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

Abstract

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 l (maximum orbital angular momentum) similar to J 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.

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

“…Indeed, quantum resonances are entirely absent in our calculations, but this is expected since the adiabatic trajectory approximation by construct prohibits energy transfer in the first step of AT-MQCT calculations and therefore disables the mechanism of Feshbach resonance formation. 47 If the resonant behavior is present in the low-energy range and happens to be important, then the full-quantum calculations are probably indispensable.…”

“…The first term in eq includes the potential coupling matrix M n n ′ with real-valued, time-independent elements that depend on R and m (these dependencies are omitted in the equations above for transparency). Efficient methodologies for generation of the initial conditions for MQCT trajectories and for constructing cross sections from the final values of probability amplitudes, a mn , have been discussed in the recent literature and will not be repeated here. ,,,, …”

“…Efficient methodologies for generation of the initial conditions for MQCT trajectories and for constructing cross sections from the final values of probability amplitudes, a mn , have been discussed in the recent literature and will not be repeated here. 22,31,39,41,47 Adiabatic Trajectory Approximation. Formulas 1−5 represent a system of coupled differential equations of the first order.…”

“…Within the framework of MQCT, the internal molecular degrees of freedom are treated quantum mechanically using the time-dependent Schrödinger equation. Therefore, this method takes into account many quantum features of molecular rotations and vibrations, such as state quantization and zero-point energy preservation, symmetry restrictions on allowed and forbidden transitions, propensities of state-to-state transition cross sections, quantum interference effects, and several other quantum features that we keep investigating. At the same time, the translational motion of colliding partners responsible for their scattering in space is treated classically within the Ehrenfest mean-field trajectory approach. , This simplification was found to work well even for the lightest collision partners such as the He atom and H 2 molecule in a broad range of collision energies. ,,, Since the quantum treatment of scattering is avoided, the mixed quantum/classical method enables an affordable computational treatment of heavier collision partners and/or at higher collision energies, the limits when the standard full-quantum methods are not affordable computationally anymore.…”

“…Within the framework of MQCT, the internal molecular degrees of freedom are treated quantum mechanically using the time-dependent Schrodinger equation. Therefore, this method takes into account many quantum features of molecular rotations and vibrations, such as state quantization and zeropoint energy preservation, 46 symmetry restrictions on allowed and forbidden transitions, 31 propensities of state-to-state transition cross sections, 36 quantum interference effects, 47 and several other quantum features that we keep investigating. At the same time, the translational motion of colliding partners responsible for their scattering in space is treated classically within the Ehrenfest mean-field trajectory approach.…”