Accurate Calculations of Rotationally Inelastic Scattering Cross Sections Using Mixed Quantum/Classical Theory

Symmetry Breaking: A Classic Example of Quantum Interference Captured by Mixed Quantum/Classical Theory

The phenomena of propensity and inverse propensity are explored using time-dependent mixed quantum classical theory, MQCT, in which the rotational motion of the molecule is treated quantum mechanically, whereas the scattering process is described classically. Good agreement with the results of accurate full-quantum calculations is reported for a closed shell approximation to the NO + Ar system. It is shown that MQCT reproduces both phenomena in a broad range of the final states of the molecule and for various initial rotational states, offering a unique time-dependent insight. It permits seeing that both propensity and inverse propensity occur due to efficient depopulation of some states at the early postcollisional stage of the scattering process, when the molecule exists in a coherent superposition of many excited states that span a very broad range of angular momentum quantum numbers, populated by an efficient stepladder process of many consecutive transitions with small Δj.

Adiabatic Trajectory Approximation: A New General Method in the Toolbox of Mixed Quantum/Classical Theory for Collisional Energy Transfer

A new version of the MQCT program is presented, which includes an important addition, adiabatic trajectory approximation (AT-MQCT), in which the equations of motion for the classical and quantum parts of the system are decoupled. This method is much faster, which permits calculations for larger molecular systems and at higher collision energies than was possible before. AT-MQCT is general and can be applied to any molecule + molecule inelastic scattering problem. A benchmark study is presented for H₂O + H₂O rotational energy transfer, an important asymmetric-top rotor + asymmetric-top rotor collision process, a very difficult problem unamenable to the treatment by other codes that exist in the community. Our results indicate that AT-MQCT represents a reliable computational tool for prediction of collisional energy transfer between the individual rotational states of two molecules, and this is valid for all combinations of state symmetries (such as para and ortho states of each collision partner).

MQCT. I. Inelastic Scattering of Two Asymmetric-Top Rotors with Application to H2O + H2O

A mixed quantum/classical theory (MQCT) for the inelastic collision of two asymmetric-top rotor molecules is developed. In this method, the quantum state-to-state transitions between the rotational states of molecules (internal) are treated quantum mechanically using the time-dependent Schrodinger equation, whereas their relative translational motion (responsible for scattering) is treated classically, using the average trajectory approach. Two versions of the formula for transition matrix elements are presented: a straightforward approach that uses numerical multidimensional quadrature over all the internal degrees of freedom and a more standard analytic approach that uses the expansion of the PES over the basis set of spherical harmonics. Adaptation to the case of identical molecules scattering is presented and is applied to the rotational excitation of two water molecules, H₂O + H₂O, using the PES from recent literature. Calculations of collisional excitation from the ground state of the system into a number of low-lying excited rotational states are carried out in a broad range of energies. Analysis of computed opacity functions shows a rather unusual scattering regime, dominated by a strong anisotropic long-range interaction (dipole-dipole). The coupled-states (CS) approximation is tested and found to agree semiquantitatively (within a factor of 2) with the fully coupled version of the method. Differential cross sections for the elastic scattering indicate a very narrow forward scattering peak.

Three sources of errors in the Ehrenfest treatment of inelastic scattering and possible ways of resolving them

In order to identify the origin of possible errors in the mixed quantum/classical approach to inelastic scattering [A. Semenov and D. Babikov, J. Chem. Phys. 140, 044306 (2014) and A. Semenov, M.-L. Dubernet, and D. Babikov, J. Chem. Phys. 141, 114304 (2014)], a simplified model is considered that consists of one intermolecular degree of freedom and two intramolecular states, coupled by a simple potential. For this system, analytic derivations are carried out to determine (i) the exact quantum mechanical solution of the inelastic scattering problem, (ii) a simplified version of it with all oscillatory terms neglected, and (iii) the Ehrenfest solution in which the translational motion is described by the mean-field trajectory while the internal molecular motion is treated by the time-dependent Schrodinger equation. It is shown that the appropriate choice of velocity for the mean-field trajectory permits to enforce microscopic reversibility and gives results in excellent agreement with full-quantum results. The average velocity method of Billing is rigorously derived as a limiting case (of this more general approach), when reversibility is enforced locally, at the initial moment of time only. It is demonstrated that errors of state-to-state transition probabilities in the Ehrenfest approach occur at lower values of total energy E if the magnitudes of excitation energy ΔE, potential energy difference between the two states ΔV, and coupling of two states V₁₂ are large. Possible ways of applying this concept to rotational transitions in real molecules are explored, using examples from CO + CO inelastic scattering.

Inelastic Scattering of Identical Molecules within Framework of the Mixed Quantum/Classical Theory: Application to Rotational Excitations in H2 + H2

Theoretical foundation is laid out for description of permutation symmetry in the inelastic scattering processes that involve collisions of two identical molecules, within the framework of the mixed quantum/classical theory (MQCT). In this approach, the rotational (and vibrational) states of two molecules are treated quantum-mechanically, whereas their translational motion (responsible for scattering) is treated classically. This theory is applied to H2 + H2 system, and the state-to-state transition cross sections are compared versus those obtained from the full-quantum calculations and experimental results from the literature. Good agreement is found in all cases. It is also found that results of MQCT, where the Coriolis coupling is included classically, are somewhat closer to exact full-quantum results than results of the other approximate quantum methods, where those coupling terms are neglected. These new developments allow applications of MQCT to a broad variety of molecular systems and processes.

Mixed Quantum/Classical Approach for Description of Molecular Collisions in Astrophysical Environments

An efficient and accurate mixed quantum/classical theory approach for computational treatment of inelastic scattering is extended to describe collision of an atom with a general asymmetric-top rotor polyatomic molecule. Quantum mechanics, employed to describe transitions between the internal states of the molecule, and classical mechanics, employed for description of scattering of the atom, are used in a self-consistent manner. Such calculations for rotational excitation of HCOOCH3 in collisions with He produce accurate results at scattering energies above 15 cm(-1), although resonances near threshold, below 5 cm(-1), cannot be reproduced. Importantly, the method remains computationally affordable at high scattering energies (here up to 1000 cm(-1)), which enables calculations for larger molecules and at higher collision energies than was possible previously with the standard full-quantum approach. Theoretical prediction of inelastic cross sections for a number of complex organic molecules observed in space becomes feasible using this new computational tool.

Mixed quantum/classical theory for inelastic scattering of asymmetric-top-rotor + atom in the body-fixed reference frame and application to the H₂O + He system

The mixed quantum/classical theory (MQCT) for inelastic molecule-atom scattering developed recently [A. Semenov and D. Babikov, J. Chem. Phys. 139, 174108 (2013)] is extended to treat a general case of an asymmetric-top-rotor molecule in the body-fixed reference frame. This complements a similar theory formulated in the space-fixed reference-frame [M. Ivanov, M.-L. Dubernet, and D. Babikov, J. Chem. Phys. 140, 134301 (2014)]. Here, the goal was to develop an approximate computationally affordable treatment of the rotationally inelastic scattering and apply it to H2O + He. We found that MQCT is somewhat less accurate at lower scattering energies. For example, below E = 1000 cm(-1) the typical errors in the values of inelastic scattering cross sections are on the order of 10%. However, at higher scattering energies MQCT method appears to be rather accurate. Thus, at scattering energies above 2000 cm(-1) the errors are consistently in the range of 1%-2%, which is basically our convergence criterion with respect to the number of trajectories. At these conditions our MQCT method remains computationally affordable. We found that computational cost of the fully-coupled MQCT calculations scales as n(2), where n is the number of channels. This is more favorable than the full-quantum inelastic scattering calculations that scale as n(3). Our conclusion is that for complex systems (heavy collision partners with many internal states) and at higher scattering energies MQCT may offer significant computational advantages.

Rotational quenching of H2O by He: mixed quantum/classical theory and comparison with quantum results

The mixed quantum/classical theory (MQCT) formulated in the space-fixed reference frame is used to compute quenching cross sections of several rotationally excited states of water molecule by impact of He atom in a broad range of collision energies, and is tested against the full-quantum calculations on the same potential energy surface. In current implementation of MQCT method, there are two major sources of errors: one affects results at energies below 10 cm(-1), while the other shows up at energies above 500 cm(-1). Namely, when the collision energy E is below the state-to-state transition energy ΔE the MQCT method becomes less accurate due to its intrinsic classical approximation, although employment of the average-velocity principle (scaling of collision energy in order to satisfy microscopic reversibility) helps dramatically. At higher energies, MQCT is expected to be accurate but in current implementation, in order to make calculations computationally affordable, we had to cut off the basis set size. This can be avoided by using a more efficient body-fixed formulation of MQCT. Overall, the errors of MQCT method are within 20% of the full-quantum results almost everywhere through four-orders-of-magnitude range of collision energies, except near resonances, where the errors are somewhat larger.