Mixed quantum/classical theory (MQCT) approach to the dynamics of molecule-molecule collisions in complex systems.

We developed a general theoretical approach and a user-ready computer code that permit study of the dynamics of collisional energy transfer and ro-vibrational energy exchange in complex molecule-molecule collisions. The method is a mixture of classical and quantum mechanics. The internal ro-vibrational motion of collision partners is treated quantum mechanically using a time-dependent Schrödinger equation that captures many quantum phenomena including state quantization and zero-point energy, propensity and selection rules for state-to-state transitions, quantum symmetry and interference phenomena. A significant numerical speed up is obtained by describing the translational motion of collision partners classically, using the Ehrenfest mean-field trajectory approach. Within this framework a family of approximate methods for collision dynamics is developed. Several benchmark studies for diatomic and triatomic molecules, such as H2O and ND3 collided with He, H2 and D2, show that the results of MQCT are in good agreement with full-quantum calculations in a broad range of energies, especially at high collision energies where they become nearly identical to the full quantum results. Numerical efficiency of the method and massive parallelism of the MQCT code permit us to embrace some of the most complicated collisional systems ever studied, such as C6H6 + He, CH3COOH + He and H2O + H2O. Application of MQCT to the collisions of chiral molecules such as CH3CHCH2O + He, and to molecule-surface collisions is also possible and will be pursued in the future.

Mixed quantum/classical calculations of rotationally inelastic scattering in the CO + CO system: a comparison with fully quantum results.

An updated version of the CO + CO potential energy surface from [R. Dawes, X. G. Wang and T. Carrington, J. Phys. Chem. A 2013, 117, 7612] is presented, that incorporates an improved treatment of the asymptotic behavior. It is found that this new surface is only slightly different from the other popular PES available for this system in the literature [G. W. M. Vissers, P. E. S. Wormer and A. Van Der Avoird, Phys. Chem. Chem. Phys. 2003, 5, 4767]. The differences are quantified by expanding both surfaces over a set of analytic functions and comparing the behavior of expansion coefficients along the molecule-molecule distance R. It is shown that all expansion coefficients behave similarly, except in the very high energy range at small R where the PES is repulsive. That difference has no effect on low collision-energy dynamics, which is explored via inelastic scattering calculations carried out using the MQCT program which implements the mixed quantum/classical theory for molecular energy exchange processes. The validity of MQCT predictions of state-to-state transition cross sections for CO + CO is also tested by comparison against full-quantum coupled-states calculations. In all cases MQCT gives reliable results, except at very low collision energy where the full-quantum calculations predict strong oscillations of state-to-state transition cross sections due to resonances. For strong transitions with large cross sections, the results of MQCT are reliable, especially at higher collision energy. For weaker transitions, and lower collision energies, the cross sections predicted by MQCT may be up to a factor of 2-3 different from those obtained by full-quantum calculations.

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.

Mixed quantum/classical theory for rotational energy exchange in symmetric-top-rotor + linear-rotor collisions and a case study of the ND3 + D2 system.

The extension of mixed quantum/classical theory (MQCT) to describe collisional energy transfer is developed for a symmetric-top-rotor + linear-rotor system and is applied to ND3 + D2. State-to-state transition cross sections are computed in a broad energy range for all possible processes: when both ND3 and D2 molecules are excited or both are quenched, when one is excited while the other is quenched and vice versa, when the ND3 state changes its parity while D2 is excited or quenched, and when ND3 is excited or quenched while D2 remains in the same state, ground or excited. In all these processes the results of MQCT are found to approximately satisfy the principle of microscopic reversibility. For a set of sixteen state-to-state transitions available from the literature for a collision energy of 800 cm-1 the values of cross sections predicted by MQCT are within 8% of accurate full-quantum results. A useful time-dependent insight is obtained by monitoring the evolution of state populations along MQCT trajectories. It is shown that, if before the collision, D2 is in its ground state, the excitation of ND3 rotational states proceeds through a two-step mechanism in which the kinetic energy of molecule-molecule collision is first used to excite D2 and only then is transferred to the excited rotational states of ND3. It is found that both potential coupling and Coriolis coupling play important roles in ND3 + D2 collisions.

Description of quantum interference using mixed quantum/classical theory of inelastic scattering.

Manifestation of the quantum interference effect in the oscillation of scattering cross section is explored using the N2 + O system as a case study. Calculations are carried out for two electronic PESs of the system, for various initial rotational states of N2, in a broad range of N2 + O collision energies and using three theoretical methods: two versions of the approximate mixed quantum/classical theory (MQCT and AT-MQCT) and the accurate full-quantum coupled-channel method (implemented in MOLSCAT). A good agreement between different methods is observed, especially at high energies. Elastic scattering cross-sections oscillate as a function of collision energy, which is the result of quantum interference. The effects of initial rotational excitation and of the PES properties are studied in detail. For the final (thermally averaged) cross sections, both MOLSCAT and MQCT calculations predict a rather regular pattern of quantum oscillations that persist through a broad range of collision energies and expand into the low-energy regime where quantum scattering resonances are common. The difference between cross sections predicted by MQCT and MOLSCAT decreases from ∼8% at low energies to ∼2% at high energies. Experimental data available at high collision energies are well reproduced.

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.

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.

Influence of the Coriolis effect on the properties of scattering resonances in symmetric and asymmetric isotopomers of ozone.

Scattering resonances above dissociation threshold are computed for four isotopically substituted ozone species: 16O18O16O, 16O16O18O, 18O16O18O and 16O18O18O, using a variational method with accurate treatment of the rotation-vibration coupling terms (Coriolis effect) for all values of the total angular momentum J from 0 to 4. To make these calculations numerically affordable, a new approach was developed which employs one vibrational basis set optimized for a typical rotational excitation (J,Λ), to run coupled rotation-vibration calculations at several desired values of J. In order to quantify the effect of Coriolis coupling, new data are contrasted with those computed using the symmetric-top rotor approximation, where the rotation-vibration coupling terms are neglected. It is found that, overall, the major properties of scattering resonances (such as their lifetimes, the number of these states, and their cumulative partition function Q) are all influenced by the Coriolis effect and this influence grows as the angular momentum J is raised. However, it is found that the four isotopically substituted ozone molecules are affected roughly equally by the Coriolis coupling. When the ratio η of partition functions for asymmetric over symmetric ozone molecules is computed, the Coriolis effect largely cancels, and this cancelation seems to occur for all values of J. Therefore, it does not seem grounded to attribute any appreciable mass-independent symmetry-driven isotopic fractionation to the Coriolis coupling effect.

Solving complex eigenvalue problems on a quantum annealer with applications to quantum scattering resonances.

Quantum computing is a new and rapidly evolving paradigm for solving chemistry problems. In previous work, we developed the Quantum Annealer Eigensolver (QAE) and applied it to the calculation of the vibrational spectrum of a molecule on the D-Wave quantum annealer. However, the original QAE methodology was applicable to real symmetric matrices only. For many physics and chemistry problems, the diagonalization of complex matrices is required. For example, the calculation of quantum scattering resonances can be formulated as a complex eigenvalue problem where the real part of the eigenvalue is the resonance energy and the imaginary part is proportional to the resonance width. In the present work, we generalize the QAE to treat complex matrices: first complex Hermitian matrices and then complex symmetric matrices. These generalizations are then used to compute a quantum scattering resonance state in a 1D model potential for O + O collisions. These calculations are performed using both a software (classical) annealer and hardware annealer (the D-Wave 2000Q). The results of the complex QAE are also benchmarked against a standard linear algebra library (LAPACK). This work presents the first numerical solution of a complex eigenvalue problem of any kind on a quantum annealer, and it is the first treatment of a quantum scattering resonance on any quantum device.

Adiabatic Trajectory Approximation within the Framework of Mixed Quantum/Classical Theory.

A hierarchy of approximate methods is proposed for solving the equations of motion within a framework of the mixed quantum/classical theory (MQCT) of inelastic molecular collisions. Of particular interest is a limiting case: the method in which the classical-like equations of motion for the translational degrees of freedom (scattering) are decoupled from the quantum-like equations for time evolution of the internal molecular states (rotational and vibrational). In practice, trajectories are pre-computed during the first step of calculations with driving forces determined solely by the potential energy surface of the entrance channel, which is an adiabatic trajectory approximation. Quantum state-to-state transition probabilities are computed in the second step of calculations with an expanded basis and very efficient step-size adjustment. Application of this method to H2O + H2 rotationally inelastic scattering shows a significant computational speedup by 2 orders of magnitude. The results of this approximate propagation scheme are still rather accurate, as demonstrated by benchmarking against more rigorous calculations in which the quantum and classical equations of motion are held coupled and against the full-quantum coupled-channel calculations from the literature. It is concluded that the AT-MQCT method (the adiabatic trajectory version of MQCT) represents a promising tool for the computational treatment of molecular collisions and energy exchange.

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.

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.

Recombination reactions as a possible mechanism of mass-independent fractionation of sulfur isotopes in the Archean atmosphere of Earth.

A hierarchy of isotopically substituted recombination reactions is formulated for production of sulfur allotropes in the anoxic atmosphere of Archean Earth. The corresponding system of kinetics equations is solved analytically to obtain concise expressions for isotopic enrichments, with focus on mass-independent isotope effects due to symmetry, ignoring smaller mass-dependent effects. Proper inclusion of atom-exchange processes is shown to be important. This model predicts significant and equal depletions driven by reaction stoichiometry for all rare isotopes: 33S, 34S, and 36S. Interestingly, the ratio of capital [Formula: see text] values obtained within this model for 33S and 36S is -1.16, very close to the mass-independent fractionation line of the Archean rock record. This model may finally offer a mechanistic explanation for the striking mass-independent fractionation of sulfur isotopes that took place in the Archean atmosphere of Earth.

The ratio of the number of states in asymmetric and symmetric ozone molecules deviates from the statistical value of 2.

Accurate calculations of vibrational states in singly and doubly substituted ozone molecules are carried out, up to dissociation threshold. Analysis of these spectra reveals noticeable deviations from the statistical factor of 2 for the ratio between the number of states in asymmetric and symmetric ozone molecules. It is found that, for the lower energy parts of spectra, the ratio is less than 2 in the singly substituted ozone molecules, but it is more than 2 in the doubly substituted ozone molecules. However, the upper parts of spectra, just below dissociation thresholds, exhibit a different behavior. In this energy range, the singly and doubly substituted ozone molecules behave similar, with the ratio of states in asymmetric and symmetric ozone molecules being more than 2 in both cases. This property may contribute to an explanation of the mysterious η-effect in the ozone forming reaction that favors the formation of the asymmetric ozone molecules.

Properties of Feshbach and "shape"-resonances in ozone and their role in recombination reactions and anomalous isotope effects.

Computational modelling of recombination reactions that form ozone require the inclusion of several quantum mechanical effects such as symmetry, zero-point energy, scattering resonances and tunneling. Major elements of theory for rigorous description of this process are reviewed, with emphasis on interpreting the famous anomalous isotope effect due to substitutions of 18O. Three reaction pathways, for the formation of symmetric and asymmetric isotopologues of ozone, are introduced and a hierarchy of theory levels is outlined. Lower levels of theory are used to account for the effects of symmetry, isotope mass, rotational excitations and vibrational zero-point energy differences. They happen to be equivalent to statistical descriptions of the process and do not show anomalous isotope effects. Properties of scattering resonances should be included at the next level of theory, and may finally explain the isotope effect. Shape resonances, trapped behind the centrifugal barrier and populated by tunneling, can be studied by neglecting couplings between the diabatic ro-vibrational states of the system. Inclusion of these couplings enables the formation of Feshbach resonances. Accurate calculations using hyper-spherical coordinates are performed to obtain resonance energies, lifetimes and wavefunctions. Differences between the shape resonances and Feshbach resonances are emphasized.

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

The essential components of theory for the description of isotope effects in recombination reaction that forms ozone are presented, including the introduction of three reaction pathways for symmetric and asymmetric isotopomers, a brief review of relevant experimental data for singly- and doubly substituted isotopologues, the definitions of ζ-effect and η-effect, and the introduction of isotopic enrichment δ. Two levels of theory are developed to elucidate the role of molecular symmetry, atomic masses, vibrational zero-point energies, and rotational excitations in the recombination process. The issue of symmetry is not trivial, since the important factors, such as 1/2 and 2, appear in seven different places in the formalism. It is demonstrated that if all these effects are taken into account properly, then no anomalous isotope effects emerge. At the next level of theory, a model is considered in which one scattering resonance (sitting right at the top of centrifugal barrier) is introduced per ro-vibrational channel. It is found that this approach is equivalent to statistical treatment with partition functions at the transition state. Accurate calculations using hyper-spherical coordinates show that no isotope effects come from difference in the number of states. In contrast, differences in vibrational and rotational energies lead to significant isotope effects. However, those effects appear to be local, found for the rather extreme values of rotational quantum numbers. They largely cancel when rate coefficients are computed for the thermal distribution of rotational excitations. Although large isotope effects (observed in experiments) are not reproduced here, this level of theory can be used as a foundation for more detailed computational treatment, with accurate information about resonance energies and lifetimes computed and included.

Calculations of Differential Cross Sections Using Mixed Quantum/Classical Theory of Inelastic Scattering.

It is shown that the mixed quantum/classical theory (MQCT) for the description of molecular scattering is considerably improved by using integer values of orbital angular momentum l, just like in quantum theory, instead of treating it as a continuous classical variable related to the impact parameter. This conclusion is justified by the excellent accuracy of the modified theory for prediction of the differential cross sections, at various values of collision energy and in both forward and backward scattering regimes. This approach requires fewer trajectories, compared to the random Monte Carlo sampling, and the only convergence parameter is lmax (maximum orbital angular momentum) similar to Jmax in the full quantum theory (maximum total angular momentum). Calculations of differential and integral cross sections for elastic and inelastic channels are presented, and the role of the scattering phase is discussed. The low-energy range is analyzed in detail to obtain insight into how the mixed quantum/classical treatment works in the scattering regime dominated by resonances. The differential cross section for rotationally inelastic scattering, computed by MQCT approach, is presented for the first time.