A universal instability of many-dimensional oscillator systems

In recent years our research group has made a systematic effort to study the validity of transition state theory (TST). We have found that the conventional theory is sometimes remarkably accurate, but in many other cases it leads to large errors. Fortunately we have found that a much more reliable theory that has many of the advantages of conventional TST can also be formulated, and it can be applied to practical problems with an effort that is much closer to that required for conventional transition state theory than to that required for quantal dynamics calculations. The two most important features in the improved approach to transition state theory state theory are the variational determination of the transition state and the incorporation of tunneling contributions by multidimensional semiclassical approximations. 13 refs.

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Variational Transition State Theory

A review of the theoretical and computational modeling of bimolecular reactions is given. The review is divided into several sections which are as follows: gas-phase thermal reactions; gas-phase state-selected reactions and product state distributions; and condensed-phase bimolecular reactions. The section on gas-phase thermal reactions covers the enthalpies and free energies of reaction, kinetics, saddle points and potential energy surfaces, rate theory for simple barrier reactions and bimolecular reactions over potential wells. The section on gas-phase state-selected reactions focuses on electronically adiabatic reactions and electronically nonadiabatic reactions. Finally, the section on condensed-phase bimolecular reactions covers reactions in liquids, reactions on surfaces and in solids and tunneling at low temperature.

Modeling the Kinetics of Bimolecular Reactions

Rather general expressions are derived which represent the semiclassical time‐dependent propagator as an integral over initial conditions for classical trajectories. These allow one to propagate time‐dependent wave functions without searching for special trajectories that satisfy two‐time boundary conditions. In many circumstances, the integral expressions are free of singularities and provide globally valid uniform asymptotic approximations. In special cases, the expressions for the propagators are related to existing semiclassical wave function propagation techniques. More generally, the present expressions suggest a large class of other, potentially useful methods. The behavior of the integral expressions in certain limiting cases is analyzed to obtain simple formulas for the Maslov index that may be used to compute the Van Vleck propagator in a variety of representations.

Integral expressions for the semiclassical time‐dependent propagator

6.2. Reactive Flux Method 3156 6.3. Model Theories 3156 6.4. Survey 3156 7. Applications 3157 7.1. Yeast Enolase 3157 7.2. Triosephosphate Isomerase 3157 7.3. Methylamine Dehydrogenase 3158 7.4. Alcohol Dehydrogenase 3158 7.5. Thermophilic Alcohol Dehydrogenase 3158 7.6. Haloalkane Dehalogenase 3159 7.7. Dihydrofolate Reductase from E. Coli 3159 7.8. Hyperthermophilic DHFR from Thermotoga Maritima 3160

https://chem.ccu.edu.tw/~hu/Web_Lib/literature/quantum/Enzyme_Tunneling_Truhlar_Chem_Rev_2006.pdf

Multidimensional tunneling, recrossing, and the transmission coefficient for enzymatic reactions.

Recent developments in mathematics and statistical mechanics indicate that efficient intramolecular energy transfer in coupled anharmonic oscillator systems can only occur above a critical energy Ec A simple method for computing Ec is proposed and is applied to the Barbanis, Henon–Heiles, Tredgold, and Thiele–Wilson systems The critical energy for these systems is shown to be easily and accurately predicted by this technique

A variational equations approach to the onset of statistical intramolecular energy transfer

Three kinds of semiclassical theory are tested against quantum mechanical results for vibrational transition probabilities and average vibrational energy transfers in collinear collisions of atoms with harmonic and Morse vibrators for the He-H 2 mass combination. The interaction potential is assumed to be a repulsive exponential function with an exponential parameter which is realistic for He-H 2 collisions. The energy range studied is total energies of 2–8 in units of ħω e . The uniform semiclassical approximations of classical S matrix theory are tested only for classically allowed transitions, i.e., for transition probabilities greater than about 0.2. They are accurate quantitatively for both harmonic and Morse vibrators. The integral expressions of classical S matrix theory are found to be quantitatively accurate for classically allowed and weakly classically forbidden transitions, i.e., for transition probabilities greater than about 0.01–0.05, and to be unreliable for strongly classically forbidden transitions. Quasiclassical trajectory methods yield qualitatively accurate results only for classically allowed transitions but the phase-averaged energy transfer in quasiclassical collisions may be accurate even when classically forbidden transition probabilities are important for the calculation of the average energy transfer. Forced quantum oscillator methods using a classical path whose initial velocity is the average of the initial and final velocities corresponding to the transition of interest are accurate for transition probabilities as small as 4 × 10 −8 for harmonic vibrators but do not seem to accurately account for the effect of anharmonicity.

Tests of semiclassical treatments of vibrational-translational energy transfer collinear collisions of helium with hydrogen molecules

Collinear quasiclassical trajectories are examined for two realistic potential energy surfaces for atom−diatomic molecule reactions for two reaction attributes: (1) vibrational energy of the products of a thermal−energy exothermic reaction; (2) threshold energy for endothermic reaction of ground−state reagents. Eight different mass combinations are studied. The potential energy surfaces differ primarily in the amount of potential energy released in an exothermic reaction before and in the region of large curvature of the minimum−energy path and in the curvature of the repulsive potential energy contours when all three atoms are close. For attribute (1), we find the results are qualitatively correlated by the theory of Hofacker and Levine although, contrary to previous work, one potential energy surface shows high mixed energy release (in the language of Polanyi and co−workers) but low excitation to product vibration for five different mass combinations. For reaction attribute (2), we find one surface has a high translational threshold (or no reaction at any energy) for six mass combinations, while the other surface shows this behavior in only three cases. Thus, this type of surface provides an exception to previous generalizations that extra vibrational energy is required for very endothermic reactions with late barriers. This demonstrates the importance of the location of the curvature of the reaction channel for such reaction attributes. Very accurate determinations of potential energy surfaces will be required to make reliable predictions of reaction attributes such as (1) and (2) for real systems. Analysis of the details of the trajectories shows that the high threshold can generally be attributed to reflection before the saddle point of the surface rather than to recrossing the saddle point region. The vibrational excitation of reagents in nonreactive collisions is also strongly effected by curvature of the minimum−energy path.

Effect of curvature of the reaction path on dynamic effects in endothermic chemical reactions and product energies in exothermic reactions

The dynamic basis for statistical behavior in classical collinear bimolecular collisions is explored using an approach suggested by recent studies in nonlinear mechanics. Studies of the dynamics on three model potential energy surfaces including the Karplus–Porter H+H2 system indicate that statistical product distributions evolve from regions of the initial phase space characterized by trajectories which exponentially separate in time from nearby neighbors by more than 3 orders of magnitude. The time scale required to achieve this degree of exponential separation and associated statistical behavior is substantially shorter than that expected from arguments based on rates of intramolecular energy transfer. Our computational results also provide evidence for the validity of several assumptions implicit in the variational equations approach to determining the critical energy for energy randomization in molecular systems.

James W. Duff

Papers

A variational equations approach to the onset of statistical intramolecular energy transfer

Tests of semiclassical treatments of vibrational-translational energy transfer collinear collisions of helium with hydrogen molecules

Effect of curvature of the reaction path on dynamic effects in endothermic chemical reactions and product energies in exothermic reactions

Exponentiating trajectories and statistical behavior in collinear atom--diatom collisions

Classical S matrix: numerical applications to classically allowed chemical reactions