We test the Franck–Condon (FC) approximation for chemical reactions by prescribing a simple construction of quasiadiabatic potential energy surfaces and evaluating numerically FC overlap integrals for the collinear case of the chemical reactions H2(D2)+F and H+Cl2. The FC model is derivable from the exact transition matrix by the use of four basic approximations: neglect of virtual transitions to excited electronic states; Born–Oppenheimer approximation; neglect of nuclear–electronic couplings; and the Franck–Condon approximation. The wave functions involved in the FC overlap are determined from quasiadiabatic potential surfaces, which were chosen to be constructed from the corresponding LEPS and anti-LEPS adiabatic surfaces for the chemical reaction in question. A coupling function which involves a single free parameter is needed to connect the quasiadiabatic surfaces. Our calculations show that the results are insensitive to this free parameter. We calculate vibrational distributions of reaction products for various initial kinetic energies of reactants and find the results to be in good qualitative agreement with both exact quantal calculations and FC models which include further approximations (with compensatory ease of calculation). Our results agree with the maximum in the vibrational distribution predicted by the other calculations and show similar trends with variation in initial relative kinetic energy and the masses (including isotopic substitution) as well as certain features of the potential surface.
We calculate planar and three-dimensional angular distributions for the products of atom–diatom chemical reactions by means of the Franck–Condon (FC) model. The wave functions on the reactant and product quasiadiabatic surfaces are expanded in partial wave series. A local uncoupling of the different degrees of freedom, as justified earlier, is assumed and consequently the individual members of the partial wave series can be separated into products of angular factors and rovibration–translation factors. To evaluate these factors, we consider the limit of weak and strong potential, and weak and strong kinematic couplings. The center of mass differential cross section is obtained by means of the T matrix formalism, where the T matrix is approximated by a generalized Franck–Condon overlap of the reactantlike and productlike wave functions. We use several further satisfactory approximations, e.g., linearization of the potential in the region of maximal overlap, and semiclassical approximation to the oscillator wave functions, beyond those of the FC model to obtain an analytic expression for the T matrix. For assumed LEPS surfaces of the systems H+H2 →H2+H, H2+F→HF+H and H+Cl2→HCl+Cl, we calculate angular distributions of reaction products in the various coupling limits for ranges of final states. The angular distributions in the strong potential coupling limits have a Gaussian shape peaked about the backscattering angle (π) (the hard sphere deflection angle for the chosen critical configuration) for each of the three reactions studied. In all three cases the 3D angular distribution is narrower than the planar (2D) angular distribution. Our calculations show no difference between the angular distributions of the weak and strong kinematic coupling limits. The angular distribution of the 2D weak potential coupling case are broader than those of the strong potential coupling. For H+H2 we find our results in the strong potential limit to be in qualitative agreement with exact quantum mechanical calculations. The angular distribution for a given product state broadens as the initial relative kinetic energy is increased, in agreement with classical trajectory calculation (F+H2). The angular distribution is also predicted to broaden as the final relative velocity increases, in agreement with experiment (H+Cl2, F+H2). Finally we introduce several simplifying approximations to our analytical model and find that, for exothermic reactions like F+H2, the radial contribution to the T matrix is dominated by certain features of the potential: the barrier width, the slope of the potential on the reactant side, and force constants in the region of maximum overlap. Our analysis provides a basis for the formulation of reduced variables which may be of use in comparing reactions. Finally we discuss some sufficient conditions for the separability of product velocity and angular distributions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.