Mixed quantum/classical theory of rotationally and vibrationally inelastic scattering in space-fixed and body-fixed reference frames

Abstract

We formulated the mixed quantum/classical theory for rotationally and vibrationally inelastic scattering process in the diatomic molecule + atom system. Two versions of theory are presented, first in the space-fixed and second in the body-fixed reference frame. First version is easy to derive and the resultant equations of motion are transparent, but the state-to-state transition matrix is complexvalued and dense. Such calculations may be computationally demanding for heavier molecules and/or higher temperatures, when the number of accessible channels becomes large. In contrast, the second version of theory requires some tedious derivations and the final equations of motion are rather complicated (not particularly intuitive). However, the state-to-state transitions are driven by realvalued sparse matrixes of much smaller size. Thus, this formulation is the method of choice from the computational point of view, while the space-fixed formulation can serve as a test of the bodyfixed equations of motion, and the code. Rigorous numerical tests were carried out for a model system to ensure that all equations, matrixes, and computer codes in both formulations are correct.

Recent Advances in Development and Applications of the Mixed Quantum/Classical Theory for Inelastic Scattering

“…This presentation summarizes derivations and generalizes results of several earlier theory papers on diatomic + atom, 54,55,57,59,60 polyatomic + atom, 58,61,62 and diatomic + diatomic 63 systems. The point we convey here is that MQCT equations have the same form for any system of two collision partners, the difference is only in the meaning of indexes and in the structure of state-to-state transition matrix.…”

“…The coupled-states (CS) approximation is obtained readily by neglecting this term. 57,60 In contrast, matrix M in eq 2 describes transitions between states n, and is computed separately for every m-component of j using the potential energy surface V(R,ω) as follows: (4) This is a potential coupling matrix and it should be computed numerically. Elements of M are real and depend on R only, which is the length of the vector Q, that itself evolves during the collision.…”

“…62 Equations of motion for classical degrees of freedom (coordinates of the vector Q) are obtained using the Ehrenfest approach. 57 The resultant differential equations for R(t) and Φ(t) also include matrixes M and V, as a commutator: 57,60,61 (5a)…”

“…For the simplest case of a diatomic + atom we have simply n ≡ {j} and Ψmn(ω) ≡ Yj m (θ,φ), i.e., spherical harmonics. 57 For an asymmetric-top rotor (general polyatomic molecule) + atom we should set n ≡ {j, ka, kc} and determine wave functions Ψmn(ω) ≡ Ψmjkakc(α,β,γ) by diagonalization of the rotational Hamiltonian in the basis set of Wigner D-functions. 61 In either case, the energy En of eigenstate depends on n only and does not depend on m, which is a projection of angular momentum vector j of the molecule onto vector Q, which plays the role of z-axis in the BF reference frame.…”

“…In our recent work, [57][58][59][60][61][62][63] summarized in this Feature Article, we carried out a systematic development of theory for this second more rigorous version of MQCT, incorporating phase information into the formalism (to compute elastic and differential over scattering angle cross sections), expanding MQCT onto the general case of an asymmetric-top rotor (suitable for description of polyatomic molecules), and including the case of molecule + molecule collisions (which makes this overall theory complete). Simultaneously, we conducted a rigorous benchmark tests of MQCT, applying it to six different molecular systems, computing elastic and inelastic, total and differential cross sections for rotational excitation and quenching, of light and heavy collision partners, with low and high levels of the initial excitation and in a broad range of collision energies from few wavenumbers up to 10 4 cm -1 .…”