Scaling Rules for Resonance Dynamics near a Saddle Point: The Pendulum as a Zero-Order Model † Matthew P. Jacobson ‡ and Mark S. Child* Physical and Theoretical Chemistry Laboratory, Oxford UniVersity, South Parks Road, Oxford OX1 3QZ, U.K. ReceiVed: December 14, 2000 The pendulum is the simplest zero-order model for an isomerizing vibrational mode (one which passes through a saddle point). We utilize the classical action/angle theory of the pendulum, for which new results are given in the appendix, to determine generic scaling laws between the quantum mechanical pendulum eigenvalue distribution and the coupling matrix elements. These scaling rules are more appropriate for isomerizing vibrational modes than are the usual harmonic oscillator scaling rules, encoded in traditional spectroscopic effective Hamiltonians, which break down catastrophically at a saddle point. As a simple example of resonant quantum dynamics in the vicinity of a saddle point, we analyze a system consisting of a pendulum model for bend/internal rotor motion, anharmonically coupled to a stretching harmonic oscillator, in qualitative agreement with the known dynamics of HCP. The dominance of just two of the infinite number of resonances, 2:1 and 4:1, at all energies including that of the saddle point, is related to the scaling properties of the zero-order pendulum model. I. Introduction The most common effective Hamiltonian models for molec- ular vibrations are expressed in terms of harmonic oscillator shift operators for the various (normal or local mode) vibrational degrees of freedom. That is, the harmonic oscillator is taken as the zero-order model, and anharmonicities are accounted for by (1) Dunham-type expansions for the zero-order energies (diagonal matrix elements) and (2) various anharmonic vibra- tional resonances, also expressed in terms of the shift operators, which couple the vibrational modes. In practice, such effective Hamiltonian models are usually derived by one of two methods. If a potential energy surface of sufficient accuracy is available, then perturbation theory (a series of successive unitary trans- formations) can be used to derive a (generally high order) effective Hamiltonian from the surface (e.g., refs 1 and 2). More commonly, the parameters are fitted directly to experimental or theoretical data. In this case, the parameters cannot be rigorously related to terms in the potential surface (i.e., they have no simple physical interpretation by themselves, although together they have predictive power), and, importantly in the context of this paper, the parameters included in the model are generally chosen without regard to consistency between the zeroth-order (diagonal) and coupling (off-diagonal) constants. The harmonic oscillator is not the only possible choice of the zero-order model. Efforts have been made to derive effective Hamiltonian models, based on Lie algebras, which implicitly use the Morse oscillator as a zero-order model (see, e.g., refs 3-6). These models, at least for certain molecules/vibrational modes, are more quickly convergent; that is, they can, in principle, provide more compact, physical models. In practical application, the primary difference between traditional effective Hamiltonians and those based on, e.g., SU(2) algebras, is the precise scaling of the matrix elements. In an anharmonic algebraic model, there is still a Dunham-like expansion (in Casimir operators) for the diagonal matrix elements, but the lowest order terms implicitly include effects of anharmonicity which are only accounted for at higher order in models based on the harmonic oscillator. Similar differences also occur in the scaling of the off-diagonal matrix elements; that is, the scaling of the off-diagonal matrix elements is also dictated by the underlying vibron model consistent with the SU(2) algebra. Neither the harmonic oscillator nor the Morse oscillator provides an adequate zero-order model for vibrational dynamics in the vicinity of a saddle point. Effective Hamiltonian models based on harmonic or anharmonic oscillators can, of course, successfully reproduce vibrational energetics/dynamics below a saddle point, and even above for those states which have negligible isomerizing character. 7,8 However, as explored here and in previous publications, 9,10 these models fail catastrophi- cally for vibrational dynamics/eigenstates that probe the vicinity of a saddle point. This paper focuses on the scaling of off-diagonal matrix elements in systems with a saddle point, which we demonstrate cannot be adequately reproduced, in a global sense, by effective Hamiltonian models based on har- monic or anharmonic oscillators. This breakdown is most evi- dent at energies near or above that of the saddle point, but harmonically coupled anharmonic oscillator models (the most common form for effective Hamiltonians) are also demonstrated to be inadequate well below the saddle-point energy. The focal point of this paper is a derivation of the scaling rules that are appropriate for a zero-order pendulum model, i.e., for motion in a simple sinusoidal potential V(1 - cos θ). This is one of the simplest models to explicitly include a local maximum and is particularly relevant to systems which can undergo bond-breaking internal rotation, 9,10 such as HCP- HPC 11 or acetylene-vinylidene. 7 The pendulum is also an attractive zero-order model because its scaling rules can be † Part of the special issue “William H. Miller Festschrift”. ‡ Present address: Department of Chemistry, Columbia University, 3000 Broadway, MC 3158, New York, NY 10027. * To whom correspondence should be addressed. 2834 J. Phys. Chem. A 2001, 105, 2834-2841 10.1021/jp0045080 CCC: $20.00 © 2001 American Chemical Society Published on Web 02/17/2001