Nonlinear dynamical effects on reaction rates in thermally fluctuating environments

Abstract

An analytical framework to scrutinize the physical origin, especially nonlinear dynamical effects, of rate constants in thermal fluctuation is developed.A framework to calculate the rate constants of condensed phase chemical reactions of manybody systems is presented without relying on the concept of transition state. The theory is based on a framework we developed recently adopting multidimensional underdamped Langevin equation in the region of rank-one saddle. The theory provides a reaction coordinate expressed as an analytical nonlinear functional of the position coordinates and velocities of the system (solute), the friction constants, and the random force of the environment (solvent). Up to moderately high temperature, the sign of the reaction coordinate can determine the final destination of the reaction in a thermally fluctuating media, irrespective of what values the other (nonreactive) coordinates may take. In this paper, it is shown that the reaction probability is analytically derived as the probability of the reaction coordinate being positive, and that the integration with the Boltzmann distribution of the initial conditions leads to the exact reaction rate constant when the local equilibrium holds and the quantum effect is negligible. Because of analytical nature of the theory taking into account all nonlinear effects and their combination with fluctuation and dissipation, the theory naturally provides us with the firm mathematical foundation of the origin of the reactivity of the reaction in a fluctuating media.

Derivation of the generalized Langevin equation in nonstationary environments

“…The other stimulating subject is the combination of the present theory and the recently developed dynamical reaction theory to extract the rigorous reaction coordinate to dominate the fate of reactions under thermal fluctuation in equilibrium. [6][7][8][9][10][11][12][13][14][15][16][17] These should provide us with great new insights into many molecular events occurring in nonstationary environments.…”

“…3 An example of such statistical property of the random force is the fluctuation-dissipation theorem, where the autocorrelation function of the random force is related to the friction kernel. It was found recently [6][7][8][9][10][11][12][13][14][15][16][17] that even though one cannot know an instantaneous value of the random force in advance since the initial condition of the bath is unknown, the statistical property enables us to analytically derive the boundary of the reaction in the state space, that is, a surface on which the system should end up with the reactant and the product with equal probability of one half. Following the pioneering works by Kramers 1 and by Grote and Hynes, 2 great progress [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] in the study of reaction dynamics in condensed phase have been made by using the GLE or the Langevin equation (a memoryless limit of GLE).…”

Phase space geometry of dynamics passing through saddle coupled with spatial rotation

“…The potential of the theories has been demonstrated not only in chemical reactions with 17,22 and without [23][24][25][26][27] time-dependent external field but also in ionization of a hydrogen atom in crossed electric and magnetic fields, [28][29][30] isomerization of clusters, [31][32][33][34][35][36] and the escape of asteroids from Mars 37,38 [Just recently the theory was also generalized to quantum Hamiltonian systems [39][40][41] and dissipative (generalized) Langevin systems. [42][43][44][45][46][47][48][49][50][51] The dimension of the phase space of an N -particle nonrigid system is (6N − 10) in the upper limit. 52 Nonrigid molecules at constant energy have ten constraints of the three coordinates of center of mass, the three conjugate momenta of center of mass, the three angular momenta (defined in the space-fixed frame), and the total energy of the system.…”

Dynamic reaction coordinate in thermally fluctuating environment in the framework of the multidimensional generalized Langevin equations

“…Recently, in order to extract the reaction coordinate and the no-return transition state for nonlinearly coupled multimode systems in a fluctuating environment, we have presented a theory [11][12][13][14] based on the concept of normal form 15 with the time-dependent formulation 16 within the framework of the multidimensional Langevin formulation. It was shown that, under certain conditions, a nonlinear coordinate transformation can be performed to provide a new reaction coordinate independent of all the other coordinates similarly to the case of Hamiltonian systems in the region of saddles.…”