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The Journal of Chemical Physics

On a Theory of Absolute Reaction Rates

Charles Alfred Coulson, Raphael David Levine

Published Aug 15, 1967• 47 citations
Publisher

Abstract

The methods of multichannel collision theory are used to introduce a yield operator Y which is an observable that determines the rate and yield of a reaction. When the reactants are characterized by a statistical operator U0, the reaction rate can be written as Tr(U0Y)/Tr(U0). By using Heisenberg's equation of motion it is possible to extend the formalism by introducing a yield operator YA that determines the rate of change of any measurable operator A in a system undergoing reactions. It is suggested to base semiempirical theories on approximations to the yield operator. Towards this aim the Arrhenius parameterization of the rate of thermal bimolecular reactions is discussed, and alternative forms for the yield operator are exhibited.

DOI
10.1063/1.1712074

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•Jan 1, 2014•1 citation
The methods of multichannel collision theory are used to introduce a yield operator Y which is an observable that determines the rate and yield of a reaction. When the reactants are characterized by a statistical operator U0, the reaction rate can be written as Tr(U0Y)/Tr(U0). By using Heisenberg's equation of motion it is possible to extend the formalism by introducing a yield operator YA that determines the rate of change of any measurable operator A in a system undergoing reactions. It is suggested to base semiempirical theories on approximations to the yield operator. Towards this aim the Arrhenius parameterization of the rate of thermal bimolecular reactions is discussed, and alternative forms for the yield operator are exhibited.
From a rigorous multichannel quantum-defect formulation of bimolecular processes, we derive a fully quantal and analytic model for the total rate of exoergic bimolecular reactions or inelastic processes that is applicable over a wide range of temperatures including the ultracold regime. The theory establishes a connection between the ultracold chemistry and the regular chemistry by showing that the same theory that gives the quantum threshold behavior agrees with the classical Gorin model at higher temperatures. In between, it predicts that the rates for identical bosonic molecules and distinguishable molecules would first decrease with temperature outside of the Wigner threshold region, before rising after a minimum is reached.
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.
Abstract Studying chemical reactions on a state-to-state level tests and improves our fundamental understanding of chemical processes. For such investigations it is convenient to make use of ultracold atomic and molecular reactants as they can be prepared in well defined internal and external quantum states. Here, we investigate a single-channel reaction of two Li2-Feshbach molecules where one of the molecules dissociates into two atoms 2AB ⇒ AB + A + B. The process is a prototype for a class of four-body collisions where two reactants produce three product particles. We measure the collisional dissociation rate constant of this process as a function of collision energy/temperature and scattering length. We confirm an Arrhenius-law dependence on the collision energy, an a4 power-law dependence on the scattering length a and determine a universal four body reaction constant.
It is shown that the quantum mechanics is capable of describing processes which have all the aspects of certain chemical reactions. It turns out that a system may change from one configuration to another when a quantum level of the one configuration has the same energy as a quantum level of the other. Among such reactions mention is made and some discussion given of the large classes of molecular rearrangements, and decompositions, including radioactive disintegration. The dependence on pressure and temperature is slightly discussed and it is indicated that the present theory quite explains the necessity of the old "activation hypothesis" of Arrhenius. The old notions should, however, be modified to take into account that it is not sufficient to have at least the "activation energy" but the molecule must be in a particular state to react. Moreover when there are several "activated" states the rate of reaction from different ones may be very different indeed. Insight is gained into the nature of certain types of catalytic action including special wall catalyses. Some features of photochemical reactions are made clear by the theory.
We present a unified formalism for describing chemical reaction rates of trapped, ultracold molecules. This formalism reduces the scattering to its essential features, namely, a propagation of the reactant molecules through a gauntlet of long-range forces before they ultimately encounter one another, followed by a probability for the reaction to occur once they do. In this way, the electric-field dependence should be readily parametrized in terms of a pair of fitting parameters (along with a C6{C}_{6}C6​ coefficient) for each asymptotic value of partial-wave quantum numbers $|L,{M}_{L}\ensuremath{\rangle}$. From this, the electric-field dependence of the collision rates follows automatically. We present examples for reactive species, such as KRb, and nonreactive species, such as RbCs.
This thesis describes the combination of two sources of cold atomic or molecular species which can be used to study a wide range of ion-molecule reactions. The challenges in forming these species and in determining the fate of reactive events are explored throughout. Reactions occur in a volume within a radio-frequency ion trap, in which ions have previously been cooled to sub-Kelvin temperatures. Ions are laser-cooled, with migration of ions slowed sufficiently to form a quasi-crystalline spheroidal structure, deemed a Coulomb crystal. Fluorescence emitted as a consequence of laser-cooling is detected; the subsequent fluorescence profiles are used to determine the number of ions in the crystal and, in combination with complementary simulations, the temperature of these ions. Motion imparted by trapping fields can be substantial and simulations are required to accurately determine collision energies. A beam of decelerated molecules is aimed at this stationary ion target. An ammonia seeded molecular beam enters a Stark decelerator, based on the original design of Meijer and co-workers. The decelerator uses time-varying electric fields to remove kinetic energy from the molecules, which exit at speeds down to 35 m/s. A fast-opening shutter and focussing elements are subsequently used to maximise the decelerated flux in the reaction volume while minimising undecelerated molecule transmission. Substantial fluxes of decelerated ammonia are obtained with narrow velocity distributions to provide a suitable source of reactant molecules. Combination of these two techniques permits studies of reactions between atomic ions and decelerated molecules that can be entirely state-specific. Changes in the Coulomb crystal fluorescence profile denote changes in the ion identities, the rate of these changes can be used to obtain rate constants. Determination of rate constants is even possible despite the fact that neither reactant nor product ions are directly observed. This work has studied reactions between sympathetically cooled Xe+ ions and guided ND3 and has obtained data consistent with prior studies. Determination of reactive events is complicated if ion identities can change without affecting the fluorescence profile, or if multiple reaction channels are possible. A range of spectroscopic techniques are discussed and considered in regards to determining rate constants and product identities. Pulsed axial excitation of trapped ions can follow rapid changes in average ion weights and subtle changes for small crystals. Time-of-flight mass spectrometry is also demonstrated using the trapping electrodes and is suitable for discrimination of ions formed within the trap.