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The Journal of Physical Chemistry

Comment on Quantum Transition State Theory

Stuart A. Rice, Soonmin Jang, Meishan Zhao

Published Jan 1, 1996• 9 citations
Publisher

Abstract

In a previous paper ( J . Phys . Chem . 1994, 98, 3444), we showed, within the framework of the reaction coordinate representation of the dynamics of a chemical reaction, that application of a complex scaling transformation to the reaction coordinate leads naturally to the definition of transition states with complex energies and to the definition of an operator whose expectation values are the lifetimes of these states. That analysis has the unphysical feature of not eliminating the dependence of the lifetime operator on the positive angle parameter of the unitary transformation that generates the scaling of the reaction coordinate. In this paper, we reexamine the approach used earlier and remove this unphysical feature of the analysis. We recover the results of our earlier analysis and provide an interpretation of other related results in the literature.

DOI
10.1021/jp9607881

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In a previous paper ( J . Phys . Chem . 1994, 98, 3444), we showed, within the framework of the reaction coordinate representation of the dynamics of a chemical reaction, that application of a complex scaling transformation to the reaction coordinate leads naturally to the definition of transition states with complex energies and to the definition of an operator whose expectation values are the lifetimes of these states. That analysis has the unphysical feature of not eliminating the dependence of the lifetime operator on the positive angle parameter of the unitary transformation that generates the scaling of the reaction coordinate. In this paper, we reexamine the approach used earlier and remove this unphysical feature of the analysis. We recover the results of our earlier analysis and provide an interpretation of other related results in the literature.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
The reaction rate constant is defined as the rate at which a reactant annihilates or a product creates in a chemical reaction. The Hamiltonian of the reaction system describing elementary reactions has already been formulated by Miller et al（J. Chem. Phys. 1980, 72, 99-112） In addition, the reaction coordinates representing the progress of the reaction are also formulated by Fukui.（J. Phys. Chem. 1970, 74, 4161-4163） In the present paper, on the basis of these two theories, we first formulated the reaction rate constants of elementary reactions using classical statistical mechanics. Then we rewrited the rate constans quantum-mechanically according to the method of Wigner representations. By this rewriting, a correction term representing the quantum effect of the reaction rate constant was naturally derived.
With this Comment, we would like to provide a general discussion on two important issues arising from a recent work by Goldfarb et al. [J. Chem. Phys.125, 231103 (2006)], which we consider to be of broad interest to the Chemical Physics community. First we introduce a discussion on trajectory-based formalisms associated with the quantum Hamilton-Jacobi approach. Then, we stress the special interest relying on how the term locality should be used in quantum mechanics, in particular, when dealing with trajectory-based formulations of this theory.
A quantum mechanical representation suitable for studying the time evolution of quantum densities in phase space is proposed and examined in detail. This representation on ℒ2 (2) phase space is based on definitions of the operators P̂ and Q̂ in phase space that satisfy various correspondences for the Liouville equation in classical and quantum phase space, as well as quantum position and momentum ℒ2 (1) spaces. The definitions presented here, P̂=p/2−iℏ∂/∂q and Q̂=q/2+iℏ∂/∂p, are related to definitions that have been recently proposed [J. Chem. Phys. 93, 8862 (1990)]. The resulting quantum phase space representation shares many of the mathematical properties of usual representations in coordinate and momentum spaces. Within this representation, time evolution equations for complex-valued functions (wave functions) and their square magnitudes (distribution functions) are derived, and it is shown that the coordinate and momentum space time evolution equations can be recovered by a simple Fourier projection. The phase space quantum probability conservation equation obtained is a good illustration of the quantization rule that requires one to replace the classical Poisson bracket between the Hamiltonian and the probability density with the quantum commutator between the corresponding operators. The possible classical analogs to quantum probabilities densities are also considered and some of the present results are illustrated for the dynamics of the coherent state.
A quantum version of a recent realization of Wigner’s transition state theory in phase space is presented. The theory developed builds on a quantum normal form which locally decouples the quantum dynamics near the transition state to any desired order in ℏ{\hbar}ℏ . This leads to an explicit algorithm to compute cumulative quantum reaction rates and the associated Gamov–Siegert resonances with high accuracy. This algorithm is very efficient since, as opposed to other approaches, it requires no quantum time propagation.