James W. Duff

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

Abstract 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.

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

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 ...

Classical S matrix: numerical applications to classically allowed chemical reactions☆☆☆

Abstract The classical S matrix theory (a theory in which all degrees of freedom are treated semiclassically) developed by Miller and Marcus is applied to the collinear H + H2 reaction on a model potential energy surface. The classical, primitive, and uniform approximations and the initial value representation integral expression for the classical S matrix are calculated for total energies in the range 15–30 kcal/mole and compared to the exact quantum mechardeal calculations of Diestler. We choose a potential energy surface for which the quasiclassical reaction probability is unity over an energy range of 6 kcal/mole in order to make the test of theory particularly meaningful. It is seen that the classical S matrix theory is in very poor agreement with the quantum mechanical results for the reaction probabilities, although the semiclassical results for reactions producing vibrationally excited products are better behaved than those for reactions producing ground state products. The collision lifetimes and the stationary trajectories show no indication at all that the semiclassical calculations include a mechanism which could account for the 75% oscillation in the quantum mechanical PR0-0

Classical probability matrix: Prediction of quantum-state distributions by a moment analysis of classical trajectories☆

Abstract A new method is presented for extracting approximate quantum mechanical state-to-state transition probabilities from the results of classical trajectory calculations. The method recognizes quantum discreteness by dealing with the quantum mechanical probability matrix, but all dynamical quantities are evaluated by classical mechanics. It is illustrated by application to the linear atom-diatom collision (vibrational excitation); it is capable of treating both classically allowed and classically forbidden processes.

Quasiclassical trajectory calculations compared to quantum mechanical reaction probabilities, rate constants, and activation energies for two different potential surfaces for the collinear reaction H2+I→H+HI, including dependence on initial vibrational state

Quantum mechanical calculations are compared to quasiclassical trajectory forward (QCT) calculations for the collinear, endoergic reaction H2(n1)+I→H+HI for two different potential energy surfaces, a rotated‐Morse‐curve (RMC) surface and the semiempirical valence‐bond surface of Raff et al. Vibrationally state‐selected reaction probabilities and rate constants and Arrhenius parameters are presented. Thermally averaged rate constants and their Arrhenius parameters are also given. For one of the potential energy surfaces, quasiclassical trajectory reverse histogram (QCTRH) calculations were also performed. The results show that classical mechanics and quantum mechanics are in significant qualitative agreement for state‐selected properties. Specifically, for the n1=0 state of the Raff et al. surface the quantum mechanical reaction probabilities are very small (less than 0.005) and the QCT method predicts this state to be totally non‐reactive. For all other states on both surfaces quantum mechanics and QCT an...

Exact quantum mechanical reaction probabilities for the collinear H + H2 reaction on a porter-karplus potential energy surface

Abstract Exact quantum mechanical calculation of the reaction probability for the collinear H + H 2 reaction on a Porter-Karplus potential energy surface are carried out by the finite-difference boundary value method at 6 energes in the threshold region and compared to close coupling, distorted wave, classical S matrix, transition state theory, and vibrational adiabatic calculations.

Use of semiclassical collision theory to compare analytic fits to the interaction potential for vibrational excitation of H2 by He

Vibrational transition probabilities for strongly classically forbidden single‐quantum and two‐quantum transitions calculated by two semiclassical methods involving real‐valued trajectories are compared to quantum mechanical close coupling for various analytic fits to the ab initio interaction potential for the He–H2 system. The final‐value‐representation integral expression from classical S matrix theory and the classical‐trajectory forced‐quantum‐oscillator method are found to be in semiquantitative agreement with the quantum mechanical calculation even for transition probabilities as small as about 10−6. Further, the semiclassical methods reproduce the important trends in the results as functions of the interaction potential. The reliability of these semiclassical calculations allows one to determine the region of the potential energy surface which is sufficient for calculation of vibrational excitation probabilities. The important region for the present calculations is in the classically allowed regio...

Classical S matrix: Application of the Bessel uniform approximation to a chemical reaction☆

Abstract The Bessel uniform approximation developed by Stine and Marcus is applied to the collinear H + H 2 reaction on Diestlers potential energy surface no. 3 to which we have previously applied other orders of approximation of classical S matrix theory. It appears that an accurate treatment of this system by classical S matrix theory requires interference of real and complex trajectories. Calculations were also performed on two other potential energy surfaces in order to more clearly understand the interrelationships of previous semiclassical and quasiclassical studies of this reaction.

Classical S matrix: Application to classically forbidden vibrational excitation for He+HBr and H+Br2

The integral expressions of classical S matrix theory are tested against quantum mechanical results and classical path-forced quantum oscillator results for vibrational transition probabilities in collinear collisions of atoms with harmonic and Morse vibrators for the H+Br2 and He+HBr mass combinations. The interaction potential is assumed to be a repulsive exponential function. The energy range studied (in units of ℏ) is 2–10 for H+Br2 and 2–6 for He+HBr. The integral expressions are found to be accurate within a factor of two for almost all transition probabilities greater than 7×10−3 but to be very inaccurate for very small transition probabilities. Quasiclassical trajectory histogram methods are found to be accurate within a factor of two only for transition probabilities greater than 0.15. Neither the integral expressions of classical S matrix theory nor the quasiclassical trajectory histogram method are found to be as generally accurate as the classical path-forced quantum oscillator results.