Kenneth G. Kay

Theory of vibrational relaxation in isolated molecules

A quantum mechanical theory of intramolecular energy transfer is presented which treats the approach to statistical equilibrium of the vibrations in isolated molecules. The theory is appropriate for nonreactive molecules which have at least 6–9 atoms and which are so highly excited that the vibrational interaction couples together many degrees of freedom at a time and causes the simultaneous exchange of a large number of quanta. A generalized master equation of the Van Hove type and a weak‐coupling master equation of the Pauli type are obtained for functions related to occupation probabilities of zero‐order states. The asymptotic behavior of the coarse‐grained occupation probabilities is examined and molecular ergodicity is proved. Convergent infinite expansions for the probabilities are also derived. These analytical expressions make it possible to study the intramolecular dynamics for all relevant times.

Some Theoretical Results for the Photochemical Decomposition of Large Molecules

The expressions describing photochemical dissociation derived by Rice, McLaughlin, and Jortner are evaluated analytically for molecules satisfying Γ ≫ e (1+r) and for times t ≪ ℏ / e, where Γ is the width of the initial state due to interaction with an intermediate manifold, e is the level spacing of the manifold, and r is essentially the ratio of the manifold level widths to e. Excitation is found to decay exponentially from the initial state with rate Γ / [(1+r) ℏ]. In contrast to the behavior originally perdicted, the decay is found to be nonsequential, and a constant ratio, equal to r, is maintained between the populations of the continuum and the manifold. The results should be applicable to large molecules having a single decomposition mode.

The Bloch equation for multiphoton absorption. I. Derivation

A Bloch equation describing infrared multiphoton absorption in an isolated polyatomic molecule is derived from first principles. The molecule is divided into a ’’system’’ mode which interacts directly with the laser field and a ’’bath’’ consisting of the remaining modes which interact with each other and the system mode via intramolecular vibrational coupling. In addition to describing the evolution of the system, the derived equation keeps track of changes in the bath state and the resulting changes in the bath–system interaction which occur as the bath gains energy. Unlike the master (or rate) equation for optical pumping, the Bloch equation is valid for arbitrary ratios W/Ω of the intramolecular relaxation rate W/h/ to the Rabi frequency of the system mode Ω/h/. The equation derived differs from certain Bloch equations previously proposed on phenomenological grounds by the appearance of off‐diagonal coupling terms. These terms may significantly reduce the vibrational dephasing rate and thus affect net ...

Decomposition of isolated molecules: A transition state treatment

It is argued that, contrary to the assumption of RRKM theory, reactant states near the critical surface frequently may not be at statistical equilibrium with the bulk of reactant states. The main objective of this work is to examine conditions under which the RRKM specific unimolecular rate constant expression nevertheless remains valid. The analysis proceeds by casting the general, time‐dependent, decomposition rate of isolated molecules into time‐independent form and applying a transition state approximation similar to that introduced by Miller for bimolecular reactions. In the process of carrying out this program, the following is achieved: (a) A potentially useful unimolecular rate expression, analogous to Miller’s transition state theory rate formula for bimolecular reactions, is obtained; (b) a criterion for choosing critical configurations similar to the Bunker–Pattengill recipe is derived when classical mechanics is obeyed; (c) features of a recent calculation of unimolecular reaction dynamics are...

Intramolecular vibrational energy transfer: A study of representations

In this paper we report a study of possible representations for the description of intramolecular energy transfer. Our ultimate goal is the definition of a scheme suitable to the description of a molecule undergoing dissociation which is valid everywhere along the reaction path. For this reason we study three sets of basis functions for dynamical calculations of intramolecular vibrational energy transfer in one‐dimensional, bound or metastable, triatomic molecules having one harmonic and one Morse bond. The different sets result from different definitions of bond coordinates and different methods for separating the bond motions. For Basis Set 1, the two bond coordinates are internuclear distances between adjacent atoms and the interbond coupling is caused by kinetic energy cross‐terms. For Basis Sets 2 and 3 the coordinates are the distance between adjacent atoms in the harmonic bond and the distance between the dissociable atom and the remaining diatomic. The separation of coordinates for Basis Set 2 is ...

Dynamical treatment of unimolecular decomposition reactions. The RRKM formula

To explore the range of validity of the RRKM theory of unimolecular reactions, we present a completely dynamical derivation of the RRKM expression for the decomposition rate of isolated molecules. This derivation avoids the usual equilibrium statistical assumptions and expresses the conditions for validity of the RRKM theory in terms of fundamental, static, molecular properties. To carry out this derivation we apply a treatment of energy transfer and decomposition which combines the Wigner–Eisenbud R‐matrix approach to scattering with a technique we previously developed for studying the internal dynamics of nonreactive molecules. We obtain a molecular dissociation rate which agrees with the predictions of microcanonical transition state theory by introducing conditions which ensure statistical equilibration of all states describing the molecular fragments in close proximity of each other. We verify that, under the conditions of our derivation, the distribution of product states is statistical, i.e., in ag...

Comparison of quantum, classical, and statistical behavior in dissociating triatomics

Time‐dependent calculations are presented which compare the dynamics of quantum and classical one‐dimensional triatomic systems undergoing intramolecular vibrational energy transfer and dissociation. The purpose of these calculations is to determine whether statistical dissociative behavior in classical systems implies similar behavior in the analogous quantum systems and to test for the presence of quantum mechanical effects that reduce the tendency of the systems to decompose statistically. The intramolecular vibrational energy transfer is monitored by computing the probability for the systems to remain in their initial, coarsely grained states, and the dissociation is followed by calculating the time‐dependent decomposition probability and the product distribution. The classical calculations are performed by a version of the quasiclassical technique while the quantum calculations are carried out by an R‐matrix method. The results show that the two forms of dynamics usually result in similar intramolecular evolution and unimolecular decay. Since the behavior of the classical systems is statistical in a well‐defined sense, it is argued that the behavior of the quantum mechanical systems can likewise be labeled as statistical in these typical cases. Important exceptions to the generally good quantum‐classical agreement occur, however, when the systems are prepared with high energy in a dissociable bond and low energy in the other bond. In such cases, the quantum behavior differs significantly from the classical behavior; the quantum dynamics of decomposition is nonstatistical even though the classical dynamics is statistical. It is found that the ‘‘quantum trapping’’ states which lead to the nonstatistical quantum mechanical behavior are associated with narrow Feshbach resonances which accumulate in certain specific energy regions. It is further concluded that these states occupy a significant proportion of the classical phase space available to molecular complexes with energy below the first vibrational threshold.Time‐dependent calculations are presented which compare the dynamics of quantum and classical one‐dimensional triatomic systems undergoing intramolecular vibrational energy transfer and dissociation. The purpose of these calculations is to determine whether statistical dissociative behavior in classical systems implies similar behavior in the analogous quantum systems and to test for the presence of quantum mechanical effects that reduce the tendency of the systems to decompose statistically. The intramolecular vibrational energy transfer is monitored by computing the probability for the systems to remain in their initial, coarsely grained states, and the dissociation is followed by calculating the time‐dependent decomposition probability and the product distribution. The classical calculations are performed by a version of the quasiclassical technique while the quantum calculations are carried out by an R‐matrix method. The results show that the two forms of dynamics usually result in similar intramolecu...

Dissociation dynamics of collinear triatomic systems by the R‐matrix method

A straightforward computational technique is developed for the quantum mechanical study of unimolecular decay. It is applied to collinear triatomic systems in which the central atom interacts with one terminal atom through a harmonic oscillator potential and with the other terminal atom through a Morse oscillator potential. Stationary state wavefunctions for these systems are generated over an energy grid by applying the Wigner R‐matrix method with Buttle correction. Projections of the stationary wavefunctions onto nonstationary wavefunctions describing metastable states of the triatomic molecule are computed from the R‐matrix basis set expansion of these functions. Time dependent state‐to‐state transition probabilities and final product distributions are then calculated from the projections by Fourier transform and quadrature techniques. The observed time evolution is analyzed in terms of contributions from bound states, resonance states, and branch cuts. Rapid nonexponential decay observed for a variety...

Extended model for photochemical dissociation reactions

The model for photodissociation reactions introduced by Rice, McLaughlin, and Jortner is generalized by allowing the excited molecule to decompose into several decay channels. In accordance with models for unimolecular reactions, intermediate resonance states having similar energies are assumed to couple to different channels and only resonances separated by a sufficiently large energy spacing, e2, are taken to couple to the same channels. The molecule is shown to decay exponentially from the initially excited state at the nonradiative rate Γ1/(1+r)ℏ, where Γ1/ℏ is the nonradiative rate in the absence of resonance‐channel coupling and r=πΓ2/2e2 measures the ratio of the nonradiative resonance widths Γ2 to the energy spacing e2. Plots of the resonance populations Pman and the channel populations Pdiss versus time consist of exponential segments linked at times t=p h/e2, p=1,2,3,…, to yield continuous but only piecewise smooth curves. For t < h/e2, decay from the initially prepared state populates the reson...

A derivation of the master equation by restricted quantum exchange

Abstract The Pauli master equation for intramolecular vibrational relaxation and the heat bath feedback Bloch equations for radiative pumping of polyatomic molecules can be derived by replacing the standard assumption of random matrix element coupling between zero-order vibrational states by an assumption that relaxation is governed by restricted quantum exchange.