Close coupling test of classical and semiclassical cross sections for rotationally inelastic Ar–N2 collisions

Rotational energy transfer in collisions of nitrogen molecules has been studied theoretically, using the N2–N2 rigid-rotor potential of van der Avoird et al. [J. Chem. Phys. 84, 1629 (1986)]. For benchmarking purposes, converged close coupling (CC) calculations have been carried out to a total energy of about 200 cm−1. Coupled states (CS) approximation calculations have been carried out to a total energy of 680 cm−1, and infinite order sudden (IOS) approximation calculations have also been carried out. The CC and CS cross sections have been obtained both with and without identical molecule exchange symmetry, whereas exchange was neglected in the IOS calculations. The CS results track the CC cross sections rather well: between 113–219 cm−1 the average deviation is 14%, with accuracy improving at higher energy. Comparison between the CS and IOS cross sections at the high energy end of the CS calculations, 500–680 cm−1, shows that IOS is sensitive to the amount of inelasticity and the results for large ΔJ transitions are subject to larger errors. State-to-state cross sections with even and odd exchange symmetry agree to better than 2% and are well represented as a sum of direct and exchange cross sections for distinguishable molecules, an indication of the applicability of a classical treatment for this system. This result, however, does not apply to partial cross sections for given total J, but arises from a near cancellation of the interference terms between even and odd exchange symmetries on summing over partial waves. In order to compare with experimental data for rotational excitation rates of N2 in the n=1 excited vibrational level colliding with ground vibrational level (n=0) bath N2 molecules, it is assumed that exchange scattering between molecules in different vibrational levels is negligible and direct scattering is independent of n so that distinguishable molecule rigid rotor rates may be used. With these assumptions good agreement is obtained. Although the IOS approximation itself is found to provide only moderately accurate values for rate constants, IOS/ECS scaling methods, especially if based on fundamental rates obtained from coupled channel results, are found to provide generally good accuracy.

A simple, direct derivation of the rotational infinite order sudden (IOS) approximation in molecular scattering theory is given. Connections between simple scattering amplitude formulas, choice of average partial wave parameter, and magnetic transitions are reviewed. Simple procedures for calculating cross sections for specific transitions are discussed and many older model formulas are given clear derivations. Total (summed over rotation) differential, integral, and transport cross sections, useful in the analysis of many experiments involving nonspherical molecules, are shown to be exceedingly simple: They are just averages over the potential angle of cross sections calculated using simple structureless spherical particle formulas and programs. In the case of vibrationally inelastic scattering, the IOSA, without further approximation, provides a well-defined way to get fully three dimensional cross sections from calculations no more difficult than collinear calculations. Integral, differential, viscosity, and diffusion cross sections for He–CO2 obtained from the IOSA and a realistic intermolecular potential are calculated as an example and compared with experiment. Agreement is good for the complete potential but poor when only its spherical part is used, so that one should never attempt to treat this system with a spherical model. The simplicity and accuracy of the IOSA make it a viable method for routine analysis of experiments involving collisions of nonspherical molecules.

Vibrationally and rotationally elastic and inelastic collisions between He and I/sub 2/ in the total energy range of 0.044-0.5 eV are treated under the vibrational/rotational infinite-order sudden approximation (VRIOSA). In this method, one simultaneously makes the vibrational energy sudden, rotational energy sudden, and centrifugal sudden approximations and generates a completely uncoupled set of equations. The solution of these equations is discussed and the integral and differential cross sections, the integral and differential cross sections summed over final rotational states, the integral cross sections summed over final rotational and vibrational states, and a type of cross section which recently has been measured experimentally are presented and compared with vibrational close-coupled infinite-order sudden approximation results. The dependence of the accuracy of the VRIOSA upon the choice of the arbitrary vibrational quantum number parameters, n vector, is also investigated. Setting n vector equal to the final vibrational quantum number gives good results for the integral and experimental cross sections, while picking n vector equal to the average of the initial and final vibrational quantum numbers gives good results for the differential cross sections. The present comparisons involving a full three-dimensional system and quantities, such as the differential cross section, which are sensitive tomore » the phase of the S matrix extend earlier, more limited studies and show that the VRIOSA should be considered convenient and reliable for light atom-heavy diatomic molecule collisions at total energies up to a few electronvolts.« less

The distorted wave infinite order sudden (DWIOS) approximation for the calculation of vibrationally inelastic molecular collision cross sections

Recent work is reviewed on the theory of angular momentum decoupling approximations in inelastic molecular scattering theory. It is shown that the failure of these approximations is more extensive than previously believed, particularly for completely state-selected m transitions. A systematic study of a number of centrifugal sudden approximations is made and discussed. A new method is presented for improving these approximations, which recouples the approximate solutions. Completely state-selected differential and integral cross sections are computed under the centrifugal decoupling approximation and its correction for Ne+HD scattering at a total energy of 31.5 meV. Five choices of the partial wave decoupling parameter (including the three known ones) are studied. The numerical results show that with the partial wave parameter chosen to be the arithmetic mean of the initial and final l-quantum numbers, the coupled states approximation gives excellent results for orientation averaged transitions, but fails dramatically for state-selected m transitions. The corrected centrifugal decoupling approximation gives significantly more accurate inelastic differential and integral state-selected cross sections than its uncorrected counterpart. The extension of this correction technique to the energy sudden and the infinite order sudden approximations is discussed.

Test of the infinite-order sudden approximation for electron scattering at intermediate energy

A classical reactive study within the infinite order sudden approximation: integral cross sections for the reactions F + H 2vi = 0) → HF( vf = 0,1,2,3) + H

Classical trajectory calculations of effective cross sections have been carried out for the N2-Ne system using the recent HFD3(N2-Ne) potential surface of McCourt et al. Forty-five such cross sections governing a wide variety of transport and relaxation phenomena in binary gaseous mixtures have been calculated at a number of temperatures ranging from 77K to 1100K. Wherever possible, we have made comparisons with much cheaper infinite-order sudden approximation (IOSA) calculations in order to assess their reliability. For cross sections relating to diffusion and viscosity, IOSA errors are less than 5·4 per cent, but, for cross sections which vanish for isotropic interactions, the IOSA is much less reliable, with possible errors of up to a factor of two. At the same time, we have compared our classical trajectory results for this surface with available experimental results. We find that the HFD3(N2-Ne) surface is too anisotropic, by as much as 20 per cent.

Classical trajectory calculations were performed using the Ar-N2 potential surface of Patterngill et al. (1979) for temperatures of 77.3 and 300K. Diffusion and viscosity cross sections, the production and relaxation cross sections governing the viscomagnetic effect and cross sections for both depolarised Rayleigh scattering and the rotational relaxation time were calculated. Comparison was made with classical energy-sudden (ES), classical infinite-order sudden (MM) and quantal infinite-order sudden (IOS) calculations. For both the viscosity and diffusion cross sections the CT and the ES, MM and IOS calculations agree within 3%, except for the IOS results at 77.3K. For the relaxation and DPR cross sections the IOS results are found to differ by about 40-80% from the CT results. The ES result for the DPR cross section is about 20% larger than the CT result, while for the production and relaxation cross sections it is very poor. The authors suggest that the main reason for these discrepancies is the breakdown of the sudden approximation for the non-sudden collisions always arising in thermal averages.

A simple procedure is developed to correct the infinite-order sudden (IOS) approximation for the finite energy separation between states. A phase-shift matrix 〈i‖Δη‖j〉 is computed using the IOS and multiplied by an energy correction factor Bij and then exponentiated to give the S matrix. The correction factor is obtained by calculating an approximate phase-shift matrix for a finite energy separation and dividing this by the approximate form with no energy separation. A simple procedure is given for choosing the energy at which the IOS phase shifts are calculated. Results for He+H2 and Ar+N2 show good agreement with accurate calculations and a substantial improvement over the IOS.

Accurate close coupling scattering calculations are presented for thermal energy HCl–He collisions. The interaction potential is obtained from the Gordon–Kim electron gas model, adjusted to have the correct long-range multipole form. A variety of phenomenological cross sections are computed from the close coupling S matrix, and these are compared with results from several commonly employed approximate methods. In particular, it is found that the total integral, total differential, and gas kinetic cross sections are accurately predicted by the central field approximation which retains just the spherical average of the interaction. Integral inelastic cross sections are represented quite accurately by the coupled states approximation of McGuire and Kouri, but only qualitatively by the effective potential method of Rabitz. Pressure broadening cross sections from the close coupling calculation are in much better agreement with experiment than either Anderson theory calculations or the classical trajectory study of Gordon. NMR spin–lattice relaxation cross sections are also presented.

We have recorded time-domain coherence decay and population decay signals for the pure rotational OCS J=0–1 transition broadened by argon and by helium using pulse methods based on Stark switching of the molecular resonance frequency. The signals have been analyzed to yield the pressure dependence of the two decay rates (1/T2 and 1/T1) and their dependence on the absorber (OCS) speed. For comparison, using a potential which has been used several times in the literature for these systems, IOS (infinite order sudden approximation) scattering results are given. These calculations give substantial agreement on the ratio of the speed dependence of the relaxation rate to the rate itself, but fail to predict the rates quantitatively, especially for helium, or the relationship between the argon and the helium results. Because the IOS is expected to be accurate for these systems, the helium potential is ruled out by these data.

A new approximate method is presented for the rapid calculation of vibrationally–rotationally inelastic molecular collision cross sections. The method treats the coupling between the rotational states, belonging to the same vibrational level, using the sudden approximation. The vibrational states are treated using an adiabatic basis. With this basis it is found that a first order distorted wave approximation gives good agreement with fully converged close coupling calculations. Some previously developed techniques are extended to permit reliable analytic expressions to be formulated for the angle fixed S matrix elements which appear in the theory. The great computational efficiency of the method is based on these analytic formulas which eliminate the need for the numerical solution of differential equations and for the numerical evaluation of difficult integrals. The angle fixed S matrices which result are then treated within the framework of the recently developed sudden approximation to yield inelastic cross sections. The method is applied to the calculation of vibrationally–rotationally inelastic cross sections for the He+H2 system and comparisons are made with close coupling calculations which are fully converged in both rotational and vibrational basis. Analysis of the results shows that for the He+H2 system the sudden approximation fails significantly within the n=1 vibrational manifold at total energies below 1.1425 eV but the validity of the approximation clearly improves with increasing energy. The He+He2 system was used in the present work as it was the only suitable one for which exact calculations were available in the literature for comparison purposes. For heavier systems the sudden approximation will be valid at much lower energies.

A quantum mechanical study is made of the D+H2(vi=0,1)→ HD(vf=0,1,2)+H reactions within the infinite order sudden approximation (IOSA) for the total energy interval 0.28≤Et≤1.28 eV. Results at various stages of the calculation are given ranging from most detailed reactive transition probabilities through opacity functions and γ-dependent cross sections to total and state-to-state integral and differential cross sections, as well as rate constants. The cross sections and rate constants are compared with other available theoretical results and experiments. It is found that the IOSA total cross sections for vi=0,1 overlap very nicely with the corresponding quasiclassical trajectory cross sections, except for the tunneling region. A less satisfactory fit is obtained with the distorted wave born approximation results. The calculated rate constants are compared with experiment and a rather good fit is obtained, in particular for rate constants from the ground state.

Differential scattering cross sections for excitation of helium by electron impact from its ground state to its $2^{1}P$ excited state have been measured at four incident electron energies in the range 26-55.5 eV for scattering angles between 10\ifmmode^\circ\else\textdegree\fi{} and 70\ifmmode^\circ\else\textdegree\fi{}, and at 81.6 eV for scattering angles between 10\ifmmode^\circ\else\textdegree\fi{} and 80\ifmmode^\circ\else\textdegree\fi{}. These differential cross sections have been placed on an absolute scale by normalizing them to the experimental absolute integral cross sections of Jobe and St. John. These experimental differential and integral cross sections have been compared with the results predicted by the Born approximation, and by several other first-order approximations in which direct excitation is calculated in the Born approximation and exchange scattering by various Ochkurlike approximations. The calculations provide reliable tests of these scattering theories since they are made using the accurate generalized oscillator strengths of Kim and Inokuti. As expected, these first-order theories are poor near threshold and exchange is important at high scattering angles for all energies. Further, the absolute magnitude of the calculated integral and small-angle differential cross sections is too large and is within 50% of experiment only at energies greater than 80 eV. These first-order models are in qualitative agreement with the experimental angular dependence at 34-81.6 eV for scattering angles between 10\ifmmode^\circ\else\textdegree\fi{} and 40\ifmmode^\circ\else\textdegree\fi{}. At higher scattering angles (corresponding to momentum transfers greater than about 1.6 a.u.), the calculated differential cross sections fall well below the experimental ones. The phase between the direct and exchange scattering amplitudes was found to be important at all energies, and is apparently not predicted correctly by any of the first-order models examined here. Some approximations for the exchange (e.g., Ochkur approximation and the post interaction form of the Ochkur-Rudge approximation) were found to be better for integral cross sections and some (e.g., prior Ochkur-Rudge approximation) were better for differential cross sections. The use of good analytic self-consistent-field (SCF) wave functions for both the ground and excited states was tested by computing generalized oscillator strengths from them and comparing these results with the calculations using the accurate generalized oscillator strengths. The SCF functions yield differential cross sections in quantitative disagreement (20%) with the accurate results, although the energy and angle dependence of the cross sections is predicted qualitatively correctly.