Transition state theory in liquids beyond planar dividing surfaces

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).…”

Variational principle for the determination of unstable periodic orbits and instanton trajectories at saddle points

“…In arbitrary dimensions at each instance in time, this structure is a normally hyperbolic invariant manifold (NHIM) 46,47 which can be obtained by perturbation theory. 26,27,30,[48][49][50] Using extremal values of the LD, we were also able to construct the DS, itself. 41,42,45 Meanwhile, at constant energy on two-dimensional stationary potentials, Pollak and Pechukas 13,51,52 showed that the nonrecrossing dividing surface is a PO associated with the nearby saddle point.…”

“…[24][25][26][27] Unstable POs are of importance in the field of classical reaction dynamics in classical systems where they form recrossingfree dividing surfaces. 2,3, 11,[28][29][30] They are important for understanding tunneling dynamics through a barrier in quantum mechanical reactions wherein the PO on the inverted potential, −V , is the instanton trajectory providing the leading contribution to the path integral. 31 In common between all of these cases, is the invariance of POs as they provide a scaffold from which to obtain other geometric structures, and thus remain objects of current interest such as in, e.g., Refs.…”

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.…”