The authors are interested in a characterization of dynamical state transitions by considering systems encoded by classical Hamiltonians, defined on phase spaces, following a point of view introduced by Wigner, and by adopting the transformation of the Hamiltonian in the Poincaré-Birkhoff normal form. In particular, the authors study the phase-space geometry associated with index 2 saddles of a potential energy surface and its influence on reaction dynamics for \(n\) degree-of-freedom Hamiltonian systems. In recent years, similar studies have been carried out for index 1 saddles of potential energy surfaces, and the phase-space geometry associated with classical transition state theory has been elucidated. In this case, the existence of a normally hyperbolic invariant manifold (NHIM) of saddle stability type has been shown, where the NHIM serves as the ‘anchor’ for the construction of dividing surfaces having the no-recrossing property and minimal flux. For the index 1 saddle case, the stable and unstable manifolds of the NHIM are co-dimension 1 in the energy surface and have the structure of spherical cylinders, and thus act as the conduits for reacting trajectories in phase space. The situation for index 2 saddles is quite different, and their relevance for reaction dynamics has not previously been fully recognized.
The authors show that NHIMs with their stable and unstable manifolds still exist, but that these manifolds by themselves lack sufficient dimension to act as barriers in the energy surface in order to constrain reactions. Rather, in the index 2 case there are different types of invariant manifolds, containing the NHIM and its stable and unstable manifolds, that act as co-dimension 1 barriers in the energy surface. These barriers divide the energy surface in the vicinity of the index 2 saddle into regions of qualitatively different trajectories exhibiting a wider variety of dynamical behavior than for the case of index 1 saddles. In particular, the authors identify a class of trajectories, which refer to as roaming trajectories, which are not associated with reaction along the classical minimum energy path (MEP).
Two examples are considered. The first involves isomerization on a potential energy surface with multiple (four) symmetry equivalent minima; the dynamics in the vicinity of the saddle enables a rigorous distinction to be made between stepwise (sequential) and concerted (hilltop crossing) isomerization pathways. The second example involves two potential minima connected by two distinct transition states associated with conventional index 1 saddles, and an index 2 saddle that sits between the two index \(1\) saddles. For the case of non-equivalent index 1 saddles, the authors’ analysis suggests a rigorous dynamical definition of non-MEP or roaming reactive events.
The paper splits in seven sections and one appendix. 1. Introduction. 2. Index 1-saddle: inverted harmonic oscillator. 3. Phase-space geometry and transport associated with a 3-degree-of-freedom quadratic Hamiltonian of a saddle-saddle-center equilibrium. 4. \(n\)-degree-of-freedom, higher-order terms in the Hamiltonian, and the Poincaré-Birkhoff normal form. 5. Comparison of phase-space geometry and trajectories for index 1 and index 2 saddles. 6. Index \(2\) saddles and isomerization dynamics. 7. Summary and conclusions. Appendix: Phase-space geometry and transport associated with an \(n\) degree-of-freedom quadratic Hamiltonian of a saddle-saddle-center-\(\cdots\)-center equilibrium point.
Remark. This paper discusses a very important subject, of great interest whether from the mathematical point of view or for its experimental implications. Really, a dynamic theory of chemical and nuclear reactions is an actual research subject under focus of the international mathematical community. Of course, since for such microscopic systems quantum effects are not neglectable ones, one can ask why to publish now a classical theory on reactions in quantum systems? The answer can be found in other works on the same subject quoted in references [H. Waalkens, R. Schubert and S. Wiggins, Nonlinearity 21, No. 1, R1–R118 (2008; Zbl 1153.81017)]. In fact, there is also a quantum version of this theory, obtained by means of a Weyl-type quantization of the classical theory.
However, let us emphasize that nowadays, after the development of the geometric theory of quantum (super) PDE’s, it is possible to give a more geometric and fundamental formulation of reactions at quantum level, taking into account also systems with number of particles changes. There the approximation obtained by means of the Weyl quantization of a classical theory does not work more. Furthermore, Weyl quantization corresponds to a linearization of the classic dynamic equation around its solutions, and hence discards nonlinear phenomena [see works on PDE’s quantization and quantum PDE’s by the reviewer of this paper, e.g. Quantized partial differential equations. Singapore: World Scientific (2004; Zbl 1067.58022)].