Benny Carmeli

Non‐Markovian theory of activated rate processes. I. Formalism

The escape of a particle from a potential well is treated using a generalized Langevin equation (GLE) in the low friction limit. The friction is represented by a memory kernel and the random noise is characterized by a finite correlation time. This non‐Markovian stochastic equation is reduced to a Smoluchowski diffusion equation for the action variable of the particle and explicit expressions are obtained for the drift and diffusion terms in this equation in terms of the Fourier coefficients of the deterministic trajectory (associated with the motion without coupling to the heat bath) and of the Fourier transform of the friction kernel. The latter (frequency dependent friction) determines the rate of energy exchange with the heat bath. The resulting energy (or action) diffusion equation is used to determine the rate of achieving the critical (escape) energy. Explicit expressions are obtained for a Morse potential. These results for the escape rate agree with those from stochastic trajectories based on the...

Effective adiabatic approximation for a two level system coupled to a bath

We consider the general problem of a two level system coupled to a fluctuating bath. We present a practical and accurate approximate theory for this type of system. Our treatment is based upon a variational perturbation theory in which the reference system is a two level system coupled to an adiabatic fluctuating bath. The methodology is easily generalized to situations involving more than two levels.

Non‐Markovian theory of activated rate processes. IV. The double well model

The transition rates associated with a particle moving in a double potential well under the influence of thermal noise and friction is considered as a generalization of Kramers’ theory of activated rate processes. We obtain expressions for these transition rates which are valid for all friction and for a general (non‐Markovian) interaction between the particle and its thermal environment. Nonthermal equilibrium effects in the steady state distribution in the well as well as effects of trajectories returning unrelaxed from the far wall are explicitly taken into account. The results reduce to all the previously obtained results of the single well model. We use the theory to analyze the experimental results of Hasha, Eguchi, and Jonas.

Random coupling models for intramolecular dynamics. I. Mathematical approach

Intramolecular dynamics in large molecules is modeled as a problem involving random coupling between manifolds of molecular levels. The random coupling model (RCM) is based on the rapid variations observed in coupling matrix elements involving highly excited bound molecular states, and on the high density of such states encountered in large molecules. The finite time and energy scales involved in real experimental situations lead to the observation that the time evolution and spectral properties characterizing the system do not depend on the detailed arrangement of states and their coupling elements but rather on low order moments of the distribution characterizing these coupling elements. This provides the basis for an approach based on ensemble averages. The coupling V is taken as a superposition V=u+v of a smoothly varying component u=〈V〉 and a randomly varying (in state space) component v=V−〈V〉. We introduce a diagrammatic expansion and averaging method to evaluate the diadic Green’s function for prob...

On a random coupling model for intramolecular dynamics

Abstract The time evolution of a system involving separable random coupling between quasi-continuous manifolds is studied. The problem is solved using ensemble averages. In the strong coupling maximum randomness case the continua are found to be effectively uncoupled on the experimentally relevant time scale.

Theory of activated rate processes: Coupled modes

Abstract The thermally activated escape rate of a classical particle out of a potential well is studied in a simple model which includes coupling between the escape (reactive) coordinate and another coordinate. The main effect of the non-reactive coordinate is to open a new (non-markovian) channel between the reactive coordinate and the thermal bath.

Random coupling models for intramolecular dynamics. II. Kinetic equations for collisionless multiphoton excitation of large molecules

Multiphoton excitation and dissociation of large molecules under collisionless conditions is discussed in terms of an intercontinuum random coupling model. The mathematical approach described in a previous paper is used to obtain the general solution for a system of consecutively coupled discrete states, quasicontinuous manifolds, and continuous (dissociative) manifolds of molecular levels (eigenstates of the total molecular Hamiltonian), where the radiative coupling matrix elements are assumed to be given as a linear combination of smoothly varying and randomly varying (over level indices in the molecular manifolds) components. In the range of discrete molecular levels the time evolution is coherent and described in terms of the optical Bloch equation. In the quasicontinuous and continuous ranges the time evolution may be described in terms of Markoffian kinetic equations for the number of photons absorbed by the molecule, provided that the intramolecular vibrational relaxation widths associated with the...

Numerical simulations of molecular multiphoton excitation models

In this paper we report the results of numerical simulations of the intramolecular dynamics of a model system for multiphoton excitation of large molecules, where the low energy range is represented by a single discrete state, while the quasicontinuum is mimicked by two or three manifolds of molecular eigenstates. The random coupling model (RCM), where the radiative coupling matrix elements are assumed to be random functions of the level indices, yields conventional rate equations describing consecutive–reversible transitions for the populations with golden rule rates. In addition, numerical simulations were conducted for a constant coupling model (CCM) and for a separable random coupling model (SRCM), confirming the counterintuitive analytical results for these model systems. The time evolution of a RCM system is determined by the distribution function of the coupling elements and not by individual coupling terms, and the intramolecular dynamics is essentially determined by the lower moments (average and...

Incubation times in the multiphoton dissociation of polyatomic molecules

The incubation period revealed in the multiphoton dissociation of molecules by intense infrared lasers is discussed. It is found experimentally that large excess of added foreign gas affects the incubation period to a much smaller degree than the overall yield. A rate equations model is presented, including both the laser intensity and collisional effects. Exact numerical solution is compared with a simple analytical approximation, based on the passage time moments method. Agreement with experimental results is quite satisfactory, indicating that the role of collisions in the case discussed (tetramethyldioxetane dissociation) is primarily vibrational relaxation of excited molecules.

First passage times and the kinetics of unimolecular dissociation

Approximate solutions for multistep master equations describing the time evolution of product formation in multiphoton or thermal unimolecular reactions are investigated. In particular, a method based on fitting the first few moments of the passage time distribution associated with the given stochastic process to proposed simple expressions for the product yield function is studied. It is shown that reasonable agreement with the exact numerical solution of the corresponding master equation is obtained with a two parameter fit (using two passage time moments) and an excellent agreement is obtained with a three parameter fit (using three passage time moments). In no case studied does a need arise for more than a three‐moment description and the quality of available experimental results makes the simpler two‐moment description sufficient in most cases. Analytical solutions for the first and second passage time moments are obtained for simple discrete and continuous master equation models. Expressions for the...