maximum impact parameter, b_max, found in the QCT calculations.^8,9 These findings imply that O + H₂⁺ collisions are very reactive and that the H product channel (proton transfer reaction) is favored with respect to the H⁺ one. We anticipate that cross sections and rate constants confirm these reactivity trends and that including the RT coupling also leads to the same tendency.

The right panels of Fig. 2 show RT reaction probabilities. The J = 0 RT couplings C^RT(2.2) are identically equal to zero, as previously indicated, and they are negligibly small at J ≥ 50. The comparison of the BO and RT reaction probabilities shows that the RT coupling term increases the energy thresholds, reduces significantly the probabilities, and opens both the H and H⁺ channels at J ∼ 10. The probability lowering occurs even at J = 0, when the two electronic states involved are uncoupled, and it is, therefore, due to the presence of the RT potentials V^RT_σ(2.1). These are, in fact, quite large in a wide region of the coordinate space, owing to the 1/sin² γ factor when OH₂⁺ is nearly linear, or due to the large L̂_R² matrix elements when OH₂⁺ bends.¹⁴ At low J values, the RT couplings C^RT(2.2) between the two electronic states are present but are rather small. Therefore, O + H₂⁺ collisions give preferentially OH⁺ + H or OH + H⁺ products when the WP starts on the ground X̃²A′′ or excited òA′ surface, respectively. In the latter case, however, the RT effects are larger owing to the linear (²Π_u) equilibrium geometry of the minimum. For J > 10, the RT diagonal terms V^RT_σ are still large but the off diagonal ones C^RT decrease and vanish at J ∼ 30.

Consequently, the two electronic states involved are, in practice, not RT-coupled at these high partial waves, as the probabilities at J = 50 show. Looking at eqn (2.2), we see that this finding is probably associated with the small 〈X̃|L̂_R|Ã〉 value at many geometries.¹⁴ Dynamic effects can be also operative, with the WP evolution being strongly influenced by the K₀ initial value, in agreement with the fast collisions occurring at high-J values. For example, K₀ = 0 might be nearly conserved during the high-J dynamics owing to the high number of available K final states opened by RT Coriolis couplings, with C^RT ∼ 0. This fact is consistent with a CS approximation that works quite well at the RT level.

We confirm this hypothesis by comparing some CC and CS reaction probabilities in Fig. 3, plotting the BO and RT X̃²A′′ probabilities P³⁰⁻_X00 at J = 30, p = −, j₀ = 0, and K₀ = 0. Within the CS approximation, the electronic states are not RT-coupled because only the K = K₀ = 0 channel is open and all C^RT couplings (eqn (2.2)) are, therefore, identically equal to zero. Other initial conditions give essentially the same scenario. Within the BO approximation, the agreement between CC and CS results is, in general, acceptable, although the CS reaction probability is, as usual, more oscillating and somewhat smaller than the CC one. This is consistent with the CC averaging effects resulting from taking into account 31 K values and the moderate and negative Coriolis couplings that enhance the CC reactivity. On the other hand, the CS approximation works quite well at the RT level because the K-diagonal matrix elements of L̂_R²/sin² γ, eqn (2.1), predominate over the Coriolis couplings. This fact inhibits more the RT CC probability than the RT CS one that are thus nearly equal, and the initial K₀ remains approximately a good quantum number. Note also that RT CC interactions with the excited òA′ electronic state, leading to the OH + H⁺ products, are fully negligible at J = 30.

Fig. 3 BO (left) and RT (right) CC (red) and CS (green) reaction probabilities. J = 30, p = −, e₀ = X̃²A′′, j₀ = 0, K₀ = 0, E_col ≥ 0.05 eV.

Fig. 4 shows how the initial K₀ value affects the O(³P) + H₂⁺ reaction probabilities, presenting BO and RT probabilities P⁵⁻_A10 and P⁵⁻_A11 at e₀ = òA′, J = 5, p = −, j₀ = 1, and K₀ = 0 and 1. The BO results are nearly equal but the RT ones present significant differences depending on the K₀ values, because they correspond to the initial Legendre functions |10〉 ∼ cos γ or |11〉∼sin γ for K₀ = 0 or 1, respectively. When K₀ = 0, the initial WP density is largest at γ = 0 and the RT linear barrier ∼〈Ã|L̂_R²|Ã〉/sin² γ strongly reduces the reactivity. In contrast, the maximum initial density of the latter WP, K₀ = 1, occurs at γ = 90°; that is to say, along a more attractive region, where the linear barrier has a minimal contribution. Contrasting the H and H⁺ reaction channels, we see that ∼15–20% of the WP jumps from the initial excited surface, òA′, to the ground one, X̃²A′′, at both K₀ values.

Fig. 4 BO (black) and RT (red and blue) CC reaction probabilities. J = 5, p = −, e₀ = òA′, j₀ = 1, K₀ = 0 (left) and 1 (right), E_col ≥ 0.05 eV.

All these results are typical of barrierless and exothermic RT collisions on bound PESs, because we found similar V^RT and K₀ effects in the C(¹D) + H₂(¹Σ_g⁺) collisions.¹⁸

IV. Cross sections and rate constants

Fig. 5 shows the BO and RT reaction cross sections σ_Xj0 and σ_Aj0 for both channels H and H⁺ and initial electronic states, at j₀ = 0 and 1, and Table 2 reports some representative numerical results for j₀ = 0. We shall first discuss BO cross sections and then show how RT effects modify the reactivity. The collision energy and rotational effects reflect the barrierless and rather isotropic character of both PESs; that is to say, the cross sections decrease with E_col without appreciable oscillations and are negligibly affected by the H₂⁺ rotation level j₀. The ground-state BO decreases with the collision energy faster than the excited-state BO . For example, Table 2 shows that the BO σ^H_X0 or is lowered by a factor of 3.2 or 1.7, respectively, from 0.05 to 0.4 eV, suggesting that for the ground PES it is easier for the system to depart from the MEP as E_col increases in comparison to the excited PES.

Fig. 5 BO (black) and RT (red and blue) CC cross sections. e₀ = X̃²A′′ (above) and òA′ (below), j₀ = 0 (left) and 1 (right), E_col ≥ 0.05 eV.

Table 2 H and H⁺ BO and RT j₀ = 0 cross sections/Ų

E col/eV	H channel			H^+ channel
	σ ^HX0	σ ^HX0	σ ^H0
	BO	RT	RT	BO	RT	RT
0.05	153.4	98.6	103.3	53.6	30.9	32.4
0.1	105.1	80.3	83.8	44.2	29.9	31.0
0.2	69.1	57.9	59.7	36.5	29.1	29.7
0.3	54.9	50.0	51.2	33.8	28.5	28.9
0.4	48.4	44.6	46.1	32.2	27.7	28.0

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The ground state is also more reactive than the excited one, up to ∼3 times at 0.05 eV, owing to its more attractive character and larger exothermicity shown in Fig. 1, and to the smaller J shifts of the reaction probabilities in Fig. 2. Therefore, more partial waves contribute to the ground-state cross sections in the energy range investigated. As E_col increases, the reactivity of the electronic states becomes progressively more similar: at j₀ = 0, e.g., the H/H⁺ BO product branching ratio decreases from 2.9 to 1.5 in the E_col range 0.05–0.4 eV. On the overall, the proton transfer reaction giving the OH⁺ + H products is preferred with respect to the hydrogen transfer one leading to OH + H⁺.

The inclusion of the RT effects confirms the behavior of the cross sections with respect to the collision energy, to the initial electronic and rotational states, and to the product channels. The results presented in Fig. 5 and Table 2 clearly point out the differences between BO and RT treatments and the different role of the contributions to RT effects, namely the adiabatic linear barriers V^RT(2.1) and the nonadiabatic couplings C^RT(2.2) where the former is by far that predominant. In this table, σ^H₀ = σ^H_X0 + σ^H_A0, , the difference σ_e00(RT) − σ_e00(BO) is mainly associated with V^RT, whereas σ^H_A0(RT) and are essentially due to C^RT. In fact, the latter cross sections correspond to NA jumps from the reactant channel on the excited òA′ PES to the product channel on the ground X̃²A′′ surface (see Fig. 1), and these transitions are induced by the 〈X̃|L̂_R|Ã〉 matrix element. RT cross sections are smoother than the BO ones, for example, the RT ratio σ^H₀(0.05)/σ^H₀(0.4) is equal to 2.2 whereas the BO one σ^H_X0(0.05)/σ^H_X0(0.4) is 3.2. In particular, RT effects decrease the reactivity of the system at low-intermediate E_col values, e.g., it is reduced up to a factor of 1.6 with respect to the BO one at 0.05 eV. This is consistent with the probability results and is due to the diagonal RT barriers V^RT. The nonadiabatic couplings C^RT are more important when the WP starts on the excited òA′ PES. At E_col = 0.05 eV, for example, the RT σ^H_A0 and σ^H_A1 contribute by 13.2 and 15.8%, respectively, to the total RT depletion cross sections of O + H₂⁺.

On the overall, at E_col ≥ 0.1 eV both V^RT and C^RT increase a little the observable H/H⁺ RT branching ratios with respect to the BO ones . The largest variations occur at 0.1 eV, from the BO ratios of 2.38 and 2.28 at j₀ = 0 and 1, respectively, to the RT values equal to 2.71 at both j₀; that is a 13.9 or 18.9% of increasing at these rotational levels. This finding is essentially associated with that correspond to òA′ → X̃²A′′ nonadiabatic transitions. For the X̃²A′′ PES the main result arising from the RT effects is the lowering of the reactivity, as a consequence of the V^RT barrier, with a negligible production of the OH + H⁺ nonadiabatic reaction channel. We close this discussion by plotting in Fig. 6 BO and RT averaged cross sections and with f_el = 1/9, at j₀ = 0 and 1. We clearly see the very small role of the H₂⁺ rotational excitation and the H/H⁺ product branching ratio in the E_col range investigated.

Fig. 6 BO (black) and RT (red and blue) CC averaged cross sections, j₀ = 0 (full) and 1 (dots), E_col ≥ 0.05 eV, f_el = 1/9.

As stated in Section II, rate constants are obtained from RWP cross sections at E_col ≥ 0.05 eV and from the extrapolated ones down to E_col = 0.005 eV viaeqn (2.9). Table 3 and Fig. 7 are related to Table 2 and Fig. 6, respectively, and present the H and H⁺ BO and RT rate constants k_e0j0 and k_j0 from 200 to 900 K, with and , with f_el = 1/9 and j₀ = 0 and 1. These data are consistent with the σ results for the H/H⁺ branching ratio, for the reactivity of the initial electronic and rotational states, and for the BO or RT findings. All rate constants increase moderately with temperature. Regarding the BO results, k^H_X0 is almost independent of T and increases by 46% from 200 to 900 K. On the other hand, RT rate constants increase for both reaction channels by 38 and 91% for k^H₀ and , respectively. As expected, the V^RT and C^RT effects on the cross sections are confirmed for the rate constants: the former nearly halves the reactivity at room T and the latter is rather small. Table 3 also shows that the RT j₀ = 0 branching ratio at 300 K is equal to 3.32 and a value of 3.10 is obtained for j₀ = 1. The differences between these values and the BO ones are of only ∼8%, i.e., less than one half of the cross-section values owing to the integration on E_col. Note that the large k^H values for the H channel imply that O(³P) + H₂⁺(²Σ_g⁺) → OH⁺(X³Σ⁻) + H(²S) is a highly reactive atom + diatom collision. In fact, at room temperature and at the RT level we have k^H₀(300 K) = k^H₁(300 K) = 2.26 × 10⁻¹⁰ cm³ s⁻¹ and this value is larger than that of similar RT reactive systems, namely 1.39 × 10⁻¹⁰ cm³ s⁻¹ for C(¹D) + H₂(¹Σ_g⁺)¹⁸ and 2.20 × 10⁻¹² cm³ s⁻¹ for N(²D) + H₂(¹Σ_g⁺).¹⁹ The present H channel reaction (1.1) is so reactive owing to barrierless and strongly attractive character of the ground PES and to the larger exothermicity.

Table 3 H and H⁺ BO and RT j₀ = 0 rate constants/10⁻¹⁰ cm³ s⁻¹ with f_el = 1/9

T/K	H channel			H^+ channel
	k ^HX0	k ^HX0	k ^H0
	BO	RT	RT	BO	RT	RT
200	3.50	1.89	2.00	1.05	0.52	0.56
300	3.67	2.14	2.26	1.19	0.64	0.68
500	3.75	2.43	2.55	1.36	0.82	0.86
700	3.74	2.58	2.69	1.46	0.95	0.98
900	3.67	2.64	2.76	1.53	1.04	1.07

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Fig. 7 BO (black) and RT (red and blue) CC rate constants, j₀ = 0 (full) and 1 (dots), f_el = 1/9.

In Fig. 8 we compare the present BO CC cross sections and rate constants for j₀ = 0 with previous CS and QCT results,⁹ with f_el = 1/9 everywhere. CC and CS data of both channels compare very well, as we have found for probabilities, whereas H-channel QCT results are slightly higher, implying that quantum effects are little important in the investigated reactions.⁹

Fig. 8 BO cross sections (left, E_col ≥ 0.05 eV) and rate constants (right), e₀ = X̃²A′′ (above) and òA′ (below), j₀ = 0, f_el = 1/9. CC (red and blue), CS (green), QCT (black). CS and QCT data from ref. 9.

Finally, we analyze the influence of the different choices (1), (2), and (3) of the O(³P) partition function presented at the end of Section II, plotting in Fig. 9 the associated RT rate constants k₀ = k_X0 + k_A0 at j₀ = 0. We remember that (1) does not consider the O(³P) fine structure, which is taken into account by (2) and (3) with energy zero at the O(³P₂) state or at the O(³P) term, respectively. At not too large T values we have f⁽²⁾_el > f⁽¹⁾_el > f⁽³⁾_el and, therefore, choices (2) and (3) present opposite effects: the former enhances whereas the latter inhibits k with respect to f⁽¹⁾_el. Moreover, f⁽²⁾_el decreases whereas f⁽³⁾_el increases with T. This behavior is confirmed in Fig. 9, where all rate constants increase with T, especially those of the H⁺ channel, choices (1) and (3) give similar results, and k due to choice (2) is significantly larger. At 300 K k^H₀ varies within (2.07–3.02) × 10⁻¹⁰ cm³ s⁻¹ and within (0.55–0.87) × 10⁻¹⁰ cm³ s⁻¹ according to the three choices of f_el, and these intervals increase at smaller T. These effects could be validated by further studies that consider the O(³P_M) fine structure at low temperatures.

Fig. 9 RT rate constants, j₀ = 0, f_el from eqn (2.8) with q_el,O(³P) from choices (1), (2), and (3).

V. Summary and conclusions

We have investigated the reactive collisions O(³P) + H₂⁺(X²Σ_g⁺) → OH⁺(X³Σ⁻) + H(²S), the H channel, and the → OH(X²Π) + H⁺, H⁺ channel, on the OH₂⁺ X̃²A′′ and òA′ BO uncoupled or RT coupled PESs. Highly accurate MRCI PESs and L̂ matrix elements, a CC quantum theory of RT atom + diatom collisions up to J = 75, and the time-dependent RWP and flux methods were employed. The electronic states are the degenerate components of a linear ²Π species and are, therefore, strongly perturbed by RT effects due to L̂, which modify the BO PESs and couple the electronic states via RT potentials V^RT(2.1) and couplings C^RT(2.2), respectively. The forms and roles of these RT terms are thoroughly discussed, pointing out that they describe the couplings among total rotation, electronic rotation, and angular motion. In particular, V^RT represents a dynamical and linear barrier added to the usual BO PES whereas C^RT couples the electronic states when K > 0.

The present RT CC work extends BO QCT and RWP CS studies on the uncoupled PESs reported recently by some of us,^8,9 showing that all the theoretical results agree rather well and give a common picture of the reaction dynamics. Thus, reaction probabilities are large and without threshold energy, the cross sections σ decrease with E_col from large threshold values, and the rate constants k show a small dependence with T, because the PESs are barrierless. The X̃²A′′ ground electronic state is more reactive up to a factor of ∼3, owing to its larger attractive character and exothermicity, the H product channel is thus favored, and the H₂⁺ rotational excitation does not practically affect the σ and k values.

Owing to the repulsive collinear potentials V^RT(2.1), the RT energy thresholds are larger and the reaction probabilities are smaller than the BO ones. This finding is more evident for the òA′ excited state, owing to its ²Π_u equilibrium geometry where the rotational constant B is large. The RT couplings C^RT(2.2) mix the electronic states up to ∼20% and are more important when the WP starts on the excited PES at low J values and K₀ = 0. These couplings, however, decrease quickly at high J values and, on the overall, the reactivity is inhibited when K₀ and/or K = 0. These results are explained by considering the expressions of the diagonal and off-diagonal RT terms, the linear values of the initial Legendre functions, and the good agreement between BO CC and CS probabilities at K₀ = 0, which implies that the collision dynamics does not appreciably modify the initial K₀ value. RT effects smooth the σ(E_col) curves with respect to the BO ones and the RT diagonal barriers V^RT nearly halve the σ and k values at low energy and room temperature. The RT nonadiabatic couplings C^RT modify these observables and the H/H⁺ product branching ratios up to ∼19% with respect to the BO ones. The collision O(³P) + H₂⁺(X²Σ_g⁺) → OH⁺(X³Σ⁻) + H(²S) is highly reactive, as it comes out from k_j0(300) = 2.26 × 10⁻¹⁰ cm³ s⁻¹.

In summary, we note that other nonadiabatic interactions could be important for the collisional dynamics in OH₂⁺ and deuterated variants at low energies. Looking at the correlation diagrams of Fig. 1 and to those more detailed in ref. 8 and 11, we see for example:

(1) A RT-induced charge-transfer quenching (q), OH(X²Π) + D⁺ → OH⁺(X³Σ⁻) + D, and an exchange-quenching (eq), OH(X²Π) + D⁺ → OD⁺(X³Σ⁻) + H, which are in competition with the exchange process (e), OH(X²Π) + D⁺ → OD(X²Π) + H⁺, but not with the formation of O(³P) + HD⁺ that is closed below E_col = 1.62 eV. Preliminary results suggest a branching ratio [(q) + (eq)]/(e) greater than one, that is significant RT effects larger than those in O(³P) + H₂⁺.

(2) A B̃²B₂ − òA₁ conical intersection (CI) in the O(³P) + H₂⁺(X²Σ_g⁺) arrangement, which might decrease the efficiency of the present reactions, mainly for the H⁺ channel, opening a third product channel, OH⁺(a¹Δ) + H(²S). We plan to investigate this point in a nonadiabatic future work considering X̃ − à − B̃ three-state CI and RT effects.

On the other hand, intersystem-crossing effects among the ã⁴A′′ quartet, shown in Fig. 1 of ref. 8 and 11, and the X̃²A′′ and òA′ doublets are not expected to be important, taking into account the O(³P,¹D) + H₂(X¹Σ_g⁺)³⁵ and O⁺(⁴S) + H₂(X¹Σ_g⁺)³⁶ dynamical studies.

Acknowledgements

This work was supported by the Spanish Ministry of Science and Innovation (MICINN projects CTQ2011-27857-C02-01 and CONSOLIDER INGENIO 2010 under grant no. CSD2009-00038 entitled “Molecular Astrophysics: the Herschel and Alma era”). Thanks are also given to the “Generalitat de Catalunya” (Autonomous Government of Catalonia; refs. 2014SGR 25, 2014SGR 1582 and XRQTC) for some help. We gratefully acknowledge H. Guo and D. Xie for the matrix elements of the electronic angular momenta.

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