Semiclassical Transition Probabilities Research Articles

Charge transfer betweenS2+and He: A comparative study of quantal and semiclassical approaches

A comparative study of charge transfer in collisions of ground-state ${\mathrm{S}}^{2+}$ ions with He has been performed within fully quantal and semiclassical molecular-orbit close-coupling approaches. The processes for capture into ${\mathrm{S}}^{+}(^{4}S^{0},^{2}D^{0},^{2}P^{0})+{\mathrm{He}}^{+}$ are taken into account. Quantal and semiclassical cross sections were evaluated, respectively, in the diabatic and adiabatic representations and found to be in good agreement. The calculations show that at collision energies below about $40\phantom{\rule{0.3em}{0ex}}\mathrm{eV}∕\mathrm{u}$, the charge-transfer processes are dominated by ${\mathrm{S}}^{2+}(^{3}P)+\mathrm{He}\ensuremath{\rightarrow}{\mathrm{S}}^{+}(^{2}D^{0})+{\mathrm{He}}^{+}$, and capture into the $^{2}P^{0}$ and $^{4}S^{0}$ states becomes comparable with that into the $^{2}D^{0}$ state above $40\phantom{\rule{0.3em}{0ex}}\mathrm{eV}∕\mathrm{u}$ and $600\phantom{\rule{0.3em}{0ex}}\mathrm{eV}∕\mathrm{u}$, respectively. The multireference single- and double-excitation configuration-interaction method was utilized to obtain adiabatic potentials and nonadiabatic coupling matrix elements. A detailed comparison of quantal and semiclassical transition probabilities is discussed. State-selective and total rate coefficients are presented with temperatures between $10\phantom{\rule{0.2em}{0ex}}000\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and $5.0\ifmmode\times\else\texttimes\fi{}{10}^{6}\phantom{\rule{0.3em}{0ex}}\mathrm{K}$.

Numerical study of the accuracy and efficiency of various approaches for Monte Carlo surface hopping calculations

A one-dimensional, two-state model problem with two well-separated avoided crossing points is employed to test the efficiency and accuracy of a semiclassical surface hopping technique. The use of a one-dimensional model allows for the accurate numerical evaluation of both fully quantum-mechanical and semiclassical transition probabilities. The calculations demonstrate that the surface hopping procedure employed accounts for the interference between different hopping trajectories very well and provides highly accurate transition probabilities. It is, in general, not computationally feasible to completely sum over all hopping trajectories in the semiclassical calculations for multidimensional problems. In this case, a Monte Carlo procedure for selecting important trajectories can be employed. However, the cancellation due to the different phases associated with different trajectories limits the accuracy and efficiency of the Monte Carlo procedure. Various approaches for improving the accuracy and efficiency of Monte Carlo surface hopping procedures are investigated. These methods are found to significantly reduce the statistical sampling errors in the calculations, thereby increasing the accuracy of the transition probabilities obtained with a fixed number of trajectories sampled.

Quantal and semiclassical calculations of charge transfer cross sections in + H collisions for impact energies of

Total and n-partial cross sections for charge transfer in collisions are calculated for collision energies between and . A molecular expansion is employed, and semiclassical and quantal calculations are carried out including a common translation factor and a common reaction coordinate, respectively. The comparison between quantal and semiclassical cross sections and transition probabilities is discussed.

Dynamic plasma screening effects on electron–ion collisional excitations in a generalized Lorenzian (kappa) distribution plasma

Dynamic plasma screening effects are investigated on 1s→2p dipole transition probabilities for electron-impact excitation of hydrogenic ions in a generalized Lorenzian (kappa) velocity distribution plasmas. The electron–ion interaction potential is obtained by introduction of the plasma dielectric function. A semiclassical straight-line trajectory method is applied to the path of the projectile electron in order to visualize the dynamic plasma screening effects on the semiclassical transition probabilities as a function of the impact parameter and the collision energy. The transition probability including the dynamic plasma screening effect is found to be always greater than that including the static plasma screening effects. It is also found that the dynamic plasma screening effects on the electron–ion collisional excitations in the Lorenzian velocity distribution plasmas are more effective than those in the Maxwellian velocity distribution plasmas.

Dynamic plasma screening effects on semiclassical inelastic electron–ion collisions in dense plasmas

In dense plasmas, dynamic plasma screening effects are investigated on 1s→2p dipole transition probabilities for electron-impact excitation of hydrogenic ions. The electron–ion interaction potential is considered by introduction of the plasma dielectric function. A semiclassical straight-line trajectory method is applied to the path of the projectile electron in order to visualize the semiclassical transition probability as a function of the impact parameter, projectile energy, and plasma parameters. The transition probability including the dynamic plasma screening effect is always greater than that including the static plasma screening effect. When the projectile velocity is smaller than the electron thermal velocity, the dynamic polarization screening effect becomes the static plasma screening effect. When the projectile velocity is greater than the electron thermal velocity, then the interaction potential is almost unshielded. The difference between the dynamic and static plasma screening effects is more significant for low-energy projectiles. It is also found that the static plasma screening formula obtained by the Debye–Hückel model overestimates the plasma screening effects on the atomic excitation processes in dense plasmas.

Semiclassical transition probabilities for interacting oscillators

Semiclassical transition probabilities characterize transfer of energy between “hard” and “soft” modes in various physical systems. We establish the boundary problem for singular euclidean solutions used to calculate such probabilities. Solutions are found numerically for a system of two interacting quartic oscillators. In the double-well case, we find numerical evidence that certain regular minkowskian trajectories have approximate stopping points or, equivalently, are approximately periodic. This property leads to estimates of tunneling excitation probabilities in that system and suggests that similar estimates may be possible in other systems with tunneling.

Semiclassical transition probabilities for the electron-impact excitation of hydrogenic ions in dense plasmas

Plasma-screening effects are investigated on semiclassical transition probabilities for electron-impact excitation of hydrogenic ions in dense plasmas. The electron–ion interaction potential is obtained by a nonspherical Debye–Hückel model of the screened Coulomb interaction for the interesting domain of the Debye length Λ/aZ≥10. Semiclassical straight-line trajectory method is used to describe the behavior of the projectile electron. The screening effect is increased with increasing the projectile energy. The plasma-screening effect on the optically allowed (dipole) transition (1s→2p) is found to be more effective than that on the optically forbidden transition (1s→2s). For 1s→2s transition, the maximum position of the transition probability is almost unchanged with increasing or decreasing the projectile energy and plasma-screening effect. However, for 1s→2p transition, the maximum position of the transition probability has receded from the nucleus as the projectile energy increases.

Instanton-like transitions at high energies in (1+1)-dimensional scalar models

Instanton-like transitions (“shadow processes”) are considered in (1+1)-dimensional models with one scalar field whose potential is a quadratic well with a cliff. The corresponding classical boundary value problem is solved, and the semiclassical transition probabilities are found in a rather wide range of energies and the numbers of initial particles.

Analytical treatment of the quadratic and nonrelativistic Renner-Teller effect for a triatomic molecule in a Π electronic state

This work completes our analytical treatment of the quadratic and non-relativistic Renner-Teller effect and deals with a situation (intermediate limit case, ILC) placed between the recently studied first and second limit cases (FLC, high energy limit, and SLC, high energy and high angular momentum limit, respectively). In the ILC the second order ordinary differential equation in the complex plane that results from a recently reported matrix-integral transformation is also of singular perturbation type, as also occurred in the FLC and SLC, but the method of matched asymptotic expansions cannot be applied in the same extent. However, thanks to the calculation of the non-adiabatic transition probability based on results of an earlier semiclassical study, it has been possible to achieve a fairly deep insight into this problem. To the best of our knowledge this is the first time that the semiclassical non-adiabatic transition probability has been determined, the results being valid for a wide interval of ener...

The Einstein equation for fluctuation probabilities of a thermodynamical system and the energy symmetrization in inelastic molecular-collision processes

Using an analogy with statistical thermodynamics and the Einstein equation for fluctuation probabilities, «fluctuation» model for the inelastic collisional interaction governing molecular energy transfers is introduced. «Symmetrized» expressions respecting the detailed balance principle are obtained for the classical and semi-classical transition probabilities. These expressions are shown to be equal—within standard approximations (saddle point for perturbation integrals)—to the corresponding quantum-mechanical WKB equations. Although here introduced for the case of vibration-translation molecular energy transfers, the model equations reveal some generality and can be applied to a number of inelastic processes in physics. A few possible applications and developments are discussed.

Application of the strong‐coupling correspondence principle to atom–triatom collinear collisions

AbstractThe strong‐coupling correspondence principle has been used to calculate the T–V transition probabilities in the collinear collision of He and Kr with CO2. For a harmonic CO2 potential the results for He agree well with published quantum mechanical probabilities. For Kr the agreement is less satisfactory but at worst the ratio of quantum to semiclassical transition probability is approximately 0.2. Introduction of anharmonicity in the CO2 potential was found to increase the semiclassical transition probabilities but this may just be due to the lowering of the vibration frequencies.

Simple semiclassical calculation of fine-structure transition probability

The construction of a semiclassical multichannel scattering matrix in the matching approximation is discussed. The accuracy of the matching approximation is demonstrated by comparison of the calculated semiclassical fine-structure transition probability P12 to 3/2/(b) with the rigorous quantum calculation of Reid and Rankin (see ibid., vol.11, p.55 (1978)) for the Li(2Pj)-He system.

A test of the quantum mechanical sudden vibrational approximation for collinear inelastic collisions

Numerical results for collinear vibrationally inelastic transition probabilities based on a quantum mechanical sudden vibrational approximation are presented. The results are compared with those obtained by replacing the vibrational wave functions by a semiclassical formula. For the exponentially repulsive potential employed, the quantal and semiclassical transition probabilities are given analytically. To test both theories transition probabilities are compared with exact results for three collision systems with different mass ratios as a function of energy. As the relative mass of the atom decreases the quantal approximation becomes quantitatively accurate whereas the semiclassical does not, independent of the energy.

A model for ground-state and excited-state microwave optical double resonance

A model is developed to describe the physical principles of both ground-state and excited-state microwave optical double resonance. This model uses a steady-state kinetic analysis to determine the populations of three quantum levels in the presence of two monochromatic radiation fields. The microwave transition rate and the laser transition rate are obtained from separate analyses using the semiclassical two-state transition probability averaged over the effects of collisions. The Doppler effect of the optical transition rate is explicitly included. The competing effects of laser power, microwave power, and pressure on signal intensity and lineshape are described. Calculated lineshapes and relative signal intensities based on this model are in good agreement with previous measurements on BaO and NO 2.

Semiclassical theory of molecular collisions: collinear atom harmonic oscillator collisions with attractive interactions

Semiclassical transition probabilities have been calculated for the collinear collision of an atom with a harmonic oscillator interacting via a Lennard-Jones (12, 6) or Morse interaction. Calculations have been made for both classically allowed and classically forbidden transitions. The uniform Airy and Bessel semiclassical approximations agree well with exact quantum results, although care is necessary for elastic transitions. The Airy approximation breaks down for near elastic collisions whilst the Bessel approximation breaks down for low quantum numbers at high collision energies (especially for 0 →0 transitions). The primitive semiclassical approximation is generally less accurate than either the Airy or Bessel approximations.

Is semi-classical scattering theory accurate for transitions from low-lying vibrational states?

Various semi-classical approximations are compared with exact transition probabilities from the lowest vibrational state for the forced harmonic oscillator. The primitive semi-classical transition probability is shown to be seriously inaccurate except, paradoxically, for the most strongly forbidden classical transitions. Bessel and Airy uniformizations lead to substantial and equivalent improvements for the 0 →n transitions. For the 0 →0 transition, the Bessel uniformization is correct in near elastic situations but the Airy uniformization is required for cases of strong interaction.

Comparison of semiclassical, quasiclassical, and exact quantum transition probabilities for the collinear H + H2 exchange reaction

Using the classical (CSC), primitive (PSC), and uniform (USC) semiclassical expressions for transition probabilities given by Miller and co-workers, we have calculated the reactive and nonreactive 0 → 0 and 0 → 1 transition probabilities for the collinear H + H2 exchange reaction. Comparison with previously calculated exact quantum and quasiclassical results for the reactive and nonreactive 0 → 0 transitions reveals that the semiclassical approximations are not very good, especially the CSC and PSC ones. All three semiclassical probabilities for the reactive 0 → 0 transition exceed unity in the collision energy range from 0.0 to 0.2 eV above the quasiclassical reaction threshold. This feature coupled with the failure of any of the semiclassical approximations to produce the marked quantum effects present in this transition causes these results to be less accurate than the corresponding quasiclassical ones. For the reactive and nonreactive 0 → 1 transitions the USC results are in qualitative agreement with the exact quantum ones and are better than the standard quasiclassical results. However, the reverse quasiclassical results are almost as good as the USC ones for these transitions. A probable reason for the inability of the USC expression to produce the strong oscillations observed in the exact quantum results is that the latter are due to interference between direct and resonant (i.e., compound state) processes whereas the present formulation of the semiclassical method does not encompass such phenomena. A comparison of the total reaction probabilities obtained by the USC and quasiclassical methods with the exact quantum one indicates that the USC result is more accurate than the quasiclassical one, except at collision energies less than 0.50 eV. This improved accuracy is due to a partial cancellation of errors in the contributing 0 → 0 and 0 → 1 USC reactive transition probabilities.

Semiclassical transition probabilities by an asymptotic evaluation of the S matrix for elastic and inelastic collisions. Bessel uniform approximation

It has been observed in the past that the usual Airy uniform approximation gives probabilities greater than one, especially for near elastic collisions. By mapping the phase onto −ζ cos y + ky + A rather than (1/3)y3 − ζy + A one obtains a uniform approximation involving Bessel functions of the first kind, which approaches unity for the elastic collision. This Bessel uniform approximation is no more complicated than the Airy and also gives good agreement with exact quantum results, even if probabilities are large.

Comparison of semi-classical, exact quantum, and quasi-classical reactive transition probabilities for the collinear H + H 2 reaction

The classical (CSC), primitive (PSC), and uniform (USC) semi-classical expressions for reactive transition probabilities of Miller and co-workers have been used to obtain reaction probabilities for the collinear H + H 2 exchange reaction. Comparisons with exact quantum results for the 0 → 0 transition reveal that these semi-classical approximations are not very good, especially the CSC and PSC ones. The USC results are better, but the quasi-classical trajectory results give in average a more accurate description. For this transition, the extra effort required in going from the quasi-classical to the semi-classical probabilities did not produce an improvement in the results. For the 0 → 1 reactive transition the USC calculations are in fairly good qualitative agreement with the exact quantum results except for an upward shift of 0.08 eV in the effective threshold energy. They are somewhat better than the microscopically reversed quasi-classical ones.

Classical S-Matrix for Vibrational Excitation of H2 by Collision with He in Three Dimensions

Complex-valued classical trajectories have been computed by direct numerical integration of the equations of motion for three-dimensional collisions of He and H2, and from such trajectories classical S-matrix elements for transitions between specific rotational-vibrational states of H2 have been constructed. At the collision energies employed (∼1–2 eV) all vibrationally inelastic transitions are classically forbidden, thus the need for analytically continued, complex-valued trajectories. Comparison with the quantum mechanical calculations of Eastes and Secrest shows excellent agreement between the quantum and uniform semiclassical transition probabilities. Since, however, for three-dimensional collision systems one is seldom concerned with individual S-matrix elements, but rather sums over many of them, an important practical feature is developed which shows how one can combine the usual Monte Carlo classical treatment of some of the internal degrees of freedom with a semiclassical state-by-state description of others; i.e., one can ``quantize'' only those degrees of freedom that are highly quantumlike (e.g., vibration). This ``partial averaging'' approach also greatly simplifies the practical aspects of applying classical S-matrix theory to systems with several internal degrees of freedom.