4 September 1998 Ž . Chemical Physics Letters 293 1998 448–454 Semiclassical treatment of resonances in the collinear O q HO exchange reaction Gennady V. Mil’nikov 1 , Chaoyuan Zhu, Hiroki Nakamura, Vladimir I. Osherov 2 DiÕision of Theoretical Studies, Institute for Molecular Science, Myodaiji, Okazaki 444, Japan Received 27 February 1998; in final form 6 July 1998 Abstract Feshbach type sharp resonances appearing in the O qHO collinear reaction are analyzed by the newly completed semiclassical theory for the Landau-Zener-Stueckelberg type curve crossings. Not only the resonance positions but also the widths are nicely reproduced in comparison with the exact numerical calculations, even when the widths are as small as 10 y8 ;10 y11 au. The semiclassical theory is extended so as to be applicable to the case that the bottom of the upper adiabatic potential is higher than the crossing point. q 1998 Elsevier Science B.V. All rights reserved. 1. Introduction The linear Landau-Zener model has been widely used for describing the nonadiabatic transitions in w x atomic, ionic and molecular collisions 1,2 . The original solutions of this model found by Landau, w x Zener and Stueckelberg 3–5 were based on the assumption of the localization of the nonadiabatic transition and the classical motion of the nuclei, i.e., the high energy approximation. Numerous attempts have been made to improve the applicability of this Ž wx old theory see for example Ref. 6 and references . therein . However, no accurate theory has been de- vised which can be used at low energies close to or lower than the crossing point. The semiclassical solution recently completed by 1 Permanent address: Institute of Structural Macrokinetics, Rus- sian Academy of Sciences, Chernogolovka, Moscow region, Rus- sia. 2 Permanent address: Institute of Chemical Physics, Russian Academy of Sciences, Chernogolovka, Moscow region, Russia. w x Ž . Zhu and Nakamura 2,7 ZN can cover a wide range of energy and coupling strength for both Lan- Ž . dau-Zener and nonadiabatic tunneling NT type transitions. For the present purpose we shall need the solution for the Landau-Zener type in which the two diabatic potential curves cross with the same sign of slopes. The reduced scattering matrix S R is ex- pressed as a function of the two adiabatic phases along adiabatic potentials, and the two diabatic pa- rameters a 2 and b 2 which can be determined from wx adiabatic potentials directly along the real axis 8 : 2 " 2 2 ' a s d y 1 2 0 0 mt y t E R y E R Ž . Ž . Ž . Ž . q y q 0 y 0 1.1 Ž . and E y E R q E R r2 Ž . Ž . Ž . q 0 y 0 2 2 ' b s d y 1 E R y E R r2 Ž . Ž . Ž . q 0 y 0 1.2 Ž . 0009-2614r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. Ž . PII: S0009-2614 98 00791-X