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# Volume 26, Number 2, 2021 Special Issue: Nonlinear Dynamics in Chemical Sciences: Part II

 In Memory of Professor Alexey V. Borisov (27.03.1965–24.01.2021) Abstract Citation: In Memory of Professor Alexey V. Borisov (27.03.1965–24.01.2021), Regular and Chaotic Dynamics, 2021, vol. 26, no. 2, pp. 113-118 DOI:10.1134/S1560354721020015
 Rashmi R.,  Yadav K.,  Lourderaj U.,  Paranjothy M. Second-order Saddle Dynamics in Isomerization Reaction Abstract The role of second-order saddle in the isomerization dynamics was investigated by considering the $E-Z$ isomerization of guanidine. The potential energy profile for the reaction was mapped using the ab initio wavefunction method. The isomerization path involved a torsional motion about the imine (C-N) bond in a clockwise or an anticlockwise fashion resulting in two degenerate transition states corresponding to a barrier of 23.67 kcal/mol. An alternative energetically favorable path ($\sim$1 kcal/mol higher than the transition states) by an in-plane motion of the imine (N-H) bond via a second-order saddle point on the potential energy surface was also obtained. The dynamics of the isomerization was investigated by ab initio classical trajectory simulations. The trajectories reveal that isomerization happens via the transition states as well as the second-order saddle. The dynamics was found to be nonstatistical with trajectories exhibiting recrossing and the higher energy second-order saddle pathway preferred over the traditional transition state pathway. Wavelet based time-frequency analysis of internal coordinates indicate regular dynamics and existence of long-lived quasi-periodic trajectories in the phase space. Keywords: transition state, second-order saddle, reaction dynamics, RRKM, potential energy surface, wavelet transform, time-frequency analysis Citation: Rashmi R.,  Yadav K.,  Lourderaj U.,  Paranjothy M., Second-order Saddle Dynamics in Isomerization Reaction, Regular and Chaotic Dynamics, 2021, vol. 26, no. 2, pp. 119-130 DOI:10.1134/S1560354721020027
 Eklund  E. C.,  Ananth N. Investigating the Stability and Accuracy of a Classical Mapping Variable Hamiltonian for Nonadiabatic Quantum Dynamics Abstract Previous work has shown that by using the path integral representation of quantum mechanics and by mapping discrete electronic states to continuous Cartesian variables, it is possible to derive an exact quantum “mapping variable” ring-polymer (MV-RP) Hamiltonian. The classical molecular dynamics generated by this MV-RP Hamiltonian can be used to calculate equilibrium properties of multi-level quantum systems exactly, and to approximate real-time thermal correlation functions (TCFs). Here, we derive mixed time-slicing forms of the MV-RP Hamiltonian where different modes of a multi-level system are quantized to different extents. We explore the accuracy of the approximate quantum dynamics generated by these Hamiltonians through numerical calculation of quantum real-time TCFs for a range of model nonadiabatic systems, where two electronic states are coupled to a single nuclear degree of freedom. Interestingly, we find that the dynamics generated by an MV-RP Hamiltonian with all modes treated classically is more stable across all model systems considered here than mixed quantization approaches. Further, we characterize nonadiabatic dynamics in the 6D phase space of our classical-limit MV-RP Hamiltonian using Lagrangian descriptors to identify stable and unstable manifolds. Keywords: nonadiabatic, path integral, mapping variables, Lagrangian descriptors, correlation functions Citation: Eklund  E. C.,  Ananth N., Investigating the Stability and Accuracy of a Classical Mapping Variable Hamiltonian for Nonadiabatic Quantum Dynamics, Regular and Chaotic Dynamics, 2021, vol. 26, no. 2, pp. 131-146 DOI:10.1134/S1560354721020039
 Crossley R.,  Agaoglou M.,  Katsanikas M.,  Wiggins S. From Poincaré Maps to Lagrangian Descriptors: The Case of the Valley Ridge Inflection Point Potential Abstract In this paper we compare the method of Lagrangian descriptors with the classical method of Poincaré maps for revealing the phase space structure of two-degree-of-freedom Hamiltonian systems. The comparison is carried out by considering the dynamics of a twodegree- of-freedom system having a valley ridge inflection point (VRI) potential energy surface. VRI potential energy surfaces have four critical points: a high energy saddle and a lower energy saddle separating two wells. In between the two saddle points is a valley ridge inflection point that is the point where the potential energy surface geometry changes from a valley to a ridge. The region between the two saddles forms a reaction channel and the dynamical issue of interest is how trajectories cross the high energy saddle, evolve towards the lower energy saddle, and select a particular well to enter. Lagrangian descriptors and Poincaré maps are compared for their ability to determine the phase space structures that govern this dynamical process. Keywords: phase space structure, periodic orbits, stable and unstable manifolds, homoclinic and heteroclinic orbits, Poincar´e maps, Lagrangian descriptors Citation: Crossley R.,  Agaoglou M.,  Katsanikas M.,  Wiggins S., From Poincaré Maps to Lagrangian Descriptors: The Case of the Valley Ridge Inflection Point Potential, Regular and Chaotic Dynamics, 2021, vol. 26, no. 2, pp. 147-164 DOI:10.1134/S1560354721020040
 Pandey P.,  Naik S.,  Keshavamurthy S. Classical and Quantum Dynamical Manifestations of Index-2 Saddles: Concerted Versus Sequential Reaction Mechanisms Abstract The presence of higher-index saddles on a multidimensional potential energy surface is usually assumed to be of little significance in chemical reaction dynamics. Such a viewpoint requires careful reconsideration, thanks to elegant experiments and novel theoretical approaches that have come about in recent years. In this work, we perform a detailed classical and quantum dynamical study of a model two-degree-of-freedom Hamiltonian, which captures the essence of the debate regarding the dominance of a concerted or a stepwise reaction mechanism. We show that the ultrafast shift of the mechanism from a concerted to a stepwise one is essentially a classical dynamical effect. In addition, due to the classical phase space being a mixture of regular and chaotic dynamics, it is possible to have a rich variety of dynamical behavior, including a Murrell – Laidler type mechanism, even at energies sufficiently above that of the index-2 saddle. We rationalize the dynamical results using an explicit construction of the classical invariant manifolds in the phase space. Keywords: reaction mechanisms, index-2 saddles, classical-quantum correspondence, dynamic Murrell-Laidler, invariant manifolds, concerted and sequential reactions Citation: Pandey P.,  Naik S.,  Keshavamurthy S., Classical and Quantum Dynamical Manifestations of Index-2 Saddles: Concerted Versus Sequential Reaction Mechanisms, Regular and Chaotic Dynamics, 2021, vol. 26, no. 2, pp. 165-182 DOI:10.1134/S1560354721020052
 Fukuda I.,  Moritsugu K.,  Fukunishi Y. On Ergodicity for Multidimensional Harmonic Oscillator Systems with Nosé – Hoover-type Thermostat Abstract A simple proof and a detailed analysis of the nonergodicity for multidimensional harmonic oscillator systems with the Nosé – Hoover-type thermostat are presented. The origin of the nonergodicity is the symmetries in the multidimensional target physical system, and it differs from that in the Nosé – Hoover thermostat with the one-dimensional harmonic oscillator. A new and simple deterministic method to recover the ergodicity is also proposed. An individual thermostat variable is attached to each degree of freedom, and all of these variables act on a friction coefficient for each degree of freedom. This action is linear and controlled by a Nosé mass matrix $\mathbf{Q}$, which is the matrix analogue of the scalar Nosé mass. The matrix $\mathbf{Q}$ can break the symmetry and contribute to the ergodicity. Keywords: ergodicity, Nosé – Hoover equation, thermostat, molecular dynamics, invariant space, multidimensional harmonic oscillator, Nosé-mass matrix, density dynamics Citation: Fukuda I.,  Moritsugu K.,  Fukunishi Y., On Ergodicity for Multidimensional Harmonic Oscillator Systems with Nosé – Hoover-type Thermostat, Regular and Chaotic Dynamics, 2021, vol. 26, no. 2, pp. 183-204 DOI:10.1134/S1560354721020064

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