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反応速度定数の量子論的定式化 : Wigner表示

Keisho Umesaki, Kei Odai

Published Mar 31, 2022• 0 citations
OpenAlex

Abstract

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.

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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.
Quantum spin dynamics (QSD) offer a novel and potentially transformative approach to chemical reaction simulation, particularly when compared to traditional methods. This paper explores the application of QSD to modeling chemical reaction rates and product formation, aiming to enhance accuracy and efficiency through precise simulation. We investigate a model framework utilizing the quantum spin states of reactants to represent the chemical reaction pathway. The core mechanism focuses on quantitatively predicting reaction outcomes and employing optimization techniques to refine reaction conditions. The current work demonstrates the potential for QSD to provide a more robust and insightful understanding of chemical reactions.
We address basic questions concerning the atomistic-level description of “chemical” materials and their “changes” in the framework of current quantum mechanical understanding of natural phenomena. In this framework, we show how modern quantum chemical analysis of chemical reaction mechanisms, based on Fukui’s intrinsic reaction coordinate (IRC) and the Pauling-like bond order descriptors of natural resonance theory (NRT), can serve to resolve the logical paradoxes of Schrödinger’s cat and Wigner’s “impossibility of life” proof in common-sense manner.
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.
A newly derived partial differential equation for the temperature-dependent spatial distribution function of quantum-statistical mechanics is applied to the case of calculating the spatial distribution function along reaction coordinates for a system at equilibrium. In the limit where Planck's constant tends to zero, it is shown that the partial differential equation yields the classical formula for the probability that the chemical complex is at a specified region of reaction coordinates. Explicit formulas are given for the quantum correction to the probability density as a function of temperature for stable chemical complexes, whose total energy is negative, and for unstable chemical complexes, whose total energy is positive.
We continue our study of Wigner path integrals. We first define a Wigner path integral for the kernel of the Wigner–Bloch equation, solve it for the free particle and harmonic oscillator cases, discuss the physical interpretation which is new and then we apply it (along with the path integral representation of the Liouville equation) to obtain a path integral representation in phase space of the quantum transition state theory of Miller. We also obtain an expression involving products of Wigner kernels for the correlation function form of the exact bimolecular rate constant.