On the theory of energy distributions of products of molecular beam reactions involving transient complexes

Theoretical energy distributions of reaction products in molecular beam systems are described for reactions proceeding via transient complexes. Loose and tight transition states are considered for the exit channel. For a loose transition state and the case of l ≫ j, the result is the same as of Safron et al. For the case of a tight transition state exit channel effects are included analogous to steric effects for the reverse reaction. It is shown how, via one mechanism, bending vibrational energy of that transition state can contribute to the translational energy of the reaction products. Expressions are derived for the energy distributions of the products when l ≫ j and j ≫ l

Theory of Semiclassical Transition Probabilities (S Matrix) for Inelastic and Reactive Collisions. Uniformization with Elastic Collision Trajectories

A canonical transformation is described for uniformizing the coordinates used in Paper I of this series. For comparison with the results of Paper III, which based a uniformization on exact trajectories, the present article describes one based on elastic collision trajectories. The question of invariance of S-matrix elements with respect to semiclassical unitary transformations is also discussed

Spiers Memorial Lecture: Interplay of theory and computation in chemistry—examples from on-water organic catalysis, enzyme catalysis, and single-molecule fluctuations

In this lecture, several examples are considered that illustrate the interplay of experiment, theory, and computations. The examples include on-water catalysis of organic reactions, enzymatic catalysis, single molecule fluctuations, and some much earlier work on electron transfer and atom or group transfer reactions. Computations have made a major impact on our understanding and in the comparisons with experiments. There are also major advantages of analytical theories that may capture in a single equation an entire field and relate experiments of one type to those of another. Such a theory has a generic quality. These topics are explored in the present lecture

Separation of Sets of Variables in Quantum Mechanics

Separation of the Schrödinger equation for molecular dynamics into sets of variables can sometimes be performed when separation into individual variables is neither possible nor for certain purposes necesary. Sufficient conditions for such a separation are derived. They are the same as those found by Stäckel for the corresponding Hamilton—Jacobi problem, with an additional one which is the analog of the Robertson condition for one‐dimensional sets. Expressions are also derived for operators whose eigenvalues are the separation constants. They provide a variational property for these constants. For use in aperiodic problems an expression is obtained for the probability current in curvilinear coordinates in an invariant form. Application of these results to reaction rate theory is made elsewhere

Enzymatic catalysis and transfers in solution. I. Theory and computations, a unified view

The transfer of hydride, proton, or H atom between substrate and cofactor in enzymes has been extensively studied for many systems, both experimentally and computationally. A simple equation for the reaction rate, an analog of an equation obtained earlier for electron transfer rates, is obtained, but now containing an approximate analytic expression for the bond rupture-bond forming feature of these H transfers. A "symmetrization," of the potential energy surfaces is again introduced [R. A. Marcus, J. Chem. Phys. 43, 679 (1965); J. Phys. Chem. 72, 891 (1968)], together with Gaussian fluctuations of the remaining coordinates of the enzyme and solution needed for reaching the transition state. Combining the two expressions for the changes in the difference of the two bond lengths of the substrate-cofactor subsystem and in the fluctuation coordinates of the protein leading to the transition state, an expression is obtained for the free energy barrier. To this end a two-dimensional reaction space (m,n) is used that contains the relative coordinates of the H in the reactants, the heavy atoms to which it is bonded, and the protein/solution reorganization coordinate, all leading to the transition state. The resulting expression may serve to characterize in terms of specific parameters (two "reorganization" terms, thermodynamics, and work terms), experimental and computational data for different enzymes, and different cofactor-substrate systems. A related characterization was used for electron transfers. To isolate these factors from nuclear tunneling, when the H-tunneling effect is large, use of deuterium and tritium transfers is of course helpful, although tunneling has frequently and understandably dominated the discussions. A functional form is suggested for the dependence of the deuterium kinetic isotope effect (KIE) on DeltaG° and a different form for the 13C KIE. Pressure effects on deuterium and 13C KIEs are also discussed. Although formulated for a one-step transfer of a light particle in an enzyme, the results would also apply to single-step transfers of other atoms and groups in enzymes and in solution

Free Energy of Non equilibrium Polarization Systems. III. Statistical Mechanics of Homogeneous and Electrode Systems

A statistical mechanical treatment is given for homogeneous and electrochemical systems having nonequilibrium dielectric polarization. A relation between the free energy of these systems and those of related equilibrium ones is deduced, having first been derived in Part II by a dielectric continuum treatment. The results can be applied to calculating polar contributions in the theory of electron transfers and in that of shifts of electronic spectra in condensed media. The effect of differences in polarizability (of a light emitting or absorbing molecule in its initial and final electronic states) on the polar term in the shift is included by a detailed statistical analysis, thereby extending Part II. Throughout, the "particle" description of the entities contributing to these phenomena is employed, so as to derive the results for rather general potential energy functions

Solvent dynamics: Modified Rice–Ramsperger–Kassel–Marcus theory. II. Vibrationally assisted case

Expressions are given for a solvent dynamics-modified Rice–Ramsperger–Kassel–Marcus (RRKM) theory for clusters. The role of vibrational assistance across the transition state region is included. The usual differential equation for motion along the slow coordinate X in constant temperature systems is modified so as to apply to microcanonical systems. A negative entropy term, –Sv(X), replaces the (1/T)∂U/∂X or (1/T)∂G/∂X which appears in canonical systems. Expressions are obtained for the RRKM-type rate constant k(X) and for the Sv(X) which appear in the differential equation. An approximate solution for steady-state conditions is given for the case that the "reaction window" is narrow. The solution then takes on a simple functional form. The validity of the assumption can be checked a posteriori. Recrossings of the transition state are included and the condition under which the treatment approaches that in Part I is described

Theory of Semiclassical Transition Probabilities for Inelastic and Reactive Collisions. V. Uniform Approximation in Multidimensional Systems

An integral semiclassical expression for the S matrix of inelastic and reactive collisions was formulated earlier in this series. In the present paper a uniform approximation for the expression is derived for the case of multidimensional systems. The method is an extension of that employed in Part II for the case of one internal coordinate. The final result, Eq. (2), is highly symmetrical, thus making some of its properties immediately clear

Relation between State‐Selected or State‐Averaged Cross Sections of Endothermic Reactions and Rate Constants of Exothermic Reactions. Application of Bimolecular Microcanonical Activated Complex Theory

An expression is derived relating state‐selected or state‐averaged molecular beam reaction cross sections of endothermic reactions to observed rate constants for the inverse (exothermic) reactions. An approximation of Anlauf, Maylotte, Polanyi, and Bernstein is used, together with bimolecular microcanonical activated complex theory [R. A. Marcus, J. Chem. Phys. 45, 2138 (1966)]