Simple semiclassical optical model of harpooning elementary proceses K + Br2

The ionic rearrangement, chemi-ionization, and dissociation processes in collisions K + Br₂ which proceed via the harpooning mechanism, are calculated in a classical trajectory approach using a semiclassical optical model for treating the electronic transition. For the transition matrix element in dependence on the nuclear configuration an approximate analytic expression, based on a one-electron model is derived, the ionic potential energy surface is represented by a semiempirical model function with polarization. Total and differential cross sections are obtained in qualitative agreement with molecular beam data. The results depend critically on potential and interaction parameters, inclusion of classically forbidden transitions is important. By an analysis of the product internal-state and translational-energy distributions the mechanism of the different processes is discussed.     

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

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.     

A uniform semiclassical representation of transition probabilities derived from factorization formulas

Factorization formulas are used to derive a uniform semiclassical approximation of transition probabilities. The latter are determined in the analytical form where the basis transition probabilities are set by the analytical formula. As an example, we consider the rigid rotor, harmonic oscillator, and Morse oscillator in collisions with structureless particles.     

Classical theory of vibration energy transfer in collinear collisions

Classical theory of collisions is cast in a form which also includes the uncertainty principle. This theory is used for analyzing the vibration energy transfer in the collinear collision which approximates the He-H₂ system. The results are compared with the quantum calculations and several classical and semiclassical approaches. Very good agreement with quantum theory is found, for all the parameters investigated.     

Vibrational energy transfer according to the morse potential: Application to self-relaxation of methane

The perturbation integral in the semiclassical theory of vibrational energy transfer is derived in closed form for a Morse potential. The temperature dependence of VT tranfer in CH₄CH₄ collisions is investigated, using a recently published numerical CH₄ potential to determine the parameters of the Morse potential.     

Semi-classical calculations of rotational/vibrational transitions in He-H2

The applicability of the classical trajectory equations in three-dimensional calculations of rot/vib transitions in the He-H₂ system has been investigated. The vibrational relaxation time is calculated and the agreement with experimental data is excellent for temperatures above 250 K. The method has been used to determine the differential cross-sections for vib/rot excitation at 1.09 eV total energy and comparison is made with recent quantum mechanical effective potential calculations.     

Triatom-triatom collisions: Fixed angle close-coupling study of vibrational excitation in the 12CO2+13CO2 system

A close-coupling approach to the calculation of quantal vibrational transition probabilities for the fixed angle scattering of a linear triatomic molecule with another linear triatomic molecule is described. The method is applied to the ¹²CO₂+¹³C0₂ collisional system. For a calculated inelastic transition probability to have an appreciable magnitude, it is found that the amount of energy transferred in a transition must be very small and just one quantum of energy must be exchanged between either the symmetric stretch or the asymmetric stretch vibrational modes of ¹²C0₂ and ¹³CO₂. For collisional energies away from threshold, the probabilities for transitions involving the symmetric stretch ¹²CO₂ and ¹³CO₂ modes are insensitive to long range multipole terms in the potential energy surface, while the probabilities for energy exchange between the asymmetric stretch modes are considerably diminished when the long range terms are removed from the potential energy surface. A brief discussion is presented on the possibilities of extending the technique to the calculation of vibrational excitation cross sections for three-dimensional triato—triatom collisions.     

Semiclassical time-correlation function approach to collisional energy transfer into many-atom systems

Cross sections for energy transfer into many-body systems can be expressed in terms of time-correlation functions (TCFS ) of transition operators. A semiclassical version is presented by treating internal motions as quantized and relative motions as classical. The time evolution of internal motions can be calculated in the Heisenberg picture and avoids expansions in target states. The decoupling of fast and slow internal motions is treated and applied to vibrational–rotational decoupling in polyatomic molecules. Results are presented for Li⁺-CO₂ collisions.     

Comparison of classical mechanics and the coupled states approximation for Li+-CO scattering on an ab initio calculated CI potential energy surface

The dynamics of rotational excitation on an ab initio calculated CI rigid rotor potential energy surface for Li⁺-CO are investigated using classical mechanics and the quantum mechanical coupled-states (CS) approximation. Transition probabilities out of the j = 0 initial level are calculated for various impact parameters between b = 0 and 40a_o for 1 eV collisions. The classical results agree well with the average of Δj-even and Δj-odd quantum transition probabilities except for a few lower impact parameters where CS seems to underestimate the large Δ transitions. No propensity rule is observed for the preference of the Δj-even versus Δj-odd transitions as might have been expected.     

Anharmonicity in rotational and vibrational excitation of H2 by Li+ collisions

Integral cross sections for pure rotational and vibrational-rotational excitation of H₂(X¹Σ⁺_g) by Li⁺(¹S) impact are computed by close-coupling methods at 0.2, 0.6, and 1.2 eV in the c.m. system using vibrational functions that are numerical solutions of the one-dimensional radial Schrödinger equation for harmonic, Morse, and adiabatically corrected Kolos-Wolniewicz (KW) potential functions. Comparison of results employing KW and Morse functions shows excellent agreement for all transitions studied. Findings using harmonic oscillator functions, however, differ noticeably from KW and Morse values for vibrational (0 → 1) and very large rotational (Δj = 10) transitions, but are satisfactory for lower order (0 → 2, 4, 6, 8) rotational transitions.     

Inaccuracy of an approximate dynamical treatment of He-H2 vibrational energy transfer

The recent semi-classical approach of Shin is applied to He-H₂ vibrationally-inelastic collisions using the Gordon-Secrest potential surface. The calculated de-excitation cross sections, when compared to accurate coupled-states results are too large, particularly at low collision energies. As a result the rate constants for vibrational relaxation are artificially enhanced at low temperature, leading to a fortuitous agreement with experiment.     

Lie algebraic approach to the collinear collisions between two diatomic molecules

We present the quantum mechanical studies on the vibrational energy transfer in the inelastic collinear collision between two diatomic molecules using a dynamic Lie algebraic method of Alhassid and Levine [Phys. Rev. A 18 , 89 (1978)] within the semiclassical approximations. A dynamical algebra h₁₅ is formed and used for calculating the transition probabilities and the expectation values of the interaction potential. Under the first-order approximation of the group parameters, the selection rules for the transitions among the vibrational levels have been obtained. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 159–165, 1997     

Semiclassical theory for diatom-diatom collisions

A semiclassical approach to diatom-diatom collisions is presented. The method involves a classical treatment of the translational and rotational motion of both molecules. The vibrational degrees of freedom are quantized using a Morse oscillator approximation. The method is used to evaluate the accuracy of previous calculations based upon the energy-corrected harmonic-oscillator approach.     

Vibrational levels and Franck–Condon factors of diatomic molecules via Morse potentials in a box

This work deals with two shortcomings in the use of Morse potentials to describe energy spectra and transitions of diatomic molecules: (1) Morse's well-known “exact” solution for purely vibrational states includes the unphysical region – ∞ < r < 0 of the internuclear separation, and (2) Franck-Condon factors are evaluated in harmonic and anharmonic approximations to the Morse potentials. The method of confining the molecule in a spherical box is developed to obtain (1) purely vibrational energy spectra and eigenvectors of Morse potentials in the physical region 0 ? r < ∞, and (2) the corresponding Franck-Condon factors without any additional approximations. The method is applied to Li₂, N₂, CN, and CO molecules. © 1995 John Wiley & Sons, Inc.     

Rotational-vibrational rainbows in impulsive electron — diatomic molecule collisions: I. state-to-state transitions

Collisional induced combined rotational-vibrational excitation of diatomic molecules is discussed in a simple quantum mechanical spectator model with applications to electron-molecule collisions at intermediate collision energies (≈ 10² eV) within the rigid rotor/harmonic oscillator approximation. Quantum mechanical transition probabilities of the rotational-vibrational excitation, which show typical vibrational and rotational rainbow patterns, are calculated and compared with the structure of classical rainbow singularities.     

Calculations of vibrational energy transfer using semi-classicals matrix theory

We utilize an extension of Miller's semi-classical S matrix theory to calculate resonant and non-resonant vibrational energy transfer probabilities. The collisions under study are collinear D₂D₂ interactions at energies below and above the classical dynamic transition threshold and collinear H₂D₂ interactions above the dynamic threshold. Below threshold we employ an initial angle representation. We find that the computed probabilities are in substantial agreement with exact quantum mechanical computations and represent a major improvement over quasi-classical results. At energies above threshold we apply the first order and the classical semi-classical versions of the theory. The results indicate fair agreement with quantum mechanical calculations, but no significant improvement over quasi-classical results.     

Quantum mechanical study of electronic transitions in collinear atom—diatom collisions

Quantum mechanical calculations are reported for model, nonreactive, collinear collision systems composed of the H₂ diatom and the halogen atom X = F, Cl, Br or I. The model involves two electronic potential energy surfaces, obtained in a diatomics-in-molecules formulation, that correspond asymptotically to the two spin-orbit states of X. On each surface the calculations include as many vibrational states of H₂ as are asymptotically allowed, up to a limiting number of five. The first two collision systems, FH₂ and ClH₂, are characterized by electronic splittings much smaller than any vibrational spacing included in the diatom spectrum, and as a result they show a high degree of vibrational elasticity with essentially all transition activity testricted to spin—orbit switching in the halogen. This pattern is broken for BrH₂ collisions, where the near-equality between electronic and vibrational quanta apparently leads to a resonant exchange of energy between the two modes. The greater spinorbit splitting in iodine (～ 2 vibrational quanta) results in largely elastic behavior in IH₂ collisions for both vibrational and electronic transitions. A modified Massey criterion is exhibited for some of the FH₂ and BrH₂ transitions.     

The role of memory effects in scholastics approach to collinear diatom—diatom collisions

The role of memory effects in the stochastic approach to vibrational—vibrational transitions in collinear diatom—diatom collisions is studied. It is shown that with the help of a new recurrence relation for Clebsch—Gordan coefficients we were able to solve exactly a non-markovian master equation for a model hamiltonian. The derived solution for probabilities of V—V transitions is compared with markovian as well as exact semiclassical results over a wide range of velocities of the colliding molecules. We have found a substantial improvement of markovian results, both qualitative and quantitative, when the non-markovian effects have been included in the stochastic theory. This is in contrast with a recent study by King and Schatz who got for a restricted V—T model more accurate probabilities from the markovian approach than from the non-markovian one. The reasons for this are also discussed.