### Session V38: Focus Session: The Transition State in Physics, Chemistry, and Astrophysics II

 Thursday, March 19, 2009 8:00AM - 8:36AM V38.00001: Quantum Transition State Theory Invited Speaker: Holger Waalkens The main idea of Wigner's transition state theory (TST) is to compute reaction rates from the flux through a dividing surface placed between reactants and products. In order not to overestimate the rate the dividing surface needs to have the no- recrossing property, i.e. reactive trajectories cross the dividing surface exactly once, and nonreactive trajectories do not cross it at all. The long standing problem of how to construct such a diving surface for multi-degree-of-freedom systems was solved only recently using ideas from dynamical systems theory. Here a normal form allows for a local decoupling of the classical dynamics which leads to the explicit construction of the phase space structures that govern the reaction dynamics through transition states. The dividing surface is spanned by a normally hyperbolic manifold which is the mathematical manifestation of the transition state as an unstable invariant subsystem of one degree of freedom less than the full system. The mere existence of a quantum version of TST is discussed controversially in the literature. The key isssue is the presence of quantum mechanical tunneling which prohibits the existence of a local theory analogous to the classical case. Various approaches have been devloped to overcome this problem by propagating quantum wavefunctions through the transition state region. These approaches have in common that they are computationally very expensive which seriously limits their applicability. In contrast the approach by Roman Schubert, Stephen Wiggins and myself is local in nature. A quantum normal form allows us to locally decouple the quantum dynamics to any desired order in Planck's constant. This yields not only the location of the scattering and resonance wavefunctions relative to the classical phase space structures, but also leads to very efficient algorithms to compute cumulative reaction probabilities and Gamov-Siegert resonances which are the quantum imprints of the transition state. Thursday, March 19, 2009 8:36AM - 9:12AM V38.00002: Transition States in a Noisy Environment Invited Speaker: Thomas Bartsch Thursday, March 19, 2009 9:12AM - 9:24AM V38.00003: Statistical Theory of Asteroid Escape Rates Charles Jaffe Transition states in phase space are identified and shown to regulate the rate of escape of asteroids temporarily captured in circumplanetary orbits. The transition states, similar to those occurring in chemical reaction dynamics, are then used to develop a statistical semianalytical theory for the rate of escape of asteroids temporarily captured by Mars. Theory and numerical simulations are found to agree to better than 1\%. These calculations suggest that further development of transition state theory in celestial mechanics, as an alternative to large-scale numerical simulations, will be a fruitful approach to mass transport calculations. Thursday, March 19, 2009 9:24AM - 10:00AM V38.00004: Exploring remnants of invariants buried in a deep potential well in chemical reactions Invited Speaker: Tamiki Komatsuzaki How the reacting system climbs through saddles from one basin to another on potential energy surface has been one of the most intriguing subjects not only in chemistry but also physics and biology. This decade significant progress has been achieved in establishing the concept of the so-called transition state (TS), that is, a hypersurface of co-dimension one through which the system passes through only once from one basin to another [1-3]. However, there exist still open problems to be resolved; 1) how the no-return TS ceases or bifurcates as the energy increases [4], 2) how the stable/unstable invariant manifolds emanating from the normally hyperbolic invariant manifold \textit{wander} in deep potential wells in many-degrees of freedom (dofs) systems [5] or how one can generalize the \textit{remnant of invariant manifolds} [6] to many-dofs systems, 3) how one can generalize the concept of no-return TS besides the region of first-rank saddles. Related to the problem 2), most of all the chemical reaction theories assume that all of the available energy redistributes \textit{statistically} through the dofs of system in the reactant well before the reaction takes place. It is implicitly expected that the ratio of the measure occupied by tori in phase space to that of the ambient space decreases exponentially as the dimensionality of the system increases. Here we present a novel technique to scrutinize the remnant of invariants buried in chaos in many-degrees of freedom systems [7]. This is regarded as the remnants of a destroyed invariant manifold that may dominate the transport in phase space even at high energy regions where most of all tori vanish. We demonstrate the potentiality of our technique for HCN isomerization, where the conventional procedure based on a finite order truncation in the coordinate transformation of canonical perturbation theory prevent us from detecting remnants of invariants. \\[4pt] [1] T. Komatsuzaki \textit{et al.}, \textit{J. Chem. Phys.} \textbf{105}, 10838(1996); \textit{ibid}. \textbf{110}, 9160 (1999) \\[0pt] [2] W.S. Koon \textit{et al.}, \textit{Chaos} \textbf{10}, 427 (2000) \\[0pt] [3] T. Uzer \textit{et al.}, \textit{Nonlinearity} \textbf{15}, 957(2002); H. Waalkens\textit{ et al.}, \textit{Nonlinearity} \textbf{21}, 1 (2008) \\[0pt] [4] C.-B. Li\textit{ et al.}, \textit{Phys. Rev. Lett. }\textbf{97}, 028302 (2006) \\[0pt] [5] R. B. Shirts \textit{et al.}, \textit{J. Chem. Phys. }\textbf{77}, 5204 (1982) \\[0pt] [6] C. Jaff\'e\textit{ et al.}, \textit{Phys. Rev. A} \textbf{60}, 3833 (1999) \\[0pt] [7] H. Teramoto \textit{et al.} \textit{J. Chem. Phys. }\textbf{129} 094302~(2008); \textit{Phys. Rev. E} \textbf{78}, 017202 (2008) Thursday, March 19, 2009 10:00AM - 10:36AM V38.00005: Intramolecular energy transfer, driving mechanisms, and reaction rates for collective motions of clusters Invited Speaker: Tomohiro Yanao Conformational transitions of molecules, clusters, and biopolymers are typically large-amplitude collective motions that involve a large number of degrees of freedom in a coherent manner. One of the major challenges in modern molecular science is to understand the general mechanism for such collective motions. This talk highlights a novel dynamical mechanism for conformational transitions of atomic clusters in terms of intramolecular energy transfer and the driving forces for large-amplitude collective motions. First, we introduce a method of hyperspherical mode analysis, which generally classifies the (3n-6) internal modes of an arbitrary n-atom molecule into three gyration-radius modes, three twisting modes, and (3n-12) shearing modes. This hyperspherical mode analysis reveals that the gyration-radius modes are the primary collective modes that need to be activated in order for the system to achieve large-amplitude conformational transitions. Moreover, it illustrates how the twisting modes and the shearing modes critically initiate and trigger the conformational transitions by inducing the essential driving forces that activate the gyration-radius modes via mode coupling. Finally, we characterize this driving mechanism for conformational transitions from the viewpoint of the phase space geometry of the low-dimensional dynamical system of the gyration-radius modes. This reduced phase space geometry clearly accounts for the origin of non-statistical reaction rate processes of the clusters. The present method of hyperspherical mode analysis as well as the driving mechanism for collective motions could be widely applicable to conformational dynamics of complex molecular systems. Thursday, March 19, 2009 10:36AM - 10:48AM V38.00006: Collective coherent control: Synchronization of polarization in ferroelectric PbTiO$_{3}$ by shaped THz fields Tingting Qi , Young-Han Shin , Ka-Lo Yeh , Nelson Keith , Rappe Andrew Coherent optical control over ultrafast molecular behavior including chemical reactions has been explored in recent years, spurred by the application of optimal control theory and related methods and by the development of femtosecond pulse shaping techniques through which complex optical waveforms have been crafted and optimized to induce specified molecular responses. Here we propose and model theoretically the extension of coherent control to collective structural change. We show that properly shaped terahertz fields, resonant with selected lattice vibrational frequencies, could be used to move ions in ferroelectric crystals from their positions in an initial domain orientation along well defined collective microscopic paths into the positions they occupy in a new domain orientation. Collective coherent control will enable direct observation of fast highly nonlinear material responses and far-from-equilibrium structures that can be harnessed in electro-optic devices and non-volatile computer memory. Thursday, March 19, 2009 10:48AM - 11:00AM V38.00007: Classical-Quantum correspondence in isomerization dynamics: quantum eigenstates and classical Arnol'd web S. Keshavamurthy Recently, there has been a renaissance of sorts in chemical dynamics with researchers critically examining the validity of the two pillars of reaction rate theory - transition state theory and the Rice-Ramsperger-Kassel-Marcus (RRKM) theory. Since both theories have classical dynamics at their foundation, advances in our understanding of nonlinear dynamics and continuing efforts to characterize the phase space structure of systems with three or more degrees of freedom are beginning to yield crucial mechanistic insights into the dynamics. This talk focuses on a mechanistic understanding of the deviations from RRKM theory for a model isomerization problem with three degrees of freedom. Several studies have established that such systems are prime candidates for observing non-RRKM behavior\footnote{D. M. Leitner, Int. J. Quant. Chem. {\bf 75}, 523 (1999).}. The model is inspired, and generalized, from a much earlier study by De Leon and Berne\footnote{N. De Leon and B. J. Berne, J. Chem. Phys. {\bf 75}, 3495 (1981).}. We try to answer two of the questions posed in this early work by studying the intramolecular vibrational energy flow in the system from both classical and quantum viewpoints. Using a wavelet-based local frequency analysis it is possible to construct a useful representation of the classical phase space (Arnol'd web) highlighting the important dynamical structures. Insights into the dynamics originate from the various nonlinear resonances and phase space traps which potentially result in quantum eigenstates of varying degree of localization\footnote{D. M. Leitner and M. Gruebele, Mol. Phys. {\bf 106}, 433 (2008).}.