Origin of the “odd” behavior in the ultraviolet photochemistry of ozone

Significance

This study solves a 30-y mystery about the origin of dramatic oscillations in the quantum product states of ¹⁶O₃ upon photodissociation. Ozone is a key component of the atmosphere and undergoes complex cycles of formation and destruction by photodissociation and reaction. These dynamics are a rich proving ground for theories of quantum reaction dynamics as well as subtleties such as nuclear spin-statistics and symmetry. A previous theory constructed to explain the observed oscillations, although plausible, was not fully consistent with experimental measurements. Our quantum model and calculations show conclusively the origin of the effect, and match new more detailed experiments.

Abstract

The origin of the even–odd rotational state population alternation in the ¹⁶O₂(a¹Δ_g) fragments resulting from the ultraviolet (UV) photodissociation of ¹⁶O₃, a phenomenon first observed over 30 years ago, has been elucidated using full quantum theory. The calculated ¹⁶O₂(a¹Δ_g) rotational state distribution following the 266-nm photolysis of 60 K ozone shows a strong even–odd propensity, in excellent agreement with the new experimental rotational state distribution measured under the same conditions. Theory indicates that the even rotational states are significantly more populated than the adjacent odd rotational states because of a preference for the formation of the A′ Λ-doublet, which can only occupy even rotational states due to the exchange symmetry of the two bosonic ¹⁶O nuclei, and thus not as a result of parity-selective curve crossing as previously proposed. For nonrotating ozone, its dissociation on the excited B¹A′ state dictates that only A′ Λ-doublets are populated, due to symmetry conservation. This selection rule is relaxed for rotating parent molecules, but a preference still persists for A′ Λ-doublets. The A′′/A′ ratio increases with increasing ozone rotational quantum number, and thus with increasing temperature, explaining the previously observed temperature dependence of the even–odd population alternation. In light of these results, it is concluded that the previously proposed parity-selective curve-crossing mechanism cannot be a source of heavy isotopic enrichment in the atmosphere.

Given the importance of the stratospheric ozone layer in protecting the earth from harmful ultraviolet (UV) rays (1), it is not surprising that the UV photodissociation of ozone has been the subject of numerous experimental (2–15) and theoretical studies (16–24). In the strongly absorbing Hartley band (200–300 nm), the photodissociation involves two low-lying excited states with the same symmetry (A′) as the ground state (X): the B state which diabatically correlates to O₂(a¹Δ_g) and O(¹D) products, and the R state which diabatically correlates to O₂(X³${\text{Σ}}_{\text{g}}^{-}$) and O(³P) products. Accordingly, photodissociation proceeds primarily through these two spin-allowed channels (16):

$${\text{O}}_3+hv\to {\text{O}}_2\left(X^3{\mathrm{\Sigma }}_{\text{g}}^{-}\right)+\text{O}\left({}^3\mathrm{P}\right)\to {\text{O}}_2\left(a^1{\mathrm{\Delta }}_{\text{g}}\right)+\text{O}\left({}^1\mathrm{D}\right).$$

As illustrated in Fig. 1, following initial excitation to the B state, O₃ molecules can undergo direct dissociation on the same state, resulting in excited singlet products, or cross to the repulsive R state, dissociating into ground-state triplet products. As singlet O(¹D) reacts with water to form the hydroxyl radical, an important species in atmospheric chemistry, the singlet/triplet branching ratio has been well studied. Throughout most of the Hartley band (wavelengths <300 nm), the singlet/triplet branching ratio is largely independent of wavelength and temperature, with ∼90% of molecules following the singlet channel, and the remainder crossing to the triplet channel (25–27).

Fig. 1.

Surface plots of the relevant PESs (22). One O–O bond is held at 2.28 atomic units (a.u.) [close to equilibrium geometries for O₃(B), O₂(X³${\text{Σ}}_{\text{g}}^{-}$), and O₂(a¹Δ_g)], while the other O–O bond distance and bending angle are varied. The B state (upper diabat) has a well corresponding to its equilibrium geometry of r₁ = 2.28 a.u., r₂ = 3.20 a.u., Θ = 108.1°, and correlates to the O₂(a¹Δ_g) + O(¹D) products. The R state (lower diabat) cuts through the B state well and correlates to the O₂(X³${\text{Σ}}_{\text{g}}^{-}$) + O(³P) products.

Many of the experimental results on the photodissociation of ozone in the Hartley band have been semiquantitatively reproduced by theory (16–24). Previous theoretical studies, however, have failed to replicate a prominent and subtle feature of the O₂(a¹Δ_g) rotational distribution. Over 30 years ago, Valentini and coworkers measured the nascent O₂(a¹Δ_g) rotational distribution following photodissociation of ozone at 300 K for multiple wavelengths in the Hartley band using Coherent Anti-Stokes Raman Spectroscopy (CARS) (28). A population alternation was observed in the measured rotational distributions for the ¹⁶O₂(a¹Δ_g) photofragments, with even rotational states being significantly more populated than the adjacent odd rotational states. However, no such population alternation was observed in the ¹⁶O¹⁸O(a¹Δ_g) rotational distribution resulting from the photodissociation of mixed isotopologs. While the vibrational distributions and overall shape of the rotational distributions have been well described by theory, the observed even–odd rotational state population alternation has been neither reproduced nor definitively explained.

Valentini and coworkers invoked a parity-selective curve-crossing model to explain the observed population alternation (28). Due to nuclear symmetry restrictions, ¹⁶O₂(X³${\text{Σ}}_{\text{g}}^{-}$) only exists in odd rotational states. Consequently, it was proposed that only molecules dissociating to form O₂ in odd rotational states could cross from the B state to the R state, selectively depleting the singlet channel of fragments in odd rotational states. Valentini et al. pointed out that the sum of the missing odd states in the ¹⁶O₂(a¹Δ_g) rotational distribution was consistent with the reported triplet-channel yield, seemingly supporting the theory that fragments in odd rotational states were crossing from the singlet to the triplet channel. The lack of alternation in the ¹⁶O¹⁸O(a¹Δ_g) rotational distribution was also attributed to the curve-crossing model. Nuclear symmetry is broken in ¹⁶O¹⁸O fragments, so that ¹⁶O¹⁸O(X³Σ_g⁻) can exist in both odd and even rotational states. Consequently, according to the curve-crossing model, both odd and even states of ¹⁶O¹⁸O can cross from the B to the R state. The lack of alternation in the ¹⁶O¹⁸O(a¹Δ_g) rotational distribution was explained by both even and odd states curve crossing at equal rates, effectively doubling the triplet-channel quantum yield. This proposed enhanced crossing rate for heteronuclear O₂ has been cited (29, 30) as a possible explanation for the observed mass-independent enrichment of heavy ozone isotopologs (containing ¹⁷O and ¹⁸O) in the atmosphere, an important puzzle that has so far defied a definitive explanation (31).

A second mechanism that has been proposed to explain the observed population alternation is a preference for the formation of the A′ Λ-doublets (18, 28, 32, 33). The O₂(a¹Δ_g) product has a nonzero electronic angular momentum quantum number (Λ = 2), so the corresponding electronic state is doubly degenerate for N = 0 (N is the rotational quantum number). Upon rotation (N > 0), the coupling between the electronic and rotational angular momenta lifts the degeneracy and leads to two closely spaced fine-structure states which are symmetric (A′) and antisymmetric (A″) with respect to the rotational plane, namely the Λ-doublets (34). The splitting of the doublets is very small (often <1 cm⁻¹), but increases with the rotational quantum number. Due to nuclear exchange restrictions in ¹⁶O₂(a¹Δ_g), importantly, the A′ and A′′ components of the Λ-doublet occupy even and odd rotational states, respectively. Consequently, the even/odd rotational state populations provide a proxy for the Λ-doublet populations and preferential formation of the A′ Λ-doublet would thus lead to a rotational distribution favoring the even rotational states. In ¹⁶O¹⁸O(a¹Δ_g), the nuclear exchange restriction is absent, and Λ-doublets can occupy rotational states of both parities, thus without preference for either odd or even rotational states.

Recent studies have provided tantalizing evidence in favor of the Λ-doublet preference mechanism over the parity-selective curve-crossing model (32, 33). Experiments in a jet-cooled molecular beam found that the degree of population alternation is strongly temperature dependent, with the relative odd-state population decreasing with decreasing temperature (32). Since the curve-crossing model proposes that the odd states missing from the singlet channel have crossed over to the triplet channel, the model implies that the triplet yield should show a corresponding increase with decreasing temperature, which is inconsistent with the well-established temperature independence of the singlet/triplet branching ratio (25–27). Therefore, the degree of even–odd alternation is not correlated to the degree of curve crossing. The observed temperature dependence is, however, consistent with a Λ-doublet preference model. A temperature-dependent Λ-doublet preference has been observed in OH fragments resulting from the UV photodissociation of water (35). Photodissociation of water in a jet-cooled molecular beam results in a strong preference for the formation of OH fragments in one Λ-doublet, while photodissociation of room-temperature water results in approximately equal yields of both Λ-doublets. This temperature dependence arises because parent rotation reduces the preference for the formation of a specific Λ-doublet (36).

In this article, we present a combined theoretical and experimental study of the photodissociation ¹⁶O₃, which resolves the controversy surrounding the even–odd propensity in the ¹⁶O₂(a¹Δ_g) product rotational distribution. Experimentally, the O₂ rotational distribution is measured upon 266-nm photodissociation of jet-cooled ¹⁶O₃. This is complemented with a fully quantum-mechanical characterization of the photodissociation dynamics on a first-principles–based potential energy surface (PES). Different from all previous theoretical studies, our model includes the O₂ electronic angular momentum and its coupling with the rotational angular momentum, thus allowing the characterization of the Λ-doublet populations. Significantly, our theoretical model reproduces the even–odd propensity in the experimentally measured ¹⁶O₂ rotational state distribution and its temperature dependence. Furthermore, it reveals the dynamical origin of the propensity.

Results and Discussion

We have remeasured the rotational distribution of the ¹⁶O₂(a¹Δ_g, v = 0) fragments resulting from the photodissociation of ¹⁶O₃ at 266 nm and 60 K. As discussed above, the even/odd rotational-state populations are identical to the A′ and A′′ Λ-doublet populations in this species. The experimental set up is described in Methods, and more details can be found in refs. 32 and 33, as well as SI Appendix.

Fig. 2A shows a representative ion image corresponding to N = 28 in the S branch (with a small contribution from N = 35 in the R branch). The position at which the image was taken is marked by the arrow in Fig. 2B. Improved experimental and analysis techniques have led to a modest shift in the experimental rotational distribution compared to the one previously reported (32). Improved background and power correction have increased the accuracy of the experimental spectrum. The current experimental spectrum was fit using Lorentzian functions, as opposed to Gaussian functions that were used previously. As can be seen in Fig. 2B, the experimental lineshapes and baseline are accurately reproduced using Lorentzian functions. The error bars shown in Fig. 2B were determined using a Monte Carlo approach. The degree of even–odd alternation in the current rotational distribution agrees well with that previously reported. Accordingly, the conclusions drawn in the previous study are not impacted by the slight shift in rotational distribution.

Fig. 2.

(A) Ion image corresponding to N = 28 in the S branch and N = 35 in the R branch. The photolysis beam is polarized parallel to the image plane, and the probe beam is polarized perpendicular to the image plane. Image position marked by an arrow on the spectrum in B. (B) 2 + 1 REMPI spectrum of O₂(a¹Δ_g, v = 0) fragments resulting from the photodissociation of ozone at 266 nm, probed via the O₂(d¹Π_g, v = 4) state. Experimental data are represented as dots, and the simulation is represented by solid lines. The S branch is shown in purple, the R branch is shown in blue, and the P branch is shown in orange. The sum of all branches is shown in black. The locations of the combs indicate the N quantum number of the O₂(a¹Δ_g) fragments.

To help interpret the experimental results and identify the origin of the even–odd population difference in the product rotational distribution, we carried out calculations of the photodissociation dynamics using a full quantum theory (details given in Methods and SI Appendix) (37). The ab-initio–based B state PES of Schinke and McBane was used (22), and the R state was ignored, as previous theoretical studies including the dissociative singlet (R) state have found no significant effect on the O₂ product-state distributions and produced no even–odd alteration in the O₂ rotational distribution (23, 24). While the dissociation dynamics are characterized by the propagation of the wave packet on the B-state PES as before, the asymptotic projection to O₂ rovibrational states is done differently from all previous theoretical studies on this system. Specifically, coupling between the rotational and electronic angular momenta of the ¹⁶O₂(a¹Δ_g) fragment and the exchange symmetry of the two ¹⁶O nuclei are explicitly considered, which allow the determination of the Λ-doublet populations in the O₂ product. All previous theoretical investigations failed to capture the even–odd propensity because they did not consider the Λ-doublets, which are correlated to the even and odd rotational states of ¹⁶O₂ as discussed above.

This quantum model leads to the reproduction of the even–odd propensity in the O₂ rotational distribution, as shown in Fig. 3 (Left), in which the experimental and calculated rotational distributions for the ¹⁶O₂(a¹Δ_g) fragments resulting from the 266-nm photolysis of ¹⁶O₃ at 60 K are compared. The magnitude of the even–odd population oscillations calculated by theory is in excellent agreement with the experimental results. At the peak of the experimental distribution, odd-state populations are ∼16% of the adjacent even-state populations, and at the peak of the calculated distribution odd-state populations are ∼10–11% of the adjacent even populations. The experimental rotational distributions peak between N = 22 and 24 while the calculated rotational distribution peaks at N = 20, but within the experimental error bounds. The calculated distribution is also broader than the experimental distribution. These discrepancies can presumably be attributed to inaccuracies in the PES, as discussed in SI Appendix.

Fig. 3.

(Left) Comparison of experimental and calculated rotational distributions for the ¹⁶O₂(a¹Δ_g, v = 0) fragments resulting from the photodissociation of 60 K ozone at 266 nm. The maximum value of the distributions is set to unity. Experimental error bars represent 1 σ. (Right) Calculated temperature dependence of the rotational distributions.

The dramatic propensity shown in Fig. 3 can be understood qualitatively from the conservation of symmetry, as described above (33). The B state of ozone has A′ symmetry and thus is symmetric with respect to the molecular plane. In the dissociation of a nonrotating parent, the O₂ rotational plane is identical to the ozone molecular plane at the time of dissociation. Accordingly, to conserve symmetry, the A′ Λ-doublet is formed exclusively, because it is also symmetric to this plane, as illustrated in Fig. 4. Out-of-plane rotation of the parent molecule tilts the fragment rotational plane with respect to the molecular plane at the time of dissociation, relaxing the aforementioned symmetry restriction and allowing for the formation of the A′′ Λ-doublet. This classical picture is borne out in quantum theory. As discussed with more detail in SI Appendix, the product rotational amplitude is modulated by a factor ${\left(-1\right)}^{\mathrm{\Lambda }}{d}_{K\mathrm{\Lambda }}^N+\epsilon d_{K-\mathrm{\Lambda }}^N$, in which Λ = 2, $K$ is the projection of the total angular momentum $\left(J^{tot}\right)$ onto the dissociation coordinate, and $d_{K\mathrm{\Lambda }}^N$ is the reduced Wigner rotational matrix. For even/odd N states of ¹⁶O₂(a¹Δ_g), as explained in SI Appendix, the reflection symmetry [ε = (−1)^N] is different, due to the exchange symmetry between the two bosonic ¹⁶O nuclei. As a result, when the parent molecule is not rotating $\left(K=J^{tot}=0\right)$, the two terms in the above factor cancel for odd N states because $d_{02}^N=d_{0-2}^N$, confirming the aforementioned classical model. For even N states, on the other hand, the two identical terms lead to doubling of the corresponding populations. For $K>0$, the selection rule is relaxed as $d_{K2}^N\ne d_{K-2}^N$. However, there is still a bias against odd states because $d_{K2}^N+d_{K-2}^N$ for even states is always larger than $d_{K2}^N-d_{K-2}^N$ for odd states. For mixed isotopomers, the indistinguishability is absent, and the reflection symmetry $\left(\epsilon \right)$ is no longer associated with the rotational quantum number N. Hence, the propensity rule no longer holds, again consistent with experimental observations (28).

Fig. 4.

illustration of the preference of the A′ Λ-doublet component of O₂ over its A″ counterpart during the dissociation of nonrotating O₃.

The dramatic propensity in the ¹⁶O₂ product can be understood as a consequence of the breakdown of the Born–Oppenheimer approximation. At the collinear geometry, the degenerate diabatic states are the eigenfunctions of the electronic angular momentum operator $\widehat{\text{Λ}}$, with eigenvalues of Λ = ±2. The corresponding adiabatic states (labeled by A′ and A″) are linear combinations of these diabatic states. As discussed above for $K=0$, the constructive interference of the two diabatic states gives rise to the doubling of population of the even-state population, while destructive interference leads to the complete cancellation for the odd states. For the dissociation on the B¹A′ state, as shown in Fig. 4, the in-plane dissociation preserves the A′ symmetry, thus leading to the even–odd propensity. We note in passing that total cross-section is not affected by the inclusion of the Λ-doublet in the dynamical treatment, as shown in SI Appendix, Fig. S1.

The insight provided by the aforementioned quantum-mechanical factor directly leads to a temperature effect, as higher-$K$ quantum states of the parent molecule are populated at higher temperatures. Consequently, the propensity becomes weakened at higher temperatures. In Fig. 3 (Right) the calculated temperature dependence of the rotational distribution is shown. The higher-temperature results were obtained by summing with the appropriate Boltzmann weights over product distributions from parent rotational states with significant populations at the experimental temperature (for detail, see SI Appendix). It is clear that the relative odd- vs. even-state populations become less pronounced at higher temperatures, in agreement with the observed temperature effect.

In addition to the rotational population alternation, even–odd alternations in the vector correlations of the O₂(a¹Δ_g) fragments have been previously measured (7, 8, 13, 15). Vector correlations describe the average angles between the transition dipole moment (μ), fragment recoil velocity vector (v), and fragment angular momentum vector (j), and can provide a window into photodissociation dynamics. For example, if v is perpendicular to j, the fragment recoils with a cartwheeling motion, while if v is parallel to j, the fragment recoils with a helicoptering motion. Hancock et al. were the first to report even–odd alternations in O₂(a¹Δ_g) photofragment vector correlations, and found that following the 270-nm photolysis of ozone at 140 K, the μ-j and v-j angles were smaller for the odd fragments than for the even fragments (7). Similar results have been reported for the photolysis of 140 K ozone at multiple wavelengths (8, 13, 15), and for the 266-nm photolysis of ozone at 70, 115, and 170 K (33). One explanation for the observed alternation is differences in the formation of the A′ and A′′ Λ-doublets. As initial out-of-plane motion will decrease the angles between the v and j and μ and j vectors, the authors suggested that parent out-of-plane motion may have a larger impact on the A′′ fragments than the A′ fragments (7), and that the A′′ fragments may be formed from a more rotationally excited parent population than the A′ fragments (13). This was later verified by temperature-dependent vector correlation measurements (33). While previous studies suggested that the even–odd rotational population alternations and differences in the vector correlations of odd and even states had different origins, recently a simple classical model based on Λ-doublet preference (33) was able to reproduce both experimental observations. The decreased μ-j and v-j angles for the A′′ fragments can be attributed to the increased probability of forming A′′ fragments with increasing out-of-plane parent rotation. A Λ-doublet propensity model successfully explains the temperature-dependent vector correlation and rotational population alternation measurements.

Conclusions

The experimentally observed even–odd population alternation in the rotational distribution of the ¹⁶O₂(a¹Δ_g) photoproducts has been reproduced by theory. This is made possible by including the O₂ electronic angular momentum in the product-state projection and allowing it to couple with its rotational angular momentum. Our quantum-mechanical theory reveals that the Λ-doublet populations are controlled by a factor that depends on $K$, the projection of the total rotational quantum number $\left(J^{tot}\right)$ onto the body-fixed axis defined as the vector from the O to the O₂ product. For $K=0$, the dissociation of ¹⁶O₃ within the molecular plane leads exclusively to A′ (even) component of the Λ-doublet of ¹⁶O₂. This selection rule can be readily understood as the dissociation of rotationless ozone in the B¹A′ state only allows the population of the A′ Λ-doublet of ¹⁶O₂, which corresponds to the even rotational states. This restriction is relaxed for $K>0$, but a preference for even rotational states persists. Thus, only with a rotationally excited parent population do the odd states become populated. The faster the starting parent rotates, the greater the relative population of the odd rotational states. This result is consistent with the experimental trend of increasing odd-state population with increasing parent ozone temperature. The success of theory in reproducing the experimental results presented in this work, as well as the observed temperature-dependent trend, provides conclusive evidence in favor of a Λ-doublet preference mechanism over parity-selective curve crossing. Consequently, the observed ¹⁷O and ¹⁸O enrichment in the atmosphere cannot be attributed to enhanced curve crossing of heteronuclear fragments.

Methods

A 5–10% mixture of ozone in helium was introduced into the ion-imaging apparatus via pulsed valve. The instrument has been describe in detail elsewhere (38, 39). Supersonic expansion into vacuum resulted in a 60 K beam of ozone. Ozone was photodissociated using the fourth harmonic (266 nm) of a neodymium-doped yttrium aluminum garnet (Nd:YAG) laser. The nascent O₂(a¹Δ_g) photoproducts were ionized using 2 + 1 resonant enhanced multiphoton ionization (REMPI) via the d¹Π_g(v = 4)←←a¹Δ_g(v = 0) transition around 302 nm. The probe wavelengths near 302 nm were generated by doubling the output of a dye laser (Rhodamine 610/640 mix) pumped by a second Nd:YAG laser. The resulting ions were accelerated toward a dual-chevron microplate and phosphor screen assembly detector. The subsequent phosphorescence was detected using a photomultiplier tube.

The photolysis wavelength of 266 nm was selected for this study because it is near the peak of the Hartley band, and photodissociation of ozone at 266 nm has been well characterized. The majority of O₂(a¹Δ_g) photoproducts resulting from the 266-nm photolysis of ozone are formed in the ground vibrational state (40, 41), so we have limited the measurement to the rotational distribution of v = 0 fragments. The nascent O₂(a¹Δ_g, v = 0) photofragments were probed using 2 + 1 REMPI via the unperturbed d¹Π_g (v = 4) state. The experimental rotational distribution was determined by fitting the resulting spectrum with a forward convolution simulation. Two photon line strengths used in the simulation were calculated according to the expressions from Bray and Hochstrasser (42), and rotational constants for the O₂(a¹Δ_g, v = 0) and O₂(d¹Π_g, v = 4) states were taken from Morrill et al. (43) Relative populations of the rotational states were adjusted to achieve the best agreement between the simulated and experimental spectra.

The quantum-dynamical calculations were performed with an approximate Hamiltonian, similar to our recent work on the Λ-doublet distribution in photodissociation of H₂O (44, 45). The model was based on a wave-packet implementation (46) of the photodissociation theory of Balint-Kurti and Shapiro (37), using the Chebyshev propagator (47). The initial wave packet on the excited B state was prepared from the ground vibrational state of O₃(X) associated with different rotational states, with the appropriate selection rules obeyed. The wave packet was propagated and projected onto the product quantum states in the asymptote at the energy determined by the experimental photon wavelength. Importantly, the product basis for the open-shell product accounts for the coupling among the O₃ rotational angular momentum, the O₂ electronic angular momentum, and O₂ rotational angular momentum, thus capable of determining the Λ-doublet state populations (48). This so-called sudden-recoupling scheme (49) is valid due to the extremely rapid nature of the direct dissociation process. The nuclear spin symmetry of the two ¹⁶O atoms was explicitly included in the model. More details of the theory and calculations are given in SI Appendix.

Data Availability.

All data discussed in the paper are available in the main text and SI Appendix.