Quantum Kramers model

As in other flux correlation function computations, f is the complex time t ——. Thus, given the Quantum Kramers model for the reaction in the complex system, and the re-summed operator expansion as a practical way to evaluate the necessary evolution operators needed for the flux autocorrelation function, the quantum rate in the complex system is reduced to a simple combination of gas phase correlation functions with simple algebraic functions. 

The Quantum Kramers Model and Proton Coupled Electron Transfer 1231... 

The standard language used to describe rate phenomena in condensed phases has evolved from Kramers one dimensional model of a particle moving on a one dimensional potential, feeling a random and a related friction force. In Section II, we will review the classical Generalized Langevin Equation (GEE) underlying Kramers model and its application to condensed phase systems. The GLE has an equivalent Hamiltonian representation in terms of a particle which is bilinearly coupled to a harmonic bath. The Hamiltonian representation, also reviewed in Section II is the basis for a quantum representation of rate processes in condensed phases. Eas also been very useful in obtaining solutions to the classical GLE. Variational estimates for the classical reaction rate are described in Section III. 

This chapter reviews the generalizations of the Kramers model that were develojjed during the past few years. The result of this effort, which we may call the generalized Kramers theory, provides a useful framework for the theoretical description of activated rate processes in general and of chemical reaction rates in condensed phases in particular. Some applications of this framework as well as its limitations are also discussed. In the last few years there has also been substantial progress in the study of the quantum mechanical Kramers model, which may prove useful for condensed phase tunneling reactions. This aspect of the problem is not covered by the present review. 

This rate of energy exchange between an oscillator and the thermal environment was the focus of Chapter 13, where we have used a quantum harmonic oscillator model for the well motion. In the y -> 0 limit of the Kramers model we are dealing with energy relaxation of a classical anharmonic oscillator. One may justifiably question the use of Markovian classical dynamics in this part of the problem, and we will come to this issue later. For now we focus on the solution of the mathematical problem posed by the low friction limit of the Kramers problem. 

Here the first two terms just give ma = Force as mass times the second time derivative of the friction equal to the F as the negative derivative of potential, y is the memory friction, and F(t) is the random force. Thus the complex dynamics of all degrees of freedom other than the reaction coordinate are included in a statistical treatment, and the reaction coordinate plus environment are modeled as a modified one-dimensional system. What allows realistic simulation of complex systems is that the statistics of the environment can in fact be calculated from a formal prescription. This prescription is given by the Fluctuation-Dissipation theorem, which yields the relation between the friction and the random force. In particular, this theory shows how to calculate the memory friction from a relatively short-time classical simulation of the reaction coordinate. The Quantum Kramers approach. 

There have been previous model studies of these systems [61]. These studies, while including the effects of environment, did not address the question of the effect of a promoting vibration. These reactions are inherently electronically nonadia-batic, while the formulation we have thus far presented included evolution only on a single Born-Oppenheimer potential energy surface. We have developed a model system to allow the extension of the Quantum Kramers methodology to such systems, and we now describe that model. 

To demonstrate the accuracy of eqn (12.26) together with eqn (12.30) and its boundary condition eqn (12.31), called the quantum Kramers-like theory, we use a symmetric spin-boson model as a concrete system. In this benchmark model, the nuclear vibrational motions are characterized by the Ohmic spectral density ... 

In this paper we shall begin by a short historical overview of the different theories of electron transfer. This overview will be of course limited, in order to emphasize the physical principles involved in electron transfer. For additional details, exhaustive reviews of the different theoretical treatments can be found in refs. Moreover, we shall restrict ourselves to theories of quantum mechanical nature. Thus, stochastic models (cf for instance the Kramers model) are not discussed here, but they are treated elsewhere in this book. We shall then focus on recent developments in intramolecular electron transfer and its solvent influence. 

This rate of energy exchange between an oscillator and the thermal environment was the focus of Chapter 13, where we have used a quantum harmonic oscillator model for the well motion. In the y 0 limit of the Kramers model we are dealing... 

Our approach to the study of the departure from equilibrium in chemical reactions and of the "microscopic theory of chemical kinetics is a discrete quantum-mechanical analog of the Kramers-Brownian-motion model. It is most specifically applicable to a study of the energy-level distribution function and of the rate of activation in unimolecular (dissociation Reactions. Our model is an extension of one which we used in a discussion of the relaxation of vibrational nonequilibrium distributions.14 18 20... 

The DIRAC package [55,56], devised by Saue and collaborators, rather than exploiting the group theoretical properties of Dirac spherical 4-spinors as in BERTHA, treats each component in terms of a conventional quantum chemical basis of real-valued Cartesian functions. The approach used in DIRAC, building on earlier work by Rosch [80] for semi-empirical models, uses a quaternion matrix representation of one electron operators in a basis of Kramers pairs. The transformation properties of these matrices, analysed in [55], are used to build point group transformation properties into the Fock matrix. 

Many important problems in computational physics and chemistry can be reduced to the computation of dominant eigenvalues of matrices of high or infinite order. We shall focus on just a few of the numerous examples of such matrices, namely, quantum mechanical Hamiltonians, Markov matrices, and transfer matrices. Quantum Hamiltonians, unlike the other two, probably can do without introduction. Markov matrices are used both in equilibrium and nonequilibrium statistical mechanics to describe dynamical phenomena. Transfer matrices were introduced by Kramers and Wannier in 1941 to study the two-dimensional Ising model [1], and ever since, important work on lattice models in classical statistical mechanics has been done with transfer matrices, producing both exact and numerical results [2]. 

However, model reactions can demcxistrate that the interferometric measurement in combination with application of the Kramer-Kronig relationship [S] allows calculation of the abscnbance-time curves and the determination of the partial photochemical quantum yield. Thus, the method makes it pos-sibile to monitor photoreactions in a solid matrix. As an application the quantum yield of the photoreaction of a 7,7a-dihydrobenzofuran derivative to the (Z)-fulgide is obtained as =0.03 [201]. 

Christiansen tried to apply the description and the model of chain reactions to different mechanisms (Christiansen 1922) and wrote a paper with Kramers in 1923, cited previously, about unimolecular reactions confronting the activation mechanism due to thermal collisions and radiation absorption. They treated the radiation mechanism with the fundamental Einstein s quantum theory about matter-radiation interaction (Einstein 1917). Other work of Einstein and Smoluchowski will be necessary later for Christiansen-Kramers approach. After the paper the collaboration probably ended and the two researchers will reconsider separately these arguments aroxmd fifteen years later. 

The relationship between rotation and refractive index or between ellipticity and absorption coefficient can be represented, in analogy to dispersion theory, by the model of coupled linear oscillators or by quantum mechanical methods. ORD and CD are related to one another by equations analogous to the Kramers-Kronig equations [34]. 

Fig. 4.4 VDOS calculated from a a two-body potential (VB) derived by Kramer et al. on a quantum-chemical calculation of an H4Si04 cluster [55], b a two-body potential (TS) derived by Tsuneyuki et al. using a Hartree-Fock calculation on SiOJ cluster [9], c a three-body (3B) potential by Sander, Leslie, and Catlow with a shell-model description and with three-body interactions [48] and d a two-body potential (KR) proposed by Kramer et al. using a mixed self-consistent field and empirical procedures [10], Note the KR potential here is the BKS potential. Figure taken from [56]...

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