Quantum mechanical reactive scattering for three-dimensional atom plus diatom systems. II. Accurate cross sections for H+H_2

Accurate three‐dimensional reactive and nonreactive quantum mechanical cross sections for the H+H_2 exchange reaction on the Porter–Karplus potential energy surface are presented. Tests of convergence in the calculations indicate an accuracy of better than 5% for most of the results in the energy range considered (0.3 to 0.7 eV total energy). The reactive differential cross sections are exclusively backward peaked, with peak widths increasing monotonically from about 32° at 0.4 eV to 51° at 0.7 eV. Nonreactive inelastic differential cross sections show backwards to sidewards peaking, while elastic ones are strongly forward peaked with a nearly monotonic decrease with increasing scattering angle. Some oscillations due to interferences between the direct and exchange amplitudes are obtained in the para‐to‐para and ortho‐to‐ortho antisymmetrized cross sections above the effective threshold for reaction. Nonreactive collisions do not show a tendency to satisfy a "j_z‐conserving" selection rule. The reactive cross sections show significant rotational angular momentum polarization with the m_j=m′_j=0 transition dominating for low reagent rotational quantum number j. In constrast, the degeneracy averaged rotational distributions can be fitted to statistical temperaturelike expressions to a high degree of accuracy. The integral cross sections have an effective threshold total energy of about 0.55 eV, and differences between this quantity and the corresponding 1D and 2D results can largely be interpreted as resulting from bending motions in the transition state. In comparing these results with those of previous approximate dynamical calculations, we find best overall agreement between our reactive integral and differential cross sections and the quasiclassical ones of Karplus, Porter, and Sharma [J. Chem. Phys. 43, 3259 (1965)], at energies above the quasiclassical effective thresholds. This results in the near equality of the quantum and quasiclassical thermal rate constants at 600 K. At lower temperatures, however, the effects of tunneling become very important with the quantum rate constant achieving a value larger than the quasiclassical one by a factor of 3.2 at 300 K and 18 at 200 K

Quantum mechanical reactive scattering for three-dimensional atom plus diatom systems. I. Theory

A method is presented for accurately solving the Schrödinger equation for the reactive collision of an atom with a diatomic molecule in three dimensions on a single Born–Oppenheimer potential energy surface. The Schrödinger equation is first expressed in body‐fixed coordinates. The wavefunction is then expanded in a set of vibration–rotation functions, and the resulting coupled equations are integrated in each of the three arrangement channel regions to generate primitive solutions. Next, these are smoothly matched to each other on three matching surfaces which appropriately separate the arrangement channel regions. The resulting matched solutions are linearly combined to generate wavefunctions which satisfy the reactance and scattering matrix boundary conditions, from which the corresponding R and S matrices are obtained. The scattering amplitudes in the helicity representation are easily calculated from the body fixed S matrices, and from these scattering amplitudes several types of differential and integral cross sections are obtained. Simplifications arising from the use of parity symmetry to decouple the coupled‐channel equations, the matching procedures and the asymptotic analysis are discussed in detail. Relations between certain important angular momentum operators in body‐fixed coordinate systems are derived and the asymptotic solutions to the body‐fixed Schrödinger equation are analyzed extensively. Application of this formalism to the three‐dimensional H+H_2 reaction is considered including the use of arrangement channel permutation symmetry, even–odd rotational decoupling and postantisymmetrization. The range of applicability and limitations of the method are discussed

Vibrational deactivation on chemically reactive potential surfaces: An exact quantum study of a low barrier collinear model of H + FH, D + FD, H + FD and D + FH

We study vibrational deactivation processes on chemically reactive potential energy surfaces by examining accurate quantum mechanical transition probabilities and rate constants for the collinear H + FH(v), D + FD(v), H + FD(v), and D + FH(v) reactions. A low barrier (1.7 kcal/mole) potential surface is used in these calculations, and we find that for all four reactions, the reactive inelastic rate constants are larger than the nonreactive ones for the same initial and final vibrational states. However, the ratios of these reactive and nonreactive rate constants depend strongly on the vibrational quantum number v and the isotopic composition of the reagents. Nonreactive and reactive transition probabilities for multiquantum jump transitions are generally comparable to those for single quantum transitions. This vibrationally nonadiabatic behavior is a direct consequence of the severe distortion of the diatomic that occurs in a collision on a low barrier reactive surface, and can make chemically reactive atoms like H or D more efficient deactivators of HF or DF than nonreactive collision partners. Many conclusions are in at least qualitative agreement with those of Wilkin’s three dimensional quasiclassical trajectory study on the same systems using a similar surface. We also present results for H + HF(v) collisions which show that for a higher barrier potential surface (33 rather than 1.7 kcal/mole), the deactivation process becomes similar in character to that for nonreactive partners, with v→v−1 processes dominating

Comment on "Theory of collisions between an atom and a diatomic molecule in the body-fixed coordinate system"

It is shown that the phase factor i^(J+j+I) omitted in a theory of atom‐diatom monreactive scattering formulated by schatz and kupperman but included in the Choi, Poe, and Tang theory does not lead to errors in the analysis of te (H,H_2) system

Quantum mechanical reactive scattering for planar atom plus H_2 diatom systems. II. Accurate cross sections for H+H_2

The results of an accurate quantum mechanical treatment of the planar H+H_2 exchange reaction on a realistic potential energy surface are presented. Full vibration–rotation convergence was achieved in the calculations, and this, together with a large number of auxiliary convergence and invariance tests, indicates that the cross sections are accurate to 5% or better. The reactive differential cross sections are always backward peaked over the range of total energies from 0.3 to 0.65 eV. Nonreactive j=0 to j′=2 cross sections are backward peaked at low energy (0.4 eV) shifting to sidewards peaking for E≳0.5 eV. Quantum symmetry interference oscillations are very significant in the j=0 to j′=2 para‐to‐para cross sections for E≥0.6 eV. Reactive integral cross sections show two distinct kinds of energy dependence. At low energy (<0.5 eV), barrier tunneling gives them a largely exponential energy dependence while above 0.5 eV (the effective threshold energy) the cross sections vary nearly linearly. Comparison of collinear and coplanar transition probabilities indicates similar 1D and 2D energy dependence but with a shift in energy from 1D to 2D due to bending motions in the transition state. An analysis of rotational distributions indicates surprisingly good correspondence with temperaturelike distributions. The results of a one‐vibration‐approximation calculation are examined, and errors of as much as three orders of magnitude are found at some energies. Shapes of angular distributions are, however, accurately predicted by this approximate method. Additional analyses include comparisons with previous distorted wave and coupled‐channel results, and calculations of thermal rate constants

Role of direct and resonant (compound state) processes and of their interferences in the quantum dynamics of the collinear H + H2 exchange reaction

The question of the relative importance of compound state (i.e., activated complex) and direct reaction mechanisms has been of central importance for the dynamical foundations of chemical kinetics [1,2]. The studies here reported indicate that in the quantum dynamics of the historically important H + H2 exchange reaction not only do both such mechanisms contribute but also that their interference plays a central role in determining the pronounced quantum oscillations of the reaction probability as a function of energy [3]. This accounts not only for the absence of such oscillations in quasiclassical calculations [4], but also for the inability of the present semiclassical formalism [5] to produce them [6]

Quantum mechanical reactive scattering: An accurate three-dimensional calculation

Accurate three–dimensional (3–D) quantum–mechanical calculations of differential and total cross sections for the H + H2 exchange reaction on the Porter and Karplus potential energy surface have been performed. These are the first such calculations which are vibrationally and rotationally converged, and the results are believed to be accurate to 5% or better. They can serve as a comparison standard against which approximate methods can be tested

Coplanar and collinear quantum mechanical reactive scattering: The importance of virtual vibrational channels in the H + H_2 exchange reaction

We have performed accurate quantum mechanical calculations for the coplanar H + H_2 exchange reaction, using sufficient rotational and vibrational basis functions in the close-coupling expansion to ensure convergence. We repeated these calculations with a converged rotational basis set but with only one vibrational basis function, in analogy to what Saxon and Light and Wolken and Karplus, respectively, did for the similar coplanar and three dimensional reaction. The vibrationally converged and one-vibration results differ substantially for the coplanar as well as the collinear reaction, indicating the crucial role played by virtual vibrational channels

Exact quantum quasiclassical, and semiclassical reaction probabilities for the collinear F+D_2 → FD+D reaction

Exact quantum, quasiclassical, and semiclassical reaction probabilities and rate constants for the collinear reaction F+D_2 → FD+D are presented. In all calculations, a high degree of population inversion is predicted with P^R_(03) and P^R(04) being the dominant reaction probabilities. In analogy with the F+H_2 reaction (preceding paper), the exact quantum 0→3 and 0→4 probabilities show markedly different energy dependence with PR03 having a much smaller effective threshold energy (E_T=0.014 eV) than P^R_(04) (0.055 eV). The corresponding quasiclassical forward probabilities P^R_(03) and P^R_(04) are in poor agreement with the exact quantum ones, while their quasiclassical reverse and semiclassical counterparts provide much better approximations to the exact results. Similar comparisons are also made in the analysis of the corresponding EQ, QCF, QCR, and USC rate constants. An information theoretic analysis of the EQ and QCF reaction probabilities indicates nonlinear surprisal behavior as well as a significant isotope dependence. Additional quantum results at higher energies are presented and discussed in terms of threshold behavior and resonances. Exact quantum reaction probabilities for the related F+HD → FH+D and F+DH → FD+H reactions are given and an attempt to explain the observed isotope effects is made