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Stephen Wiggins - One of the best experts on this subject based on the ideXlab platform.
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tuning the branching ratio in a symmetric potential energy surface with a post transition state bifurcation using external time dependence
Chemical Physics Letters, 2020Co-Authors: Victor J Garciagarrido, Matthaios Katsanikas, M Agaoglou, Stephen WigginsAbstract:Abstract Chemical selectivity, as quantified by a branching ratio, is a phenomenon relevant for many organic chemical reactions. It may be exhibited on a potential energy surface (PES) that features a valley-ridge inflection point (VRI) in the region between two sequential index-1 Saddles, with one Saddle having higher energy than the other. Reaction occurs when a trajectory crosses the region of the higher energy Saddle (the “entrance channel”) and approaches the lower energy Saddle. On both sides of the lower energy Saddle, there are two wells and the question we address is that given an initial ensemble of trajectories, what is the relative fraction of trajectories that enter each well. For a symmetric PES this fraction is 1 : 1 . We consider a symmetric PES subject to a time-periodic forcing characterized by an amplitude, frequency, and phase. In this letter we analyse how the branching ratio depends on these three parameters.
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phase space structure and transport in a caldera potential energy surface
arXiv: Chaotic Dynamics, 2018Co-Authors: Matthaios Katsanikas, Stephen WigginsAbstract:We study phase space transport in a 2D caldera potential energy surface (PES) using techniques from nonlinear dynamics. The caldera PES is characterized by a flat region or shallow minimum at its center surrounded by potential walls and multiple symmetry related index one Saddle points that allow entrance and exit from this intermediate region.We have discovered four qualitatively distinct cases of the structure of the phase space that govern phase space transport. These cases are categorized according to the total energy and the stability of the periodic orbits associated with the family of the central minimum, the bifurcations of the same family, and the energetic accessibility of the index one Saddles. In each case we have computed the invariant manifolds of the unstable periodic orbits of the central region of the potential and the invariant manifolds of the unstable periodic orbits of the families of periodic orbits associated with the index one Saddles. We have found that there are three distinct mechanisms determined by the invariant manifold structure of the unstable periodic orbits govern the phase space transport. The first mechanism explains the nature of the entrance of the trajectories from the region of the low energy Saddles into the caldera and how they may become trapped in the central region of the potential. The second mechanism describes the trapping of the trajectories that begin from the central region of the caldera, their transport to the regions of the Saddles, and the nature of their exit from the caldera. The third mechanism describes the phase space geometry responsible for the dynamical matching of trajectories originally proposed by Carpenter and described in Collins et al. (2014) for the two dimensional caldera PES that we consider.
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index k Saddles and dividing surfaces in phase space with applications to isomerization dynamics
Journal of Chemical Physics, 2011Co-Authors: Peter Collins, Gregory S Ezra, Stephen WigginsAbstract:In this paper, we continue our studies of the phase space geometry and dynamics associated with index k Saddles (k > 1) of the potential energy surface. Using Poincare-Birkhoff normal form (NF) theory, we give an explicit formula for a "dividing surface" in phase space, i.e., a codimension one surface (within the energy shell) through which all trajectories that "cross" the region of the index k Saddle must pass. With a generic non-resonance assumption, the normal form provides k (approximate) integrals that describe the Saddle dynamics in a neighborhood of the index k Saddle. These integrals provide a symbolic description of all trajectories that pass through a neighborhood of the Saddle. We give a parametrization of the dividing surface which is used as the basis for a numerical method to sample the dividing surface. Our techniques are applied to isomerization dynamics on a potential energy surface having four minima; two symmetry related pairs of minima are connected by low energy index 1 Saddles, with the pairs themselves connected via higher energy index 1 Saddles and an index 2 Saddle at the origin. We compute and sample the dividing surface and show that our approach enables us to distinguish between concerted crossing ("hilltop crossing") isomerizing trajectories and those trajectories that are not concerted crossing (potentially sequentially isomerizing trajectories). We then consider the effect of additional "bath modes" on the dynamics, by a study of a four degree-of-freedom system. For this system we show that the normal form and dividing surface can be realized and sampled and that, using the approximate integrals of motion and our symbolic description of trajectories, we are able to choose initial conditions corresponding to concerted crossing isomerizing trajectories and (potentially) sequentially isomerizing trajectories.
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index k Saddles and dividing surfaces in phase space with applications to isomerization dynamics
arXiv: Statistical Mechanics, 2011Co-Authors: Peter Collins, Gregory S Ezra, Stephen WigginsAbstract:In this paper we continue our studies of the phase space geometry and dynamics associated with index k Saddles (k > 1) of the potential energy surface. Using normal form theory, we give an explicit formula for a "dividing surface" in phase space, i.e. a co-dimension one surface (within the energy shell) through which all trajectories that "cross" the region of the index k Saddle must pass. With a generic non-resonance assumption, the normal form provides k (approximate) integrals that describe the Saddle dynamics in a neighborhood of the index k Saddle. These integrals provide a symbolic description of all trajectories that pass through a neighborhood of the Saddle. We give a parametrization of the dividing surface which is used as the basis for a numerical method to sample the dividing surface. Our techniques are applied to isomerization dynamics on a potential energy surface having 4 minima; two symmetry related pairs of minima are connected by low energy index one Saddles, with the pairs themselves connected via higher energy index one Saddles and an index two Saddle at the origin. We compute and sample the dividing surface and show that our approach enables us to distinguish between concerted crossing ("hilltop crossing") isomerizing trajectories and those trajectories that are not concerted crossing (potentially sequentially isomerizing trajectories). We then consider the effect of additional "bath modes" on the dynamics, which is a four degree-of-freedom system. For this system we show that the normal form and dividing surface can be realized and sampled and that, using the approximate integrals of motion and our symbolic description of trajectories, we are able to choose initial conditions corresponding to concerted crossing isomerizing trajectories and (potentially) sequentially isomerizing trajectories.
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phase space geometry and reaction dynamics near index 2 Saddles
Journal of Physics A, 2009Co-Authors: Gregory S Ezra, Stephen WigginsAbstract:We 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 (DoF) 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. We 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, we can identify a class of trajectories, which we refer to as 'roaming trajectories', which are not associated with reaction along the classical minimum energy path (MEP). We illustrate the significance of our analysis of the index 2 Saddle for reaction dynamics with two examples. 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, our analysis suggests a rigorous dynamical definition of 'non-MEP' or 'roaming' reactive events.
Mithat Unsal - One of the best experts on this subject based on the ideXlab platform.
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toward picard lefschetz theory of path integrals complex Saddles and resurgence
Annals of Mathematical Sciences and Applications, 2017Co-Authors: Alireza Behtash, Gerald V Dunne, Thomas Schafer, Tin Sulejmanpasic, Mithat UnsalAbstract:We show that the semi-classical analysis of generic Euclidean path integrals necessarily requires complexification of the action and measure, and consideration of complex Saddle solutions.We demonstrate that complex Saddle points have a natural interpretation in terms of the Picard–Lefschetz theory. Motivated in part by the semi-classical expansion of QCD with adjoint matter on $\mathbb{R}^3 \times S^1$, we study quantum-mechanical systems with bosonic and fermionic (Grassmann) degrees of freedom with harmonic degenerate minima, as well as (related) purely bosonic systems with harmonic nondegenerate minima. We find exact finite action non-BPS bounce and bion solutions to the holomorphic Newton equations. We find not only real solutions, but also complex solution with non-trivial monodromy, and finally complex multi-valued and singular solutions. Complex bions are necessary for obtaining the correct nonperturbative structure of these models. In the supersymmetric limit the complex solutions govern the ground state properties, and their contribution to the semiclassical expansion is necessary to obtain consistency with the supersymmetry algebra. The multi-valuedness of the action is either related to the hidden topological angle or to the resurgent cancellation of ambiguities. We also show that in the approximate multi-instanton description the integration over the complex quasi-zero mode thimble produces the most salient features of the exact solutions. While exact complex Saddles are more difficult to construct in quantum field theory, the relation to the approximate thimble construction suggests that such solutions may be underlying some remarkable features of approximate bion Saddles in quantum field theories.
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deconstructing zero resurgence supersymmetry and complex Saddles
arXiv: High Energy Physics - Theory, 2016Co-Authors: Gerald V Dunne, Mithat UnsalAbstract:We explain how a vanishing, or truncated, perturbative expansion, such as often arises in semi-classically tractable supersymmetric theories, can nevertheless be related to fluctuations about non-perturbative sectors via resurgence. We also demonstrate that, in the same class of theories, the vanishing of the ground state energy (unbroken supersymmetry) can be attributed to the cancellation between a real Saddle and a complex Saddle (with hidden topological angle pi), and positivity of the ground state energy (broken supersymmetry) can be interpreted as the dominance of complex Saddles. In either case, despite the fact that the ground state energy is zero to all orders in perturbation theory, all orders of fluctuations around non-perturbative Saddles are encoded in the perturbative E(N, g). We illustrate these ideas with examples from supersymmetric quantum mechanics and quantum field theory.
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what is qft resurgent trans series lefschetz thimbles and new exact Saddles
Proceedings of The 33rd International Symposium on Lattice Field Theory — PoS(LATTICE 2015), 2016Co-Authors: Gerald V Dunne, Mithat UnsalAbstract:This is an introductory level review of recent applications of resurgent trans-series and PicardLefschetz theory to quantum mechanics and quantum field theory. Resurgence connects local perturbative data with global topological structure. In quantum mechanical systems, this program provides a constructive relation between different Saddles. For example, in certain cases it has been shown that all information around the instanton Saddle is encoded in perturbation theory around the perturbative Saddle. In quantum field theory, such as sigma models compactified on a circle, neutral bions provide a semi-classical interpretation of the elusive IR-renormalon, and fractional kink instantons lead to the non-perturbatively induced gap, of order of the strong scale. In the path integral formulation of quantum mechanics, Saddles must be found by solving the holomorphic Newton’s equation in the inverted (holomorphized) potential. Some Saddles are complex, multi-valued, and even singular, but of finite action, and their inclusion is strictly necessary to prevent inconsistencies. The multi-valued Saddles enter either via resurgent cancellations, or their phase is tied with a hidden topological angle. We emphasize the importance of the destructive/constructive interference effects between equally dominant Saddles in the Lefschetz thimble decomposition. This is especially important in the context of the sign problem.
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complexified path integrals exact Saddles and supersymmetry
Physical Review Letters, 2016Co-Authors: Gerald V Dunne, Mithat Unsal, Alireza Behtash, Thomas Schafer, Tin SulejmanpasicAbstract:In the context of two illustrative examples from supersymmetric quantum mechanics we show that the semiclassical analysis of the path integral requires complexification of the configuration space and action, and the inclusion of complex Saddle points, even when the parameters in the action are real. We find new exact complex Saddles, and show that without their contribution the semiclassical expansion is in conflict with basic properties such as the positive semidefiniteness of the spectrum, as well as constraints of supersymmetry. Generic Saddles are not only complex, but also possibly multivalued and even singular. This is in contrast to instanton solutions, which are real, smooth, and single valued. The multivaluedness of the action can be interpreted as a hidden topological angle, quantized in units of π in supersymmetric theories. The general ideas also apply to nonsupersymmetric theories.
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decoding perturbation theory using resurgence stokes phenomena new Saddle points and lefschetz thimbles
arXiv: High Energy Physics - Theory, 2014Co-Authors: Aleksey Cherman, Daniele Dorigoni, Mithat UnsalAbstract:Resurgence theory implies that the non-perturbative (NP) and perturbative (P) data in a QFT are quantitatively related, and that detailed information about non-perturbative Saddle point field configurations of path integrals can be extracted from perturbation theory. Traditionally, only stable NP Saddle points are considered in QFT, and homotopy group considerations are used to classify them. However, in many QFTs the relevant homotopy groups are trivial, and even when they are non-trivial they leave many NP Saddle points undetected. Resurgence provides a refined classification of NP-Saddles, going beyond conventional topological considerations. To demonstrate some of these ideas, we study the $SU(N)$ principal chiral model (PCM), a two dimensional asymptotically free matrix field theory which has no instantons, because the relevant homotopy group is trivial. Adiabatic continuity is used to reach a weakly coupled regime where NP effects are calculable. We then use resurgence theory to uncover the existence and role of novel `fracton' Saddle points, which turn out to be the fractionalized constituents of previously observed unstable `uniton' Saddle points. The fractons play a crucial role in the physics of the PCM, and are responsible for the dynamically generated mass gap of the theory. Moreover, we show that the fracton-anti-fracton events are the weak coupling realization of 't Hooft's renormalons, and argue that the renormalon ambiguities are systematically cancelled in the semi-classical expansion. Our results motivate the conjecture that the semi-classical expansion of the path integral can be geometrized as a sum over Lefschetz thimbles.
Brian Swingle - One of the best experts on this subject based on the ideXlab platform.
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note on entropy dynamics in the brownian syk model
Journal of High Energy Physics, 2021Co-Authors: Shaokai Jian, Brian SwingleAbstract:We study the time evolution of Renyi entropy in a system of two coupled Brownian SYK clusters evolving from an initial product state. The Renyi entropy of one cluster grows linearly and then saturates to the coarse grained entropy. This Page curve is obtained by two different methods, a path integral Saddle point analysis and an operator dynamics analysis. Using the Brownian character of the dynamics, we derive a master equation which controls the operator dynamics and gives the Page curve for purity. Insight into the physics of this complicated master equation is provided by a complementary path integral method: replica diagonal and non-diagonal Saddles are responsible for the linear growth and saturation of Ŕenyi entropy, respectively.
Vincent S Min - One of the best experts on this subject based on the ideXlab platform.
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euclidean black Saddles and ads 4 black holes
Journal of High Energy Physics, 2020Co-Authors: Nikolay Bobev, Anthony M Charles, Vincent S MinAbstract:We find new asymptotically locally AdS4 Euclidean supersymmetric solutions of the STU model in four-dimensional gauged supergravity. These “black Saddles” have an S1 × $$ {\Sigma}_{\mathfrak{g}} $$ boundary at asymptotic infinity and cap off smoothly in the interior. The solutions can be uplifted to eleven dimensions and are holographically dual to the topologically twisted ABJM theory on S1 × $$ {\Sigma}_{\mathfrak{g}} $$ . We show explicitly that the on-shell action of the black Saddle solutions agrees exactly with the topologically twisted index of the ABJM theory in the planar limit for general values of the magnetic fluxes, flavor fugacities, and real masses. This agreement relies on a careful holographic renormalization analysis combined with a novel UV/IR holographic relation between supergravity parameters and field theory sources. The Euclidean black Saddle solution space contains special points that can be Wick-rotated to regular Lorentzian supergravity backgrounds that correspond to the well-known supersymmetric dyonic AdS4 black holes in the STU model.
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euclidean black Saddles and ads _4 black holes
arXiv: High Energy Physics - Theory, 2020Co-Authors: Nikolay Bobev, Anthony M Charles, Vincent S MinAbstract:We find new asymptotically locally AdS$_4$ Euclidean supersymmetric solutions of the STU model in four-dimensional gauged supergravity. These "black Saddles" have an $S^1\times \Sigma_{\mathfrak{g}}$ boundary at asymptotic infinity and cap off smoothly in the interior. The solutions can be uplifted to eleven dimensions and are holographically dual to the topologically twisted ABJM theory on $S^1\times \Sigma_{\mathfrak{g}}$. We show explicitly that the on-shell action of the black Saddle solutions agrees exactly with the topologically twisted index of the ABJM theory in the planar limit for general values of the magnetic fluxes, flavor fugacities, and real masses. This agreement relies on a careful holographic renormalization analysis combined with a novel UV/IR holographic relation between supergravity parameters and field theory sources. The Euclidean black Saddle solution space contains special points that can be Wick-rotated to regular Lorentzian supergravity backgrounds that correspond to the well-known supersymmetric dyonic AdS$_4$ black holes in the STU model.
Gregory S Ezra - One of the best experts on this subject based on the ideXlab platform.
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index k Saddles and dividing surfaces in phase space with applications to isomerization dynamics
Journal of Chemical Physics, 2011Co-Authors: Peter Collins, Gregory S Ezra, Stephen WigginsAbstract:In this paper, we continue our studies of the phase space geometry and dynamics associated with index k Saddles (k > 1) of the potential energy surface. Using Poincare-Birkhoff normal form (NF) theory, we give an explicit formula for a "dividing surface" in phase space, i.e., a codimension one surface (within the energy shell) through which all trajectories that "cross" the region of the index k Saddle must pass. With a generic non-resonance assumption, the normal form provides k (approximate) integrals that describe the Saddle dynamics in a neighborhood of the index k Saddle. These integrals provide a symbolic description of all trajectories that pass through a neighborhood of the Saddle. We give a parametrization of the dividing surface which is used as the basis for a numerical method to sample the dividing surface. Our techniques are applied to isomerization dynamics on a potential energy surface having four minima; two symmetry related pairs of minima are connected by low energy index 1 Saddles, with the pairs themselves connected via higher energy index 1 Saddles and an index 2 Saddle at the origin. We compute and sample the dividing surface and show that our approach enables us to distinguish between concerted crossing ("hilltop crossing") isomerizing trajectories and those trajectories that are not concerted crossing (potentially sequentially isomerizing trajectories). We then consider the effect of additional "bath modes" on the dynamics, by a study of a four degree-of-freedom system. For this system we show that the normal form and dividing surface can be realized and sampled and that, using the approximate integrals of motion and our symbolic description of trajectories, we are able to choose initial conditions corresponding to concerted crossing isomerizing trajectories and (potentially) sequentially isomerizing trajectories.
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index k Saddles and dividing surfaces in phase space with applications to isomerization dynamics
arXiv: Statistical Mechanics, 2011Co-Authors: Peter Collins, Gregory S Ezra, Stephen WigginsAbstract:In this paper we continue our studies of the phase space geometry and dynamics associated with index k Saddles (k > 1) of the potential energy surface. Using normal form theory, we give an explicit formula for a "dividing surface" in phase space, i.e. a co-dimension one surface (within the energy shell) through which all trajectories that "cross" the region of the index k Saddle must pass. With a generic non-resonance assumption, the normal form provides k (approximate) integrals that describe the Saddle dynamics in a neighborhood of the index k Saddle. These integrals provide a symbolic description of all trajectories that pass through a neighborhood of the Saddle. We give a parametrization of the dividing surface which is used as the basis for a numerical method to sample the dividing surface. Our techniques are applied to isomerization dynamics on a potential energy surface having 4 minima; two symmetry related pairs of minima are connected by low energy index one Saddles, with the pairs themselves connected via higher energy index one Saddles and an index two Saddle at the origin. We compute and sample the dividing surface and show that our approach enables us to distinguish between concerted crossing ("hilltop crossing") isomerizing trajectories and those trajectories that are not concerted crossing (potentially sequentially isomerizing trajectories). We then consider the effect of additional "bath modes" on the dynamics, which is a four degree-of-freedom system. For this system we show that the normal form and dividing surface can be realized and sampled and that, using the approximate integrals of motion and our symbolic description of trajectories, we are able to choose initial conditions corresponding to concerted crossing isomerizing trajectories and (potentially) sequentially isomerizing trajectories.
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phase space geometry and reaction dynamics near index 2 Saddles
Journal of Physics A, 2009Co-Authors: Gregory S Ezra, Stephen WigginsAbstract:We 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 (DoF) 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. We 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, we can identify a class of trajectories, which we refer to as 'roaming trajectories', which are not associated with reaction along the classical minimum energy path (MEP). We illustrate the significance of our analysis of the index 2 Saddle for reaction dynamics with two examples. 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, our analysis suggests a rigorous dynamical definition of 'non-MEP' or 'roaming' reactive events.
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phase space geometry and reaction dynamics near index two Saddles
arXiv: Chaotic Dynamics, 2009Co-Authors: Gregory S Ezra, Stephen WigginsAbstract:We 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 (DoF) Hamiltonian systems. For index 1 Saddles of potential energy surfaces (the case of classical transition state theory), 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 one in the energy surface, and act as conduits for reacting trajectories in phase space. The situation for index 2 Saddles is quite different. We 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. Rather, there are different types of invariant manifolds, containing the NHIM and its stable and unstable manifolds, that act as co-dimension one 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, we can identify a class of trajectories, which we refer to as "roaming trajectories", which are not associated with reaction along the classical minimum energy path (MEP). We illustrate the significance of our analysis of the index 2 Saddle for reaction dynamics with two examples.
