On the Analytical Mechanics of Chemical Reactions. Quantum Mechanics of Linear Collisions

The analytical quantum mechanics of chemically reactive linear collisions is treated in the vibrationally near‐adiabatic approximation. The "reaction coordinate" in this approximation is found to be the curve on which the classical local vibrational and internal centrifugal forces balance. Expressions are obtained for the calculation of transmission coefficients for these nonseparable systems. Some implications for tunneling calculations in the literature are noted. Expressions for nonadiabatic corrections are derived, the latter being associated with vibrational transitions undergone by the transmitted and reflected waves. When the system does not have enough energy to react, the last results refer to the vibration—translation energy‐transfer problem in linear collisions. Two novel features are the introduction of an actual coordinate system which passes smoothly from one suited to reactants to one suited to products and the introduction of an adiabatic‐separable method, a method which includes curvilinear effects. Extensions to collisions in higher dimensions are given in later papers

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

Dissociation and Isomerization of Vibrationally Excited Species. III

The equations of Part I for the specific and over‐all unimolecular reaction‐rate constants are extended slightly by including centrifugal effects in a more detailed way and by making explicit allowance for possible reaction‐path degeneracy (optically or geometrically isomeric paths). The expression for reaction‐path degeneracy can be applied to other types of reactions in discussions of statistical factors in reaction rates

High‐Order Time‐Dependent Perturbation Theory for Classical Mechanics and for Other Systems of First‐Order Ordinary Differential Equations

A time‐dependent perturbation solution is derived for a system of first‐order nonlinear or linear ordinary differential equations. By means of an ansatz, justified a posteriori, the latter equations can be converted to an operator equation which is solvable by several methods. The solution is subsequently specialized to the case of classical mechanics. For the particular case of autonomous equations the solution reduces to a well‐known one in the literature. However, when collision phenomena are treated and described in a classical “interaction representation” the differential equations are typically nonautonomous, and the more general solution is required. The perturbation expression is related to a quantum mechanical one and will be applied subsequently to semiclassical and classical treatments of collisions

Existence of Black Holes Due to Concentration of Angular Momentum

We present a general sufficient condition for the formation of black holes due to concentration of angular momentum. This is expressed in the form of a universal inequality, relating the size and angular momentum of bodies, and is proven in the context of axisymmetric initial data sets for the Einstein equations which satisfy an appropriate energy condition. A brief comparison is also made with more traditional black hole existence criteria based on concentration of mass.Comment: 8 pages; final versio

Symmetry or asymmetry of k_(ET) and i_(STM) vs. potential curves

The symmetry or asymmetry of STM current vs. bias potential and of electron transfer (ET) rate vs. overpotential curves is discussed for ET and for STM patterns across ordered monolayers. The superexchange expression for the electronic coupling matrix element, the Fermi–Dirac distribution and, for the ET reaction, the reorganization, are included. A mean potential approximation is assumed for the effect of bias or overpotential on the electronic orbitals or the ordered monolayer. Consequences for the symmetry vs. asymmetry of the ln(k_(ET))vs. overpotential and for the ln(i_(STM)) and pattern vs. bias are described. Examples of some relevant experiments are considered