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

Abstract

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 quantumlike 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 H 2 O + H 2 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.

Mixed Quantum/Classical Theory for Collisional Quenching of PAHs in the Interstellar Media

“…Our standard full-coupled CC-MQCT calculations, which are expected to serve as a reference and thus need to be perfectly converged, were carried out in the usual way, using a fourth-order Runge–Kutta integrator with a constant time step set to a rather small value, Δ t = 50 a.u. The convergence of the approximate AT-MQCT calculations with the adiabatic step-size predictor was also rigorously checked, by varying the value of accuracy ϵ (see ref for details). The results presented in Figure are found to be entirely converged when ϵ = 10 –3 .…”

“…Here we briefly summarize the equations of motion propagated within framework of the mixed quantum/classical theory. A rigorous derivation of these formulas is available in the literature. − …”

“…During the second run the internal molecular basis set size is increased to the desired value, and only the quantum equation () for the evolution of probability amplitudes a mn is propagated, using the values of classical variables saved during the first run. The limiting case of this approach, when the first run is done using only the basis set of degenerate m states of the initial j channel, corresponds to the rotationally adiabatic trajectory method, which we call AT-MQCT . This method was found to be sufficiently accurate and very efficient for H 2 O + H 2 and for H 2 O + H 2 O collisions and will be tested below for the C 6 H 6 + He collisions.…”

“…This replaces a four-dimensional sampling of the initial conditions by one Monte Carlo sampling, which is very efficient. Even more computational speed can be obtained by decoupling 28 the propagation of quantum eq 5 from the classical eqs 1−4. This can be done by first propagating the whole system of coupled eqs 1−5 together, but using a small basis set of the internal molecular states (to make these calculations fast).…”

“…During the past decade we developed a mixed quantum/classical approach for the description of molecule + atom and molecule + molecule collisions. − In our method the internal (rotational and vibrational) states of molecules are treated quantum mechanically, which permits a prediction of observables for quantum transitions, such as state-to-state specific cross sections. At the same time, the translational motion of collision partners is described classically using the mean-field trajectories, which captures the most important information about the scattering process, enabling massive parallelism and thus offering a very substantial computational advantage.…”