Novel experimental methods have allowed for the routine production of ultracold (sub-Kelvin) atoms and small molecules. This has facilitated the study of chemical reactions involving only a small number of partial waves, allowing for unprecedented control over ultracold chemical reactions. This thesis describes work towards a new set of theories, based on Wigner's R-matrix methodology, which are adapted for so-called heavy particle scattering, and in particular atom-atom scattering. From these new theories a new set of methods are constructed to accurately simulate scattering observables such as scattering lengths, cross-sections, and resonances for atom-atom scattering events at ultracold temperatures by producing high resolution plots of these observables. The methods utilise software built for high-accuracy diatomic spectra, such as Duo, to provide molecular eigenenergies and wavefunctions of the bound system at short internuclear distances (in a region known as the inner region), only requiring as input a matrix of diatomic internuclear potential energy curves and couplings. These methods then act as 'harnesses', allowing this information to be used to perform an R-matrix propagation at long internuclear distances (in a region known as the outer region) using R-matrix propagation codes such as PFARM. The result of this propagation is then used to produce the aforementioned scattering observables. In this work these new R-matrix methods are applied to the case of a particle scattering off a Morse potential, to elastic argon-argon collisions, and to the intramultiplet mixing of oxygen when impacted by helium. This work also serves as a basis for the future simulation of more complex scattering events, such as atom-diatom collisions and higher polyatomic collisions.
Results for elastic atom-atom scattering are obtained as a first practical application of RmatReact, a new code for generating high-accuracy scattering observables from potential energy curves. RmatReact has been created in response to new experimental methods which have paved the way for the routine production of ultracold (μK) atoms and molecules, and hence the experimental study of chemical reactions involving only a small number of partial waves. Elastic scattering between argon atoms is studied here. There is an unresolved discrepancy between different Ar2 potential energy curves which give different numbers of vibrational bound states and different scattering lengths for the Ar2 dimer. Depending on the number of bound states, the scattering length is either large and positive or large and negative. Scattering observables, specifically the scattering length, effective range, and partial and total cross-sections, are computed at low collision energies and compared to previous results. In general, good agreement is obtained, although our full scattering treatment yields resonances which are slightly lower in energy and narrower than previous determinations using the same potential energy curve.
Over the past decade, and particularly the past five years, a quiet revolution has been building at the border between atomic physics and experimental quantum chemistry. The rapid development of techniques for producing cold and even ultracold molecules without a perturbing rare-gas cluster shell is now enabling the study of chemical reactions and scattering at the quantum scattering limit with only a few partial waves contributing to the incident channel. Moreover, the ability to perform these experiments with nonthermal distributions comprising one or a few specific states enables the observation and even full control of state-to-state collision rates in this computation-friendly regime: This is perhaps the most elementary study possible of scattering and reaction dynamics.
Results for elastic atom-atom scattering are obtained as a first practical application of RmatReact, a new code for generating high-accuracy scattering observables from potential energy curves. RmatReact has been created in response to new experimental methods which have paved the way for the routine production of ultracold (μK) atoms and molecules, and hence the experimental study of chemical reactions involving only a small number of partial waves. Elastic scattering between argon atoms is studied here. There is an unresolved discrepancy between different Ar2 potential energy curves which give different numbers of vibrational bound states and different scattering lengths for the Ar2 dimer. Depending on the number of bound states, the scattering length is either large and positive or large and negative. Scattering observables, specifically the scattering length, effective range, and partial and total cross-sections, are computed at low collision energies and compared to previous results. In general, good agreement is obtained, although our full scattering treatment yields resonances which are slightly lower in energy and narrower than previous determinations using the same potential energy curve.
Studying chemical reactions at very low temperatures is of importance for the understanding of fundamental physical and chemical processes. At very low energies, collisions are dominated by only a few partial waves. Thus, studies in this regime allow the characterization of quantum effects which depend on the collisional angular momentum, e.g., reactive scattering resonances and tunneling through centrifugal barriers. Additionally, the dynamics of ultralow-energy collisions is dominated by long-range interactions, i.e., “universal” chemical forces. Thus, experiments in this domain probe the details of intermolecular interactions which is of general relevance for the understanding of chemical processes. Moreover, the increasingly more precise experiments provide valuable data for benchmarking theoretical models and quantum chemical calculations. Studies of cold ion–neutral reactions in the laboratory rely on techniques for the generation of cold ions and cold neutrals. Over the last decades, the technology in this domain has made impressive progress. Many of these techniques allow not only the translational cooling of the molecules, but also a precise preparation of their internal quantum state. Thus, by an accurate preparation of both the energy and state of the reaction partners as well as by the application of external electric, magnetic or optical fields, prospects open up to study and control chemical processes at an unprecedented level of accuracy. In this chapter, we review salient theoretical concepts and recent experimental developments for studies of ion–molecule reactions at low temperatures. We first discuss the theoretical background for the description of low-energy ion–molecule reactions. Next we present key experimental methods for laboratory studies of cold ion–molecule reactions. The chapter concludes with a review of illustrative results and an outlook on future directions.
The last decade has seen substantial advances in the exploration of inelastic and reactive collisions at collision energies in the cold ( <
< 1 K) and ultracold ( < 1
<1 mK) regime.1-13 These advances ushered in an understanding of cold and ultracold collisions that is impacting research areas ranging from molecular spectroscopy to quantum simulation and computation to quantum control of chemical reaction dynamics. Cold collisions are dominated by resonances, as amply manifest in most of the systems studied.1, 14 Resonances are also present in the thermal ( >
> 100 K) and hyperthermal ( >
> 1000 K) collision energy regimes,15 but obscured by background (non-resonant) scattering and the presence of many partial waves pertaining to higher angular momenta that blur the partial waves' individual contributions. Moreover, the outcome of thermal and hyperthermal collisions is somewhat insensitive to the details of the interaction potential – the potential energy surface (PES) – especially to its attractive long-range region. Only in a handful of cases, such as the F+HD exchange reaction, resonances in the thermal regime have been experimentally detected and theoretically accounted for by quantum mechanical scattering calculations.16, 17 In contrast, in the cold regime, collisions are dominated by only a few partial waves whose effects on the integral and differential scattering cross-sections can be singled out and linked to particular resonances. The cold collision energy regime has been attained by making use of crossed molecular beams that were either Stark- or Zeeman-decelerated or whose intersection angle could be reduced all the way down to the merging-beams limit. A beam emerging from a Stark or Zeeman decelerator has a variable, well-defined laboratory velocity. When crossed with a secondary beam at small angles (5 ∘
∘ –10 ∘
∘ ), collision energies in the 0.3–15 K could be readily achieved, as demonstrated by the group of van de Meerakker.9, 11 The technique of merged beams makes use of the ability to bend one of the beams by a magnetic or electric field such that the crossing angle becomes zero, thereby affording collision energies in the range 0.1 K–10 K. Merging beams have been used in studies of Penning ionization by the groups of Narevicius5, 12 and Osterwalder18 and made it possible to unravel shape and Feshbach resonances arising in this process. The techniques mentioned so far require the reactants to be polar or paramagnetic. For reactants that are neither, collision energies as low as 5 K could be achieved by taking advantage of a favorable combination of low intersection angles and low reduced mass of the reactants. This approach has been demonstrated by Bergeat, Costes, Naulin, and coworkers in a series of reactive3, 19 and inelastic collision experiments8, 20 of H 2
2 . Yet, another approach to achieving low relative collision velocities relies on the coexpansion of the collision partners in a single molecular beam.21 If their masses are similar, both partners will end up with nearly the same velocities, within a few ms − 1
−1 . Using this technique, Zare, Mukherjee, and coworkers carried out a series of experiments in which rotational quenching of HD by H 2
2 , D 2
2 , and He was investigated. Moreover, in these experiments, the HD molecule was prepared in v
v =1 and j
j =2 using Stark-field-induced adiabatic Raman passage that allowed, in addition, to align the rotational angular momentum of the HD reactant molecule with respect to the initial relative velocity of the HD + He reactants.22-24 Collisional energy transfer between rotation and translation, in which a molecule is excited or relaxed via scattering by an atom or another molecule, has been of particular significance for developing our understanding of inelastic collisions. Most notably, the scattering of NO molecules by rare gas (Rg) atoms has served as a benchmark system for both experimentalists and theorists for decades. In its ground electronic state, NO is a 2 Π
2Π molecule with a single unpaired electron in an antibonding π ∗
π∗ orbital. In this state, the rotational levels are split into two spin–orbit manifolds, defined by the projection | Ω |
∣Ω∣ ( = 1 2
=21 or 3 2
23 ) of the electronic angular momentum (orbital and spin) on the internuclear axis. The rotational levels with the same j
j in the two manifolds are separated by approximately 123 cm − 1
−1 for v
v =0 (117 cm − 1
−1 for v
v =10). In addition, each rotational level within each manifold is further split into two Λ
Λ -doublet states ( ε = ± 1
ϵ=±1 , e/f), which are nearly degenerate, but differ in parity, p = ( − 1 ) j − ε / 2
p=(−1)j−ϵ/2 . As the Rg atom approaches the NO molecule, the degeneracy of the Λ
Λ doublets is lifted, giving rise to two PESs, A ′
A′ and A ′ ′
A′′ , in which the orbital of the unpaired electron lies, respectively, in the plane of the three atoms or perpendicular to it. In computational practice, it is more convenient to use linear combinations (sums and differences) of the A ′ ′
A′′ and A ′
A′ PESs. Using hexapole focusing or Stark deceleration combined with detection via velocity ion imaging, fully state-to-state-resolved integral and differential cross sections for rotationally elastic and inelastic collisions could be measured with high resolution.9, 11, 25, 26 More than that, the orientation and alignment of the scattered (outgoing) NO molecules could be measured as a function of the scattering angle27-30 and, quite recently, the effects of orientation of the reactant (incoming) NO molecules on the differential cross section in collisions with Rg, mostly Ar, could be established.31-35 On the theory side, quantum mechanical close coupling (QMCC) scattering calculations on accurate two-dimensional ab initio PESs using the benchmark coupled-cluster, CCSD(T), method with the NO internuclear distance fixed at its equilibrium value36 were performed and found to account for all the details revealed by the experiments. Although state of the art, most of these experiments except for those described in Refs. 9, 11 were carried out at thermal energies for NO molecules in their lowest level, | v = 0 , j = 1 2 , Ω = 1 2 , f ⟩
∣v=0,j=21,Ω=21,f⟩ . Scattering of highly vibrationally excited molecules in the cold energy regime remained an unexplored frontier until the recent work by the groups of Arthur Suits, Ad van der Avoird, Gerrit Groenenboom, and Hua Guo who added spin–orbit and rotational state selection and looked for spin- and rotational-state changing collisions to boot. Published in the current issue of Natural Sciences,37 this work is a culmination of a 3-year collaboration that included both implementation of new experimental techniques and boosting of theoretical methods. Working with the NO+Ar system, the authors made use of stimulated emission pumping via the electronically excited A 2 Σ +
A2Σ+ state to promote NO( 2 Π 1 2
2Π21 ) molecules to the vibrationally excited state with v = 10
v=10 and a selected spin–orbit state | Ω |
∣Ω∣ and rotational state j
j , | v = 10 , j , Ω , f ⟩
∣v=10,j,Ω,f⟩ .38, 39 Low (cold) collision energies were achieved in a novel experimental setup of two beams (dual-beam arrangement) crossing at 4 ∘
∘ and intersected by the pump and dump laser beams of the stimulated-emission pumping scheme. Downstream, the collision products were probed using 1+1 ′
′ REMPI with the ionization laser tuned just above the ionization threshold. Fully state-resolved angular distributions of the scattered NO molecules were measured by velocity ion slice imaging and detected by a charge-coupled device (CCD). In order to be able to detect the product (outgoing) molecules resulting from the cold NO + Ar collisions, some excess energy had to be made available to the products. This came in the form of the initial spin–orbit and/or rotational state energy. In the study highlighted herein,37 angular distributions leading to the spin–orbit changing transition | v = 10 , Ω = 3 2 , j = 3 2 f ⟩ → | v ′ = 10 , Ω ′ = 1 2 , j ′ = 9 2 , e ⟩
∣v=10,Ω=23,j=23f⟩→∣v′=10,Ω′=21,j′=29,e⟩ were measured at energies in the 2 K–5 K range. Although intrabeam scattering could have been used as an alternative as it had been in previous work,21 the authors demonstrated that a dual-beam setup provides a superior angular and energy resolution. The highlighted experiment challenged the existing theoretical treatments of the NO+ Rg systems: first, the approximation of fixing NO at its equilibrium distance was bound to fail for NO in v = 10
v=10 due to the concomitant elongation of the NO bond. Second, for such a highly excited vibrational state, both the long- and short-range interactions are affected, triggering nonadiabatic effects involving rotational and spin–orbit degrees of freedom. The above rendered the single-reference coupled-cluster method used in previous treatments of NO( 2 Π 1 2 , v = 0
2Π21,v=0 ) inadequate. Therefore, the authors of the highlighted paper37 constructed a new coupled-cluster PES as well as a multireference configuration-interaction (MRCI) PES and carried out full-fledged (3D) QMCC calculations on both PESs. These tour-de-force calculations reproduced quite well the experimental differential cross sections and defined a new state of the art in scattering calculations. However, some discrepancies remain as at low collision energies, and the accuracy of the long-range potential needed is a challenge for any ab initio method. Moreover, treating spin-changing collisions that are dominated by the essentially attractive difference potential, V A ′ ′ − V A ′
VA′′−VA′ , puts additional demands on the accuracy of the A ′ ′
A′′ and A ′
A′ PESs. The calculated integral cross section in the 3–4 cm − 1
−1 energy interval exhibits a bump that may indicate the presence of resonances that show up as backscattering in the measured differential cross section. A partial-wave analysis points to overlapping resonances associated with contributions from several of the highest partial waves involved. In any case, tracing the resonances may require additional improvements of the existing PESs. All in all, the highlighted combined experimental and theoretical study sets a new standard for our understanding of cold collisions. There is no conflict of interest. The peer review history for this article is available at https://publons.com/publon/10.1002/ntls.20210088. This Highlight has been internally reviewed and edited by Bretislav Friedrich and Gerard Meijer. No data available.