On the Bethe–Wigner–Shapiro limit of the rate coefficient for the capture of a rotating quadrupolar polarisable diatom by an ion

ABSTRACT The low-energy (Bethe–Wigner–Shapiro) behaviour of the rate coefficient for capture of a polarisable quadrupolar rotationally excited diatom by an ion is calculated. The multichannel character of capture shows itself in the dependence of the effective attractive potential on the intrinsic angular momentum of the diatom and in a nonadiabatic diagonal correction to the potential. The predicted energy dependence of the rate coefficients is compared with previous numerically accurate results for the capture of H2 molecules by H2+ ions.


Introduction
In 1935, Bethe [1] has shown that in the limit of low energies, when the scattering is dominated by s-waves, the inelastic (reactive) cross section is inversely proportional to the collision velocity v. In 1948, Wigner [2] discussed details of the threshold energy dependence of the partial inelastic amplitudes reviewed comprehensively in [3]. As indicated by Shapiro [4], a certain property of the scattering amplitude allows one to derive the first general correction, in terms of the wave vector k, to the 1/v behaviour of the reactive cross section. An elegant derivation of this result, based on the unitary property of the scattering matrix, is given in the Landau-Lifshitz textbook ( [5], Section 143). Written in terms of the energy-dependent reaction rate coefficient K(k) (the product of the velocity and the reaction cross section), the Bethe-Wigner (BW) and the Bethe-Wigner-Shapiro (BWS) results read CONTACT J. Troe shoff@gwdg.de K(k)| k→0 → K BW = 4π |a | μ , where a is the (negative) imaginary part of the scattering length and μ is the reduced mass of the collision partners. For the capture of isotropically polarisable neutrals by ions in the field of the charge-induced dipole (cid) potential the quantity |a | was calculated by Vogt and Wannier [6] (VW) as |a | = R VW = μq 2 α/ , (1.3) where q is the charge of the ion and α is the polarisability of the neutral. It turns out that in this particular case the zero-energy BW rate coefficient is twice the high-energy Langevin (L) rate coefficient K L , K(k)| k→0 = K L , with The aim of the present note is to generalise the expressions of Equation (1.1) for the capture of rotationally excited diatomic quadrupolar and polarisable molecules by ions and to compare the BWS linear k dependence of the K BWS with previously calculated accurate data. Accordingly, the plan of this note is the following. Section 2 describes the basic assumptions underlying the calculation of the scattering length. Section 3 presents a general derivation of the latter accounting for diagonal nonadiabatic corrections that arise from the dependence of the perturbed s-wave function on R. In Section 4, we compare the BWS linear rate coefficients with numerically accurate rate coefficients for the capture of H 2 (j = 0,1) by H 2 + [7,8] across the energy range where the difference between the results for j = 0 and j = 1 is clearly seen. Section 5 concludes this note.

General strategy for the calculation of |a |
In line with our previous work on the low-energy capture of molecules by ions at low energies [9], we adopt the following approximations: (1) We employ the perturbed rotor approximation. This is valid at collision energies much lower than the energy of the rotational excitation of the rotor. This approximation permits one to regard the quantum number j of the intrinsic rotation of the diatom as a good quantum number, i.e. j is assumed to be conserved during the capture event.
(2) We also employ a perturbation approach with respect to the charge-quadrupole interaction in the capture channel with asymptotic orbital quantum number = 0 of the relative rotational motion. This is valid at large distances between the partners R where the incoming wave is increasingly reflected above the drop of the attractive interaction potential. (3) We also employ an adiabatic approximation with respect to the radial relative motion of the partners in the capture channel = 0. This is valid because the nonadiabatic coupling with other channels > 0 is weak at low collision energies. However, it does not imply that the diagonal nonadiabatic correction for the channel = 0 is negligible. (4) As noted earlier [9], the effective long-range potential for the capture channel = 0 is of R −4 behaviour. We, therefore, use the results of Vogt and Wannier [6] who calculated the quantity |a | for this kind of interaction.

Bethe-Wigner and Bethe-Wigner-Shapiro limits of the rate coefficients
For the case under discussion, the Hamiltonian of the collision partners, besides μ, α, q, depends also on the quadrupole moment Q of the diatom. The interaction energy reads where ϑ is the angle between the collision and the diatom axes in the body-fixed (BF) frame. In general, the potential V (R, ϑ ) should be supplemented by an anisotropic cid interaction. We ignore it here because the anisotropic part of the cid interaction proportional to R −4 , being handled here in the second order with respect isotropic to relative motion in the field of isotropic potential (see below), results in the correction proportional to R −6 which is neglected in comparison to V cid (R) ∝ R −4 On the basis of free functions |J, j, ; (where J is the total angular momentum of the collision complex and is the set of angles describing the orientation of the collision and the rotor axis axes in a space-fixed (SF) frame), the adiabatic approximation with respect to the radial relative motion is defined through the perturbed secondorder functions |J, j, ; , R . In particular, for the perturbed state with asymptotic quantum number = 0, the perturbed function (accurate through second order in the perturbation parameter ζ j (R)) reads and The numerator in Equation (3.4) represents the offdiagonal element of the cq interaction and the denominator is the difference in the relative rotational energy for the = 0 and = 2 zero-order states. Since the former is proportional to R −3 , and the latter to R −2 , the perturbation parameter ζ j (R) ∝ 1/R. The matrix element in the numerator of the right-hand side of Equation (3.4), in which the functions J, j, ; | are defined in an SF frame while the cq interaction depends on the angle in a BF frame, can be calculated by standard formulae [10]. The effective j-dependent interaction potential V After integrating with respect to the four angular coordinates that enter into the set , we get (3.7) Taking into account that ζ j ∝ 1/R, we have (ζ j /R) 2 = (dζ j /dR) 2 , so that the last term at the right-hand side of Equation (3.7) equals −1/6 of the middle term (this is already seen from the ratio of numerical factors in front of these two terms). Moreover, the R-dependence of these terms is the same as that of the first term. This implies that the effective potential can be written as , (3.8) where β = Q 2 μ/α 2 . Comparing Equation (3.8) with Equation (1.2) leads to the following generalisation of R VW : VW | j=0 = R VW , which corresponds to the absence of the cq interaction in the spherically symmetrical state j = 0. For a rotationally excited diatom ( j ≥ 1), R ( j) VW decreases with increasing j, tending, for j 1, to the flywheel (FW) limit that corresponds to the interaction of an ion with a space-fixed (i.e. stationary) axially symmetric quadrupolar field [11]. The latter is created by a fast rotation of the diatom about the space-fixed classical vector j = J. The respective FW potential V FW cq (R, θ ) reads where θ is the angle between the vectors R and J. The correspondence between V FW cq (R, θ ) and V cq (R, ϑ ) is established in reference configuration, when the diatom classically rotates about R (i.e. ϑ = π/2) which coincides with J (i.e. θ = 0): Corresponding to the expressions in Equations (3.9), the BW and the BWS rate coefficients are

Comparison of the Bethe-Wigner-Shapiro approximation with accurate results for H 2 (j = 0,1) capture by H 2 +
In a comparison of the BWS limit with accurate numerical results, it is simpler to use the capture rate coefficients K ( j) (k) ( j = 0, 1) scaled to the Langevin rate coefficient K L , and consider the ratios χ ( j) = K ( j) /K L as functions of the scaled wave vector κ = R VW k. Then, the expressions of Equation (3.10) assume the form . (2) The larger deviation (with increasing κ) of BWS rate coefficients from their accurate counterparts for j = 1 compared to j = 0 is due to an earlier contribution of higher partial waves to the χ ( j=1) acc , a noticeable contribution from p-waves occurs beyond the upper limit of the κscale shown; this explains the drop of χ ( j=0) acc in this limit below its asymptotic Langevin value. At low temperatures, χ (4) Across the energy range of Figure 1, the SACM approximation fails badly for j = 1 (see Figure  7 from [7]) since it ignores the rotational nonadiabatic coupling. In the present treatment, it is included in the zero-order approximation.

Conclusion
The calculation of the generalised VW radii R ( j) VW shows that the rotation of the diatom has a modest effect on the BWS rate coefficients which change from R ( j=1) 1 + 2β/45 down to R ( j 1) VW = 2 √ 1 + β/36. These two values bracket the BWS coefficients in a rather narrow range. The dependence of the BWS rate coefficient on j across this range reflects the multistate character of capture; a further manifestation of the latter is the modification of the attractive capture potential that arises from the diagonal non-adiabatic correction. This s-wave limit at zero energy and the classical Langevin limit at high energies provide a useful estimation of the possible variation of the capture rate coefficients for different j across a broad range of collision energies. Finally, we note that the energy range where the BWS approximation still satisfactorily performs for H 2 (j = 0,1) capture by H 2 + corresponds to E/k B < 10 −3 K for j = 0 and E/k B < 10 −4 K for j = 1. These temperatures, however, are still much lower than attainable experimentally [12] in studying bimolecular chemical reactions dynamics.