# Additional WKB Inversion Relations for Bound‐State and Scattering Problems

15 May 1971-Journal of Chemical Physics (American Institute of PhysicsAIP)-Vol. 54, Iss: 10, pp 4174-4177

TL;DR: In this paper, new inversion relations for the diatom system (elastic scattering of two atoms by a single potential, or a bound diatomic molecule) are presented, which are exact within the WKB approximation for the relevant quantities and are closely related to the well-known RKR expressions.

Abstract: New inversion relations for the diatom system (elastic scattering of two atoms by a single potential, or a bound diatomic molecule) are presented. They are exact within the WKB approximation for the relevant quantities and are closely related to the well‐known RKR expressions. Explicit formulas are given for bound‐state and scattering input.

## Summary (1 min read)

### II. Inversion from Eigenvalues

- The principal disadvantage o~ Equation (9) is that one cannot solve these equations to obtain e)..'":plicit e>.:pressions for r< and r> (due to the logarithms).
- For bound-state problems there is no apparent reason >vhy any pair of equa.tions should be easier to use than the usual RKR pair [Equations (10) and (11) ].
- Equations (12) and (13) do give alternate inversionrelations, however, v.rhich may be of use in some situations.

### b. Curve Crossing Probability

- In the semiclassical treatment 1 7 of' electronic transitions in atom-atom collisions which take place via a crossing of the two potentj.al curves, the transition probability is the product of a slm-rly varying factor and an oscillatory factor.
- The phase of the oscillatory fae;tor is a phase integral; more specifically, it is the difference in phase integrals on the initial and final potential curves from their respective classical turning points to the crossing point.
- If the E and £ dependence of this phase integral can be extracted from the differential and/or total cross section for the electronic transition, then the inversion formulae of Section II can be utilized to construct the two potentials involved.

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TL;DR: In this paper, the authors reviewed the practical applicability of such procedures in molecular physics rather than on the question of existence and uniqueness, and the procedures which had been applied to the determination of spherically symmetric, interatomic potentials by the inversion of actual scattering data are critically surveyed and illustrated by approprite examples.

Abstract: Solutions of the inverse problem of scattering are reviewed. Quantum mechanical, semiclassical, and classical methods in the high-energy limit are discussed for both the step from the cross section to the phase shifts or the deflection function and the step from these functions to the potential. The emphasis is on the practical applicability of such procedures in molecular physics rather than on the question of existence and uniqueness. The procedures which had been applied to the determination of spherically symmetric, interatomic potentials by the inversion of actual scattering data are critically surveyed and illustrated by approprite examples.

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TL;DR: In this article, a physically realistic two-parameter potential model is fitted to the inversion result, giving in a unique way the entire helium pair potential function, which supports a bound state for 4He2 very near to the dissociation limit.

Abstract: Measured backward glory oscillations of integral 4He2 and 3He2 scattering cross sections are evaluated by use of an improved semiclassical backward glory formula yielding the energy dependence of the s phases, which allows the calculation of the He2 potential in the region 1.83–2.12 A via Miller’s semiclassical inversion method. A physically realistic two‐parameter potential model, which uses all ab initio data available with sufficient accuracy, is fitted to the inversion result, giving in a unique way the entire helium pair potential function. A well with a depth of 10.74 K at 2.975 A is obtained, which supports a bound state for 4He2 very near to the dissociation limit. Via construction, the potential model also gives results for individual interaction energy terms in the symmetry adapted perturbation scheme for the He2 interaction. Calculations with the determined potential reproduce the various experimental data available for helium. The best ab initio He2 potentials available today converge towards that potential.

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TL;DR: In this article, the concept of mass-reduced quantum numbers is introduced and discussed for a diatomic molecule, and the isotopically combined Rydberg-Klein-Rees (RKR) method is applied to the isotopic mercury hydrides.

Abstract: The concept of mass‐reduced quantum numbers is introduced and discussed. For a diatomic molecule, the mass‐reduced vibrational quantum number η= (v+1/2)/(μ)1/2 and mass‐reduced rotational quantum number ξ=J (J+1)/μ are used and exemplified. Isotopically combined methods of potential determination from spectroscopic data follow from introduction of these quantum numbers, e.g., the isotopically combined Rydberg–Klein–Rees (RKR) method. These concepts are applied in detail to the isotopic mercury hydrides. It is shown that mass‐reduced functions exist which accurately describe, for example, vibrational spacings [ΔG (η)(=dG/dη) = μ1/2ΔG (v+1/2)], rotational constants [B (η) =dE (η,ξ)/ dξ=μB (v)], and centrifugal distortion constants [D (η) =μ2D (v)]. The vibrational and rotational mass‐reduced functions are used to construct an isotopically combined RKR potential. This potential, when joined to the long‐range potential and extrapolated at short range reproduces the energy levels fairly well. In a companion pa...

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Abstract: It is shown how established local solutions of the barrier penetration and curve-crossing problems may be linked together to give simple uniform analytical descriptions of a wide variety of spectroscopic phenomena. Applications to inversion doubling, restricted rotation, predissociation by rotation and by curve-crossing, continuum interactions and molecular perturbations are described. In each case the necessary computations may be reduced to one-dimensional quadratures.

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TL;DR: A review of recent applications of various theories of nonreactive collisions between diatomic molecules, from semiclassical to quantal, are surveyed in light of experimental data as discussed by the authors, providing the theoretician with a background of those features of the collision problems that require more accurate treatments, and to provide the experimentalist with a spectrum of models available for handling his data.

Abstract: The rapid progress in research on chemical lasers has led to increased interest in the development of theoretical models of molecular relaxation processes. In this review, recent applications of various theories of nonreactive collisions between diatomic molecules, from semiclassical to quantal, are surveyed in light of experimental data. The intention is to provide the theoretician with a back-ground of those features of the collision problems that require more accurate treatments, and to provide the experimentalist with a spectrum of models available for handling his data.

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##### References

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01 Jan 1966

TL;DR: In this paper, the authors present a survey of various approaches to the solution of three-particle problems, as well as a discussion of the Efimov effect, and the general approach to multiparticle reaction theory.

Abstract: Much progress has been made in scattering theory since the publication of the first edition of this book fifteen years ago, and it is time to update it. Needless to say, it was impossible to incorporate all areas of new develop- ment. Since among the newer books on scattering theory there are three excellent volumes that treat the subject from a much more abstract mathe- matical point of view (Lax and Phillips on electromagnetic scattering, Amrein, Jauch and Sinha, and Reed and Simon on quantum scattering), I have refrained from adding material concerning the abundant new mathe- matical results on time-dependent formulations of scattering theory. The only exception is Dollard's beautiful "scattering into cones" method that connects the physically intuitive and mathematically clean wave-packet description to experimentally accessible scattering rates in a much more satisfactory manner than the older procedure. Areas that have been substantially augmented are the analysis of the three-dimensional Schrodinger equation for non central potentials (in Chapter 10), the general approach to multiparticle reaction theory (in Chapter 16), the specific treatment of three-particle scattering (in Chapter 17), and inverse scattering (in Chapter 20). The additions to Chapter 16 include an introduction to the two-Hilbert space approach, as well as a derivation of general scattering-rate formulas. Chapter 17 now contains a survey of various approaches to the solution of three-particle problems, as well as a discussion of the Efimov effect.

4,044 citations

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01 Jan 1949

TL;DR: The perturbation theory has been applied to many-body problems and applications, such as electron collisions with atoms, collisions between atomic systems, nuclear collisions, and certain aspects of two-body systems under relativistic collisions.

Abstract: Volume II of this work covers many-body problems and applications of the theory to electron collisions with atoms, collisions between atomic systems, nuclear collisions, certain aspects of two-body systems under relativistic collisions, and the use of time-dependent perturbation theory. Despite the amount of work carried out since this book was first published, the underlying theory presented here remains both sound and of practical value to all theoretical physicists.

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TL;DR: In this article, a simple, physically intuitive, expression has been obtained for the scattering phase shift δ(l, E) in the case that the effective radial potential possesses a maximum.

Abstract: A simple, physically intuitive, expression has been obtained for the scattering phase shift δ (l, E) in the case that the effective radial potential possesses a maximum. The result [Eq. (3)] is seen to modify the usual WKB phase shift [Eq. (5)] by replacing a step function by a particular “smooth step function.” On the basis of this result, resonances in the energy dependence of the total elastic cross section are discussed. From qualitative arguments it is seen that metastable states more than ∼0.6 ℏωB energy units above the top of the barrier are too short lived to have physical significance (ωB is the harmonic frequency related to the inverted barrier). Application is also made to the eigenvalues of a hindered rotor.

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TL;DR: In this article, the uniform semiclassical wave function of Langer is used to obtain a modified quantum condition, which has the integral quantum numbers replaced by known, nonintegral constants and gives considerably better results for the quartic oscillator.

Abstract: The uniform semiclassical wavefunction of Langer is used to obtain a modified quantum condition. The result [contained in Eqs. (1) and (2)] has the integral quantum numbers replaced by known, nonintegral constants and gives considerably better results for the example of the quartic oscillator. An approximation for the Franck‐Condon factor is obtained which is qualitatively correct for all values of the parameters involved. The result, contained in Eqs. (4) and (5), reduces to the known semiclassical results in the appropriate regions and is uniformly valid in the transition regions. The method used by Carrier for evaluating integrals by the method of stationary phase in the case of more than one point of stationary phase is used to extend the semiclassical scattering amplitude of Ford and Wheeler to include the glory and rainbow regions uniformly. The result is contained in Eq. (6) of the text.

120 citations