Physical Chemistry Chemical Physics Detecting reactive islands in a system-bath model of isomerization Journal: Physical Chemistry Chemical Physics Manuscript ID CP-ART-03-2020-001362.R1 Article Type: Paper Date Submitted by the 09-May-2020 Author: Complete List of Authors: Naik, Shibabrat; University of Bristol Faculty of Science, Mathematics Wiggins, Stephen; University of Bristol Faculty of Science, Mathematics Page 1 of 24 Physical Chemistry Chemical Physics Detecting reactive islands in a system-bath model of isomerization† Shibabrat Naik,∗a Stephen Wiggins,b‡ In this article, we study the conformational isomerization in a solvent using a system-bath model where the phase space structures relevant for the reaction dynamics are revealed. These phase space structures are an integral part of understanding the reaction mechanism, that is the path- ways that reactive trajectories undertake, in the presence of a solvent. Our approach involves detecting the analogs of the reactive islands first discussed in the works by Davis, Marston, De Leon, Berne and coauthors 1–4 in the system-bath model using Lagrangian descriptors. We first present the structure of the reactive islands for the two degrees of freedom system modelling isomerization in the absence of the bath using direct computation of cylindrical (tube) manifolds and verify the Lagrangian descriptor method for detecting the reactive islands. The hierarchy of the reactive islands as indicated in the recent work by Patra and Keshavamurthy 5 is shown to be related to the temporal features in committor probabilities. Next, we investigate the influence of the solvent on the reactive islands that we previously revealed for the two degrees of freedom system and discuss the use of the Lagrangian descriptor in the high-dimensional phase space of the system-bath model. 1 Introduction cosity was obtained in 1980s by Mel’nikov, Meshkov 17 and Grote, Isomerization is an important reaction in atmospheric, medical, Hynes, Pollak, Grabert, Hänggi 18,19 and showed the turnover of ∗ 6–9 and industrial chemistry . On one hand, the influence of var- reaction rates with the increasing viscosity of the solvent. Further- ious solvents on the rate constant of conformational isomerization more, Pollak’s derivation 20 connected the Kramers theory and the 10–14 has been pursued for specific molecules using statistical me- transition state theory for condensed phase reactions which has chanics. On the other hand, dynamical systems theory has been been useful for calculations of rate expressions 21. These develop- used to develop a systematic approach for identifying trajectories ments established the Langevin equation of motion for a particle that go from reactants to products and reside inside phase space trapped in a potential well with a barrier height V ‡ as a model for structures called reactive islands in two degrees of freedom Hamil- studying a reaction in the condensed phase. In this set-up, the re- 1–4 tonian models . In this article, we extend the reactive islands action (or the system) is defined by the crossing of the barrier and approach to N > 2 degrees of freedom model for the isomerization the influence of the medium (or the bath) is studied by coupling reaction in a bath. the bath coordinates to the system coordinates. This framework Following the derivation of the Langevin equations and its gen- has received much attention in the literature where the system 15 16 eralized system dynamics in a heat bath , Kramers derived dynamics can be obtained explicitly using a one-dimensional po- the expression for the escape rate of a Brownian particle trapped tential well and the bath is modelled using a large number of in a potential well for small and large values of the viscosity. He harmonic oscillators with mass, frequency, and viscosity param- also showed that only for a specific potential barrier the escape eters 22–25. In this set-up, the bath coordinates are coupled with rate for all values of viscosity is equal to the one given by the the system coordinates via the viscosity (or the friction) param- transition state theory. However, the solution for all values of vis- eter in the form of a bilinear term in the potential energy. This gives rise to the Hamiltonian which is written as a sum of kinetic energy and potential energy in the system and bath coordinates. a School of Mathematics, University of Bristol Fry Building, Woodland Road, Bristol BS8 1UG, United Kingdom. Tel: XX XXXX XXXX; However, the dynamics of a two or more degrees of freedom E-mail: [email protected] system coupled with bath modes has not received a global anal- b [email protected]. ysis from a dynamical systems perspective of reactions. In this y Electronic Supplementary Information (ESI) available: Derivation of equations of motion and the computational approach to obtain the cylindrical manifolds. direction, the first step would be to consider a two degrees of free- ∗ https://www.sciencedirect.com/topics/engineering/isomerization dom Hamiltonian system with well-understood quantities such as Journal Name, [year], [vol.], 1–23 | 1 Physical Chemistry Chemical Physics Page 2 of 24 the reaction coordinate, and to which the harmonic bath modes the NHIM are structurally stable, that is, stable under perturba- can be coupled to represent the reaction in a condensed phase. tion 36. For two DOF systems, the NHIM is an unstable PO, and In this work, we will adopt the two degrees of freedom isomer- for an N > 2 DOF system at a fixed energy, the NHIM has the ization model of De Leon and Berne who studied the chemical re- topology of a (2N − 3)-dimensional sphere and is the equator of a action dynamics extensively using a dynamical systems perspec- (2N −2)-dimensional sphere which is the DS. This DS can then be tive 4,26–30. This Hamiltonian model of isomerization describes used to divide the (2N − 1)-dimensional energy surface into two the conformational change by the motion of an internal angle parts, reactants and products 41–45. An elementary description of where the isomers are represented by the wells in a double po- the role of the NHIM in reaction dynamics is given in 46 along tential well separated by a barrier. The two degrees of freedom in with description of their geometry using quadratic normal form the model correspond to the bond that undergoes rotation (struc- Hamiltonians. Fundamental theorems assure the existence of the tural change of a molecule) for total energies between the isomer- phase space structures — NHIM and its invariant manifolds —for ization and dissociation energy and to the bond that splits above a range of energies above that of the saddle 36. However, the pre- dissociation energy. Typically, the isomerization energy is lower cise extent of this range, as well as the nature and consequences than the dissociation energy of the molecule, and the activation of any bifurcations of the phase space structures that might occur energy for the isomerization is imparted by molecular collisions as energy is increased, is not known and is a topic of continuing or photoexcitation. research 47–52. Traditionally, the construction of a dividing surface (DS) was Thus, calculation of reaction rate (or flux) based on the geome- focused on critical points of the potential energy surface (PES), try of phase space structures requires identifying trajectories that that is, in the configuration space describing the molecular sys- start in the reactant well, cross the dividing surface constructed tem 31. Critical points on the PES do have significance in phase from the NHIM, and reach the product well. This dividing sur- space since they are the equilibrium points for zero momentum. face has been shown to be the appropriate (locally no-recrossing) But they continue to have influence for nonzero momentum for a surface that reactive trajectories must cross since the calculated range of energies above the energy of the equilibrium point. The reaction rates do not need correction due to recrossings 34. This construction of a DS separating the phase space into two parts, construction is in contrast to the “standard” transition state theory that is reactants and products, has been a focus from the dynam- (TST) for constructing the dividing surface which is only exact in ical systems point of view in recent years 32–34. In phase space, gas phase unimolecular reactions and when ergodicity of trajec- the role of the saddle equilibrium point is played by an invariant tories in the phase space holds 19. As is now established, the no manifold of saddle type stability called the normally hyperbolic recrossing (locally) property of a dividing surface is a contribution invariant manifold (NHIM) 35,36. In order to fully appreciate the of the phase space perspective of chemical reactions 33. While the NHIM and its role in reaction rate theory, it is useful to begin with standard TST relies on a recrossing free surface for calculating a precursor concept — the periodic orbit dividing surface or PODS. reaction flux, a dividing surface constructed in the configuration For systems with two DOF described by a natural Hamiltonian, ki- space violates this condition in the case of a solvent 19,53, and thus netic plus potential energy, the problem of constructing the DS in the TST based reaction rate is not exact. This violation of the re- phase space was solved during the 1970s by McLafferty, Pechukas crossing property when the DS is constructed in the configuration and Pollak 37–40. They demonstrated that the DS at a specific en- space of a reaction in a high viscosity solvent also follows from ergy is related to an invariant phase space structure, an unstable the Kramers’ diffusion model, Langevin equation, of chemical re- periodic orbit (UPO) which defines (it is the boundary of) the bot- actions 19.