Several Levels of Theory for Description of Isotope Effects in Ozone: Symmetry Effect and Mass Effect

Quantum calculation of the quasi-equilibrium constant involved in the formation of 49O3

At the beginning of the 1980s, measurements in the stratosphere revealed an enrichment of the heavy isotopes 17O and 18O in ozone O3, described as anomalous because it was much higher than statistically expected. In addition, this was roughly equal between 17O and 18O. This effect, called mass-independent fractionation (MIF), remains unexplained despite numerous theoretical and experimental studies. However, it has been shown that the origin of MIF is directly linked to strong isotopic effects found in the recombination reaction O + O2 + M → O3 + M, where M is a third body that absorbs excess energy. Here, we are interested in one of the elementary processes involved in this reaction, namely the rapid quasi-equilibrium process O + O2⇌O3*, where O3* is metastable ozone. A fully quantum approach was used to calculate the quasi-equilibrium constant of this process, giving the most accurate value of this quantity to date for the 49O3 isotopic variant of ozone. The present work represents an important step toward a better understanding of ozone formation and may help contribute to unraveling the mystery of MIF.

Statistical adiabatic channel model for termolecular reactions

In this work, we present a statistical adiabatic channel model for termolecular reactions, A + B + C → Products. Our approach relies on hyperspherical coordinates, where the adiabatic channels are readily defined in the hyper-radius after averaging the hyperangular degrees of freedom. In this way, we find a general expression for termolecular rate constants. We focus on ion-neutral association reactions to test our approach's accuracy and predictive power, finding a good agreement between theory and experiment, especially in those reactions' temperature dependence.

Triple oxygen isotope constraints on atmospheric O2 and biological productivity during the mid-Proterozoic

Reconstructing the history of biological productivity and atmospheric oxygen partial pressure (pO₂) is a fundamental goal of geobiology. Recently, the mass-independent fractionation of oxygen isotopes (O-MIF) has been used as a tool for estimating pO₂ and productivity during the Proterozoic. O-MIF, reported as Δ'¹⁷O, is produced during the formation of ozone and destroyed by isotopic exchange with water by biological and chemical processes. Atmospheric O-MIF can be preserved in the geologic record when pyrite (FeS₂) is oxidized during weathering, and the sulfur is redeposited as sulfate. Here, sedimentary sulfates from the ∼1.4-Ga Sibley Formation are reanalyzed using a detailed one-dimensional photochemical model that includes physical constraints on air-sea gas exchange. Previous analyses of these data concluded that pO₂ at that time was <1% PAL (times the present atmospheric level). Our model shows that the upper limit on pO₂ is essentially unconstrained by these data. Indeed, pO₂ levels below 0.8% PAL are possible only if atmospheric methane was more abundant than today (so that pCO₂ could have been lower) or if the Sibley O-MIF data were diluted by reprocessing before the sulfates were deposited. Our model also shows that, contrary to previous assertions, marine productivity cannot be reliably constrained by the O-MIF data because the exchange of molecular oxygen (O₂) between the atmosphere and surface ocean is controlled more by air-sea gas transfer rates than by biological productivity. Improved estimates of pCO₂ and/or improved proxies for Δ'¹⁷O of atmospheric O₂ would allow tighter constraints to be placed on mid-Proterozoic pO₂.

Efficient Method for an Approximate Treatment of the Coriolis Effect in Calculations of Quantum Dynamics and Spectroscopy, with Application to Scattering Resonances in Ozone

A numerical approach is developed to capture the effect of rotation-vibration coupling in a practically affordable way. In this approach only a limited number of adjacent rotational components are considered to be coupled, while the couplings to other rotational components are neglected. This partially coupled (PC) approach permits to reduce the size of Hamiltonian matrix significantly, which enables the calculations of ro-vibrational states above dissociation threshold (scattering resonances) for large values of total angular momentum. This method is employed here to reveal the role of the Coriolis effect in the ozone formation reaction at room temperature, dominated by large values of total angular momentum states, on the order of J = 24 and 28. We found that, overall, the effect of ro-vibrational coupling is not minor for large J. Compared to the results of symmetric top rotor approximation, where the ro-vibrational coupling is neglected, we found that the widths of scattering resonances, responsible for the lifetimes of metastable ozone states, remain nearly the same (on average), but the number of these states increases by as much as 20%. We also found that these changes are nearly the same in symmetric and asymmetric ozone isotopomers ¹⁶O¹⁸O¹⁶O and ¹⁶O¹⁶O¹⁸O. Therefore, based on the results of these calculations, the Coriolis coupling does not seem to favor the formation of asymmetric ozone molecules and thus cannot be responsible for symmetry-driven mass-independent fractionation of oxygen isotopes.

Inelastic scattering in isotopologues of O2-Ar: the effects of mass, symmetry, and density of states

The two species considered here, O2 (oxygen molecule) and Ar (argon-atom), are both abundant components of Earth's atmosphere and hence familiar collision partners in this medium. O2 is quite reactive and extensively involved in atmospheric chemistry, including Chapman's cycle of the formation and destruction of ozone; while Ar, like N2, typically plays the nevertheless crucial role of inert collider. Inert species can provide stabilization to metastable encounter-complexes through the energy transfer associated with inelastic collisions. The interplay of collision frequency and energy transfer efficiency, with state lifetimes and species concentrations, contributes to the rich and varied chemistry and dynamics found in diverse environments ranging from planetary atmospheres to the interstellar and circumstellar media. The nature and density of bound and resonance states, coupled electronic states, symmetry, and nuclear spin-statistics can all play a role. Here, we systematically investigate some of those factors by looking at the O2-Ar system, comparing rigorous quantum-scattering calculations for the 16O16O-40Ar, 18O16O-40Ar, and 18O18O-40Ar isotope combinations. A new accurate potential energy surface was constructed for this purpose holding the O2 bond distance at its vibrationally averaged distance.

Direct time delay computation applied to the O + O2 exchange reaction at low energy: Lifetime spectrum of O3 * species

We report full quantum dynamical calculations for lifetimes of scattering resonances, among which are true metastable states, of the intermediate heavy ozone complex ⁵⁰O₃ ^* of the ¹⁸O + ¹⁶O¹⁶O reaction, for any value of the total angular momentum quantum number J. We show that computations for nonzero values of J are mandatory in order to properly analyze resonances and time delays, with a view to establish a somewhat comprehensive eigenlife spectrum of the complex O₃ ^*. Calculations have been performed in a given low to moderate energy range, including the interval between zero-point energies (ZPEs) of reagents and product species. Quasi-bound states tend to be more numerous, and eigenlifetimes themselves are seen to increase with J, reaching unusually large values for J = 30. A very dense forest of O₃ ^* species is pictured already for J greater than 20, especially at the highest energies considered, leading to a quasi-continuum of metastable states. On the contrary, they appear as rather sparse and isolated at J = 0 and lower energies, including the domain between ¹⁸O¹⁶O and ¹⁶O¹⁶O ZPEs, embedded among many overlapping resonances that turn out to be not long-lived enough to be associated with genuine metastable states.

Four Isotope-Labeled Recombination Pathways of Ozone Formation

A theoretical approach is developed for the description of all possible recombination pathways in the ozone forming reaction, without neglecting any process a priori, and without decoupling the individual pathways one from another. These pathways become physically distinct when a rare isotope of oxygen is introduced, such as ¹⁸O, which represents a sensitive probe of the ozone forming reaction. Each isotopologue of O₃ contains two types of physically distinct entrance channels and two types of physically distinct product wells, creating four recombination pathways. Calculations are done for singly and doubly substituted isotopologues of ozone, eight rate coefficients total. Two pathways for the formation of asymmetric ozone isotopomer exhibit rather different rate coefficients, indicating large isotope effect driven by ΔZPE-difference. Rate coefficient for the formation of symmetric isotopomer of ozone (third pathway) is found to be in between of those two, while the rate of insertion pathway is smaller by two orders of magnitude. These trends are in good agreement with experiments, for both singly and doubly substituted ozone. The total formation rates for asymmetric isotopomers are found to be somewhat larger than those for symmetric isotopomers, but not as much as in the experiment. Overall, the distribution of lifetimes is found to be very similar for the metastable states in symmetric and asymmetric ozone isotopomers.

The role of rotation-vibration coupling in symmetric and asymmetric isotopomers of ozone

A theoretical framework and a computer code (SpectrumSDT) are developed for accurate calculations of coupled rotational-vibrational states in triatomic molecules using hyper-spherical coordinates and taking into account the Coriolis coupling effect. Concise final formulas are derived for the construction of the Hamiltonian matrix using an efficient combination of the variational basis representation and discrete variable representation methods with locally optimized basis sets and grids. First, the new code is tested by comparing its results with those of the APH3D program of Kendrick et al. [Kendrick, Pack, Walker, and Hayes, J. Chem. Phys. 110, 6673 (1999)]. Then, accurate calculations of the rovibrational spectra are carried out for doubly substituted symmetric (¹⁸O¹⁶O¹⁸O) and asymmetric (¹⁸O¹⁸O¹⁶O) ozone isotopomers for the total angular momentum up to J = 5. Together with similar data recently reported for the singly substituted symmetric (¹⁶O¹⁸O¹⁶O) and asymmetric (¹⁶O¹⁶O¹⁸O) ozone isotopomers, these calculations quantify the role of the Coriolis coupling effect in the large mass-independent isotopic enrichment of ozone, observed in both laboratory experiments and the atmosphere of the Earth. It is found that the Coriolis effect in ozone is relatively small, as evidenced by deviations of its rotational constants from the symmetric-top-rotor behavior, magnitudes of parity splittings (Λ-doubling), and ratios of rovibrational partition functions for asymmetric vs symmetric ozone molecules. It is concluded that all of these characteristics are influenced by the isotopic masses as much as they are influenced by the overall symmetry of the molecule. It is therefore unlikely that the Coriolis coupling effect could be responsible for symmetry-driven mass-independent fractionation of oxygen isotopes in ozone.

Theoretical Treatment of the Coriolis Effect Using Hyperspherical Coordinates, with Application to the Ro-Vibrational Spectrum of Ozone

Several alternative methods for the description of the interaction between rotation and vibration are compared and contrasted using hyperspherical coordinates for a triatomic molecule. These methods differ by the choice of the z-axis and by the assumption of a prolate or oblate rotor shape of the molecule. For each case, a block-structure of the rotational-vibrational Hamiltonian matrix is derived and analyzed, and the advantages and disadvantages of each method are made explicit. This theory is then employed to compute ro-vibrational spectra of singly substituted ozone; roughly, 600 vibrational states of ¹⁶O¹⁸O¹⁶O and ¹⁶O¹⁶O¹⁸O isomers combined, with rotational excitations up to J = 5 and both inversion parities (21600 coupled ro-vibrational states in total). Splittings between the states of different parities, so-called K-doublings, are calculated and analyzed. The roles of the asymmetric-top rotor term and the Coriolis coupling term are determined individually, and it is found that they both affect these splittings, but in the opposite directions. Thus, the two effects partially cancel out, and the residual splittings are relatively small. Energies of the ro-vibrational states reported in this work for ¹⁶O¹⁸O¹⁶O and ¹⁶O¹⁶O¹⁸O are in excellent agreement with literature (available for low-vibrational excitation). New data obtained here for the highly excited vibrational states enable the first systematic study of the Coriolis effect in symmetric and asymmetric isotopomers of ozone.

Rotationally inelastic scattering of O3–Ar: state-to-state rates with the multiconfigurational time dependent Hartree method

The rates of state-changing collisions are compared for different isotopologues of ozone from quantum scattering calculations with the MCTDH method.

The Role of Ozone Vibrational Resonances in the Isotope Exchange Reaction 16O16O + 18O → 18O16O + 16O: The Time-Dependent Picture

We consider the time-dependent dynamics of the isotope exchange reaction in collisions between an oxygen molecule and an oxygen atom: ¹⁶O¹⁶O + ¹⁸O → ¹⁶O¹⁸O + ¹⁶O. A theoretical approach using the multiconfiguration time-dependent Hartree method was employed to model the time evolution of the reaction. Two potential surfaces available in the literature were used in the calculations, and the results obtained with the two surfaces are compared with each other as well as with results of a previous theoretical time-independent approach. A good agreement for the reaction probabilities with the previous theoretical results is found. Comparing the results obtained using two potential energy surfaces allows us to understand the role of the reef/shoulder-like feature in the minimum energy path of the reaction in the isotope exchange process. Also, it was found that the distribution of final products of the reaction is highly anisotropic, which agrees with experimental observations and, at the same time, suggests that the family of approximated statistical approaches, assuming a randomized distribution over final exit channels, is not applicable to this case.

Use of Isotope Effects To Understand the Present and Past of the Atmosphere and Climate and Track the Origin of Life

Stable isotope ratio measurements have been used as a measure of a wide variety of processes, including solar system evolution, geological formational temperatures, tracking of atmospheric gas and aerosol chemical transformation, and is the only means by which past global temperatures may be determined over long time scales. Conventionally, isotope effects derive from differences of isotopically substituted molecules in isotope vibrational energy, bond strength, velocity, gravity, and evaporation/condensation. The variations in isotope ratio, such as ¹⁸ O/¹⁶ O (δ¹⁸ O) and ¹⁷ O/¹⁶ O (δ¹⁷ O) are dependent upon mass differences with δ¹⁷ O/δ¹⁸ O=0.5, due to the relative mass differences (1 amu vs. 2 amu). Relations that do not follow this are termed mass independent and are the focus of this Minireview. In chemical reactions such as ozone formation, a δ¹⁷ O/δ¹⁸ O=1 is observed. Physical chemical models capture most parameters but differ in basic approach and are reviewed. The mass independent effect is observed in atmospheric species and used to track their chemistry at the modern and ancient Earth, Mars, and the early solar system (meteorites).

Development of a potential energy surface for the O3–Ar system: rovibrational states of the complex

Collisional stabilization is an important step in the process of atmospheric formation of ozone.