Chemical Physics 76 (1983) 203-218 North-Holland Publishing Company zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA LOCAL NORMAL MODES AND VlBRATlONAL ADIABATIC POTENTIALS -- zyxwvutsrqponmlkjihgfed Noam AGMON * Received 25 January 1982: in final form 3 November 1957 Local normal-mode analysis for a collinear potential energy surface g?nrratss a system of curvilinear coordinaieh. zyxwvutsrqponmlkj which ;\re orthogonal in the mass-skewed system. The motion is locally separable in these coordinatrs. We compars the utility ai one of the normal modes as a transversal-vihralion coordinate. with the convcnrional choicr of the dire&on perpendicular w the reaction coordinate in the mass-skewed system. The comparison is done for IWO commonly used reaction coordinates: BEBO and the steepest-descent path. Results differ for dlfferent choices of directions and reaction conrdinarss. Fururr work ~hcwld concentrate on a choice of a reaction coordinate which is itself one of the normal modes. 1. Introduction The assumption of separability of the motion parallel and transverse to the reaction coordinate (RC) of a coliinear potential energy surface (PES) is essential for defining the vibrational frequencies for the transversal motion. Such an assumption is usually made at the saddle point in transition state theory (TST) [l-5], and all along the RC in evaluating vibrational adiabatic potentials (i.e. potential energy plus the energy of the uth trans- versal vibrational state [6]). The notion of vibra- tional adiabatic potentials (VAPs) is basic in the adiabatic theory of reactions [7-121 and in varia- tional TST [6,13-181, where the adiabatic barriers, rather than the saddle-point energy, determine the reaction probability. VAPs are also essential in procedures for evaluating tunneling corrections. which use the ground-state VAP as the tunneling potential [19], and in discussing resonances as +asi-bound states in VAP wells [20]. In applications, results may depend on two assumptions: * Work supporred by National Science Foundation grant num- her DMR-8107494. * Haim Wcizxnann Fellow for 1981-1982. Permansm address: Department of Physical Chemis@‘. The Hebrew Uniwrsity of Jerusalem. Jerusalem 91904. Israel. (a) The choice of the reaction coordinate. (b) The choice of direction for the transversal vibration_ The assumptions most often encountered in rhe literature are: (a) The reaction coordinate is taken as the steepest-descent path (SDP) [ 19.2 l] in mass-skewed coordinates (L4SC) [Z]. (b) The transversal vibration is perpendicular (in MSC) to the SDP (“ natural collision coordi- nates” [KS]). These two assumptions seem complsrel?_ ad hoc: why are these the correct choices for a dynamical calculation? In particular rhe basic separability property is not guaranteed to hold (the mised second derivative of the potential at rhc SDP along the directions parallel and perpendicular IO ir does not generally vanish). Can one propose a better justified procedure for calculating VAPs for a given PES? _A possible answer is to use [23_24] periodic Irajectories [Xl_ for which the motion is separable by definition (but only for a constant total energy!). The peri- odic trajectories were shown [23.X] to correspond to cstrema in VXPs q_ Hence they may be utilized to evaluate adiabatic barriers. but not rhs \vhols * Which is just another way of saying Ihat rhsy correspond 10 ellrema in the number of x~ws slang the RC [75]. 0301-0104/83/0000-0000/$03.00 6 19S3 North-Holland