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

Semiclassical S-matrix theory. VI. Integral expression and transformation of conventional coordinates

Sometimes, as in reactive systems, action‐angle variables are not conveniently defined at all points of the trajectory and recourse must be made to conventional coordinates. A simple canonical transformation converts the latter to coordinates of which one is time and the remainder are constant along the trajectory. The transformation serves to remove the singularities of the semiclassical wavefunction at the turning points of the trajectory. It yields, thereby, an integral expression for the S matrix by having produced wavefunctions which can be integrated over all space. The result supplements that of Paper III [R. A. Marcus, J. Chem. Phys. 56, 311 (1972)], which was derived for systems for which action‐angle variables could be defined throughout the collision

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

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

The classical mechanics of chemically reactive linear collisions is investigated for vibrationally near‐adiabatic reactions. A coordinate system which passes smoothly from one suited to the reactants to one suited to the products is used. The Hamilton—Jacobi equation is then solved in the adiabatic approximation by introduction of an "adiabatic‐separable" method. Nonadiabatic corrections, which describe the probability of vibrational transitions, are also calculated. They involve the Fourier component of local internal centrifugal and vibration frequency‐change terms. The reaction coordinate for the adiabatic system is shown to be that curve on which local vibrational and internal centrifugal forces balance pointwise. Applications can be made to the role of translational—vibrational energy interchange in reactions, reaction‐cross‐section theory, bobsled effect, and other topics. The results may be compared with electronic computer calculations as they become available

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

Fourth Picture in Quantum Mechanics

A quantum‐mechanical counterpart to the classical mechanical variation of constants method is derived, with initial values of coordinates and momenta as “constants.” Use is made of a formal operator solution for nonautonomous or autonomous systems in classical mechanics, which we published earlier, and of the correspondence between Poisson brackets and commutators. An alternative unified Lie‐algebraic derivation is also given. It is shown that the Schrödinger, Heisenberg, and interaction pictures in quantum mechanics do not correspond directly to the method of classical mechanical variation of these “constants.” A fourth picture, termed “mixed interaction,” is introduced and shown to so correspond. It complements the previous three in a symmetrical manner, bearing the same relation to the Heisenberg picture that the Schrödinger picture bears to the interaction one. The group‐theoretic relationship to the interaction picture is noted, as is the relation to the usual variation‐of‐constants method in wave mechanics. For completeness, the classical counterparts of the Heisenberg and interaction pictures are also given. The present results arose from a comparison of quantum and classical treatments of collisions

On the Theory of Chemical-Reaction Cross Sections. I. A Statistical-Dynamical Model

A statistical‐dynamical model is formulated for total chemical‐reaction cross sections as a function of the relative velocity and the vibrational and the rotational state of the reactants. It is derived for reactions for which activated‐complex configurations can be defined; reactions with or without steric and activation barriers. A quasiequilibrium is postulated between reacting pairs and activated complexes of the same energy and angular momentum. An integral equation is obtained which is solved for the reaction cross section by introduction of a second postulate: The reaction probability is a function of the excess initial energy along the reaction coordinate (in excess of potential energy barrier, centrifugal potential barrier, and vibrational adiabatic requirements). A possible dynamical origin of the postulates is considered in later papers