Rigorous quantum rate theory

Beyond Transition State Theory Rigorous Quantum Approaches for Determining Chemical Reaction Rates... 

Miller, W. H. Beyond transition state theory a rigorous quantum theory of chemical reaction rates, Acc. Chem. Res., 26 (1993), 174-181... 

Rigorous Quantum Rate Theory Versus the Quantized ARRKM Theory A Semiclassical Approximation to the Rigorous Quantum Rate Theory Effective Hamiltonian Approach to Unimolecular Dissociation Wave Packet Dynamics Approach VII. Quantum Transport in Classically Chaotic Systems... 

D. Rigorous Quantum Rate Theory Versus the Quantized ARRKM Theory... 

Equations (359) and (361) indicate that the reason that the flux-flux autocorrelation formalism gives exact quantum reaction rate constants is simply that all the dynamical information from time zero to time inflnity has been included. Indeed, as shown by Eqs. (360) and (362), in the classical limit the flux-flux autocorrelation formalism requires us to follow all classical trajectories until t = +00 so as to rigorously tell which trajectory is reactive and which trajectory is nonreactive. Evidently, then, the flux-flux autocorrelation formalism is not a statistical reaction rate theory insofar as no approximation to the reaction dynamics is made. 

To gain more insight into quantum reaction rate theory, we below make a detailed comparison between the rigorous quantum rate theory and the quantized ARRKM theory. It is significant that the rigorous quantum results [Eqs. (359) and (361)] are very similar to the results from the quantized ARRKM theory [Eqs. (346) and (351)]. In particular, with the three assumptions... 

Eq. (351) can be transformed to Eq. (359). Further identifying Ns with 2ti p( ), Eq. (346) becomes identical with Eq. (361). Hence, under certain circumstances the quantized ARRKM theory is equivalent to the rigorous quantum reaction rate theory. A number of remarks are in order. First, assumption (a) is automatically satisfied by definition. Second, assumption (b) implies that Fw in the quantized ARRKM theory be the direct analog of the quantum flux operator in the flux-flux autocorrelation formalism. Third, assumption (c) requires that the action of the operator 0jy(V5 v) at any particular time, say at time zero, is equivalent to the action of the projector P i) at time infinity. Regarding 0vi (V5 v) as the analog... 

E. A Semiclassical Approximation to the Rigorous Quantum Rate Theory... 

The final and most fundamental impact of Wigner s paper lies in the impetus that it gave to formulating a rigorous quantum theory of the chemical rate constant, without the need to solve the full state to state scattering problem. Early work by McLafferty and Pechukas concluded that can be no... 

Summary. Rate constants of chemical reactions can be calculated directly from dynamical simulations. Employing flux correlation functions, no scattering calculations are required. These calculations provide a rigorous quantum description of the reaction process based on first principles. In addition, flux correlation functions are the conceptual basis of important approximate theories. Changing from quantum to classical mechanics and employing a short time approximation, one can derive transition state theory and variational transition state theory. This article reviews the theory of flux correlation functions and discusses their relation to transition state theory. Basic concepts which facilitate the calculation and interpretation of accurate rate constants are introduced and efficient methods for the description of larger systems are described. Applications are presented for several systems highlighting different aspects of reaction rate calculations. For these examples, different types of approximations are described and discussed. 

The survey below begins with classical mechanical expressions for reaction rates and then shows how the transition state approximation emerges. It is then shown how TST is quantized in an approximate fashion. Finally a rigorous quantum theory of reaction rates is described, one that is fully equivalent to equations (2)-(4) above but that avoids the necessity of obtaining the 5-matrix. 

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