Christoph Schlier

How accurate is the Rice–Ramsperger–Kassel–Marcus theory? The case of H+3

The classical Rice–Ramsperger–Kassel–Marcus formula is tested at an accuracy level of a few percent by comparing results of numerical phase space integration with lifetimes deduced from trajectory calculations. The test object is HD+2; the calculation has been done for total energies of 0.5, 1.0, and 1.5 eV above dissociation, and for total angular momenta of 0–60ℏ. Presupposing that the trajectory calculations show the true classical dynamics, we find systematic deviations of up to 40% of the RRKM results. They can be fully explained by the influence of ‘‘direct trajectories,’’ a special kind of nonergodic behavior of the system. After correction for this phenomenon, both methods agree to within the accuracy of the calculations, which is about 3%. We also verified that the discrepancy vanishes when the energy approaches the dissociation energy.

Accurate specific molecular state densities by phase space integration. II. Comparison with quantum calculations on H+3 and HD+2

The semiclassical determination of N(E;J) and ρ(E;J), the specific number and density of quantum states at energy E, and fixed total angular momentum J, by Monte Carlo integration of phase space is compared to recent exact quantum calculations on H+3 and HD+2, which yielded lists of up to 900 quantum states for single values of J. This allows for the first time tests of such a procedure to be made without assuming anything about separability or harmonicity of the potentials. The excellent agreement between semiclassical and quantum state counts shows that the semiclassical numerical computation is a viable and simple method for the determination of state numbers and densities in small molecules with a precision of the order of 1%. For J=0, the procedure has been extended to state numbers for the different symmetry species occuring in H+3 and HD+2.

Accurate specific molecular state densities by phase space integration. I. Computational method

The semiclassical determination of the specific density of quantum states, ρ(E;J), at energy E with fixed total angular momentum J is discussed for small molecules. Monte Carlo integration allows the accurate numerical determination of the phase space volume of systems with J>0 and arbitrary anharmonicity. The corresponding semiclassical number of states can be corrected for the effects of zero point motion in analogy to the well‐known Whitten–Rabinovitch procedure. In this paper, the procedures are tested by comparison with rigid rotor harmonic oscillator models, while a comparison with recent exact quantum calculations on H+3 and HD+2 is described in the following paper. We conclude that, if the intramolecular potential is known or assumed, this numerical semiclassical procedure is a viable and simple way to get state densities of a much improved accuracy.

Monte Carlo integration with quasi-random numbers: some experience

Abstract We report on our general experience and on some test calculations with quasi-random numbers of the Halton type applied to Monte Carlo integration in several (4–8) dimensions. Compared with the traditional use of (pseudo-)random numbers we find that, at a prescribed level of accuracy, at least one order of magnitude in computing time may be saved even for a step function integrand.

Complex formation in proton-hydrogen collisions. II. Isotope effects

Abstract Classical trajectory calculations have been performed on the DIM potential energy surface of H+H2 for collision energies between 20 meV and 2 eV. Complex formation cross sections have been determined for many combinations of projectile and target masses, showing that capture behind the centrifugal barrier is a necessary but by no means sufficient condition for the formation of a long-lived complex. Its probability depends on energy and masses in a way which suggests that the initial energy loss of the projectile on its approach to, and first encounter with the target plays a crucial role in the “trapping” process. H/D isotope effects exceed 20% at energies above 1 20 of the potential well depth, and reach more than 200% at higher energies. At collision energies below 1 100 of the well depth the isotope effect disappears, and the complex formation cross section becomes equal to the capture cross section if the target has no (classical) internal excitation. If, on the contrary, the target is internally excited this equality is invalidated, and an isotope effect of a few percent due to different zero point energies remains. Microreversibility arguments show that these effects should have a perceptible influence on the results of a “dynamically biased” phase space theory.

The combination of complex scaling and the Lanczos algorithm

Abstract We present a symbiosis of three very powerful mathematical tools: the discrete variable representation (DVR), complex scaling, and the Lanczos algorithm, by which we demonstrate that the widespread negative view on combining the Lanczos algorithm with the complex coordinate method for large systems may be too pessimistic. We apply a complex version of the DVR method suitable for homogeneous operators in the DVR, such as x and make use of a Mobius transform inducing a spectral shift, which enhances significantly the efficiency of the complex symmetric matrix adapted Lanczos algorithm. This is done without any loss of the advantages provided by each method on its own. A numerical application to a collinear A—B—A molecule is given.

Orbiting trajectories and the adiabatic approximation to the capture cross section

For the potentialV(R)=CR−4(1+ε(cos2ϑ−1/2)) and a planar geometry we compute classical trajectories, which are a good approximation to those trapped orbits which separate collisions with “capture” from those without. Two possible types of such trajectories (rotating or librating in the rotating reference frame) and the transition between both types are discussed. From the limiting impact parameters accurate capture cross sections may be calculated. They can be compared with the popular approximation known as ADO — (average dipole orientation) — theory, but an analysis of the latter shows that it is theoretically not justified. Instead, the adiabatic approximation for planar collisions is developed with and without angular momentum conservation. It fits numerical calculations very well as long as the transition from rotation to libration lies inside the centrifugal wall, i.e. the separating orbit is a rotation. In the opposite case only qualitative agreement is obtained. Finally, a simple approximation to get 3D capture cross sections from planar calculations is discussed.

Angular momentum dependence of unimolecular decay in a triatomic system

Classical trajectory calculations were performed to determine the lifetime τ of long-lived collision complexes in H+-D2 or D+-HD collisions. For the same complex characterized by total energy and total angular momentum, the lifetimes are independent of (a) the kind of energy supply (i.e. translational or vibrational), and (b) the distribution of the three masses into projectile and target. It was found that τ(J) increases monotonously with J, first linearly, then faster, finally proportional to (J max - J)-1. The shape of τ(J) can be understood in terms of statistical theory if the overall rotation is included in the assumed quasi-equilibrium.

Collinear collisions in a well potential

Abstract Collinear collisions on the H3+ potential energy surface for different mass combinations m, M, m are investigated by means of classical trajectory computations. Both, the probability for reaction, Pt, and for complex formation, Pe, depend periodically on the mass parameter (1 + 2m/M) 1 2 . This can easily be understood in terms of vibrational excitation of the symmetric and asymmetric modes of the reaction complex. If the collinear constraint is lifted, none of these features survive.

How much can we learn from nearest neighbor distributions

Nearest neighbor distributions of molecular spectra can, in principle, be used to learn from quantum spectra about the classical dynamics of a system, i.e., whether it is regular or irregular (chaotic). However, the predictive power of this method is limited due to the generally small number of spectral lines available for analysis, and the ambiguities of the procedures used. This is demonstrated here for the determination of the shape of nearest neighbor distributions in terms of a Brody parameter, which was determined from fits to samples from a Brody distribution and fits to simulated molecular spectra. The procedures are also applied to computed spectra of NO2 and SO2.