Inelastic Scattering of He Atoms and NO(X2Π) Molecules: The Role of Parity on the Differential Cross Section

A general formalism to describe the stereodynamics of bond axis orientation in the scattering of a linear molecule with an atom

One of the aims of the chemist is to obtain the greatest possible level of control over the outcome of a reaction. A factor that can influence such outcomes is the so-called steric effect. The underpinning idea of this effect is that the mutual orientation of the collision partners at the moment of collision may impact the nature of the products. The steric effect has been studied in a variety of ways, notably using optical methods, as well as making use of both magnetic and electric fields, to orient or align reactants. Here, we present a general framework for interpreting and evaluating steric effects in collisions of open shell linear molecules with an atom in the presence of an electric field. While in previous studies, the theory has been limited to the specific system of interest, such as for the end-on collisions of NO(X), this new formalism provides a fundamental basis for examining any system of this type. Some examples of the utility and power of this formalism are also provided. This theory may then be built on further in the future to provide greater insights into the stereodynamics of collisions and, hence, provide the foundation for deeper study into how the steric effect may be harnessed for control.

Correlated rotational excitations in NO–CO inelastic collisions

We present a joint experimental and theoretical study of rotationally inelastic collisions between NO ( X2Π1/2, ν = 0, j = 1/2, f) radicals and CO ( X1Σ+, ν = 0, j = 0) molecules at a collision energy of 220 cm−1. State-to-state scattering images for excitation of NO radicals into various final states were measured with high resolution by combining the Stark deceleration and velocity map imaging techniques. The high image resolution afforded the observation of correlated rotational excitations of NO–CO pairs, which revealed a number of striking scattering phenomena. The so-called “parity-pair” transitions in NO are found to have similar differential cross sections, independent of the concurrent excitation of CO, extending this well-known effect for collisions between NO and rare gas atoms into the realm of bimolecular collisions. Forward scattering is found for collisions that induce a large amount of rotational energy transfer (in either NO, CO, or both), which require low impact parameters to induce sufficient energy transfer. This observation is interpreted in terms of the recently discovered hard collision glory scattering mechanism, which predicts the forward bending of initially backward receding trajectories if the energy uptake in the collision is substantial in relation to the collision energy. The experimental results are in good agreement with the predictions from coupled-channels quantum scattering calculations based on an ab initio NO–CO potential energy surface.

Mapping partial wave dynamics in scattering resonances by rotational de-excitation collisions

One of the most important parameters in a collision is the 'miss distance' or impact parameter, which in quantum mechanics is described by quantized partial waves. Usually, the collision outcome is the result of unavoidable averaging over many partial waves. Here we present a study of low-energy NO-He collisions that enables us to probe how individual partial waves evolve during the collision. By tuning the collision energies to scattering resonances between 0.4 and 6 cm^-1, the initial conditions are characterized by a limited set of partial waves. By preparing NO in a rotationally excited state before the collision and by studying rotational de-excitation collisions, we were able to add one quantum of angular momentum to the system and trace how it evolves. Distinct fingerprints in the differential cross-sections yield a comprehensive picture of the partial wave dynamics during the scattering process. Exploiting the principle of detailed balance, we show that rotational de-excitation collisions probe time-reversed excitation processes with superior energy and angular resolution.

Steric Effects in the Inelastic Scattering of NO(X) + Ar: Side-on Orientation

The rotationally inelastic collisions of NO(X) with Ar, in which the NO bond-axis is oriented side-on (i.e., perpendicular) to the incoming collision partner, are investigated experimentally and theoretically. The NO(X) molecules are selected in the |j = 0.5, Ω = 0.5, ε = -1, f⟩ state prior to bond-axis orientation in a static electric field. The scattered NO products are then state selectively detected using velocity-map ion imaging. The experimental bond-axis orientation resolved differential cross sections and integral steric asymmetries are compared with quantum mechanical calculations, and are shown to be in good agreement. The strength of the orientation field is shown to affect the structure observed in the differential cross sections, and to some extent also the steric preference, depending on the ratio of the initial e and f Λ-doublets in the superposition determined by the orientation field. Classical and quantum calculations are compared and used to rationalize the structures observed in the differential cross sections. It is found that these structures are due to quantum mechanical interference effects, which differ for the two possible orientations of the NO molecule due to the anisotropy of the potential energy surface probed in the side-on orientation. Side-on collisions are shown to maximize and afford a high degree of control over the scattering intensity at small scattering angles (θ < 90°), while end-on collisions are predicted to dominate in the backward scattered region (θ > 90°).

Observation of Rainbows in the Rotationally Inelastic Scattering of NO with CH4

Using a combination of velocity-map imaging and resonance-enhanced multiphoton ionization detection with crossed molecular beam scattering, the dynamics of rotational energy transfer have been examined for NO in collisions with CH₄ at a mean collision energy of 700 cm^-1. The images of NO scattered into individual rotational (j_NO^') and spin-orbit (Ω) levels typically exhibit a single broad maximum that gradually shifts from the forward to the backward scattering direction with increasing rotational excitation (i.e., larger Δj_NO). The rotational rainbow angles calculated with a two-dimensional hard ellipse model show reasonable agreement with the observed angles corresponding to the maxima in the differential cross sections extracted from the images for higher Δj_NO transitions, but there are clear discrepancies for lower Δj_NO (in particular, final rotational levels with j_NO^' = 7.5 and 8.5). The sharply forward scattered angular distributions for these lower Δj_NO transitions better agree with the predictions of an L-type rainbow model. The more highly rotationally excited NO appears to coincide with low rotational excitation of the co-product CH₄, indicating a degree of rotational product-pair anticorrelation in this bimolecular scattering.

Differential steric effects in the inelastic scattering of NO(X) + Ar: spin-orbit changing transitions

Spin-orbit changing transitions for bond-axis oriented collisions of NO(X) with Ar have been investigated with full quantum state selection via a crossed molecular beam experiment at collision energies of 532 cm^-1 and 651 cm^-1. NO(X) molecules were selected in their ground rotational state (Ω = 0.5, j = 0.5, f) before being adiabatically oriented using a static electric field, such that either the N- or O-end of the molecule was directed towards the incoming Ar atom. After collision, NO(X, Ω' = 1.5, j', e) molecules were probed quantum state specifically using velocity-map ion imaging, coupled with resonantly enhanced multi-photon ionization. Differences were observed between the experimental ion images and differential cross sections for collisions occurring at the two ends of the molecule, with results that could largely be accounted for by quantum mechanical scattering calculations. The bond-axis oriented data for the spin-orbit changing collisions are compared with similar results obtained previously for spin-orbit conserving transitions, and for field free scattering of NO(X) with Ar.

Energy dependent parity-pair behavior in NO + He collisions

Colliding molecules behave fundamentally differently at high and low collision energies. At high energies, a collision can be described to a large extent using classical mechanics, and the scattering process can be compared to a billiard-ball-like collision. At low collision energies, the wave character of the collision partners dominates, and only quantum mechanics can predict the outcome of an encounter. It is, however, not so clear how these limits evolve into each other as a function of the collision energy. Here, we investigate and visualize this evolution using a special feature of the differential cross sections for inelastic collisions between NO radicals and He atoms. The so-called "parity-pair" transitions have similar differential cross sections at high collision energies, whereas their cross sections are significantly different in the quantum regime at low energies. These transitions can be used as a probe for the quantum nature of the collision process. The similarity of the parity-pair differential cross sections at high energies could be theoretically explained if the first-order Born approximation were applicable. We found, however, that the anisotropy of the NO-He interaction potential is too strong for the first-order Born approximation to be valid, so higher-order perturbations must be taken into account.

Collisional excitation of interstellar PO(X2Π) by He: new ab initio potential energy surfaces and scattering calculations

The inelastic scattering of PO (X, v = 0) has been investigated by quantum scattering calculations using a new potential energy surface.

Stereodynamics in NO(X) + Ar inelastic collisions

The effect of orientation of the NO(X) bond axis prior to rotationally inelastic collisions with Ar has been investigated experimentally and theoretically. A modification to conventional velocity-map imaging ion optics is described, which allows the orientation of hexapole state-selected NO(X) using a static electric field, followed by velocity map imaging of the resonantly ionized scattered products. Bond orientation resolved differential cross sections are measured experimentally for a series of spin-orbit conserving transitions and compared with quantum mechanical calculations. The agreement between experimental results and those from quantum mechanical calculations is generally good. Parity pairs, which have previously been observed in collisions of unpolarized NO with various rare gases, are not observed due to the coherent superposition of the two j = 1/2, Ω = 1/2 Λ-doublet levels in the orienting field. The normalized difference differential cross sections are found to depend predominantly on the final rotational state, and are not very sensitive to the final Λ-doublet level. The differential steric effect has also been investigated theoretically, by means of quantum mechanical and classical calculations. Classically, the differential steric effect can be understood by considering the steric requirement for different types of trajectories that contribute to different regions of the differential cross section. However, classical effects cannot account quantitatively for the differential steric asymmetry observed in NO(X) + Ar collisions, which reflects quantum interference from scattering at either end of the molecule. This quantum interference effect is dominated by the repulsive region of the potential.

Integral steric asymmetry in the inelastic scattering of NO(X2Π)

The integral steric asymmetry for the inelastic scattering of NO(X) by a variety of collision partners was recorded using a crossed molecular beam apparatus. The initial state of the NO(X, v = 0, j = 1/2, Ω=1/2, ϵ=-1,f) molecule was selected using a hexapole electric field, before the NO bond axis was oriented in a static electric field, allowing probing of the scattering of the collision partner at either the N- or O-end of the molecule. Scattered NO molecules were state selectively probed using (1 + 1') resonantly enhanced multiphoton ionisation, coupled with velocity-map ion imaging. Experimental integral steric asymmetries are presented for NO(X) + Ar, for both spin-orbit manifolds, and Kr, for the spin-orbit conserving manifold. The integral steric asymmetry for spin-orbit conserving and changing transitions of the NO(X) + O₂ system is also presented. Close-coupled quantum mechanical scattering calculations employing well-tested ab initio potential energy surfaces were able to reproduce the steric asymmetry observed for the NO-rare gas systems. Quantum mechanical scattering and quasi-classical trajectory calculations were further used to help interpret the integral steric asymmetry for NO + O₂. Whilst the main features of the integral steric asymmetry of NO with the rare gases are also observed for the O₂ collision partner, some subtle differences provide insight into the form of the underlying potentials for the more complex system.

Comparative stereodynamics in molecule-atom and molecule-molecule rotational energy transfer: NO(A(2)Σ(+)) + He and D2

We present a crossed molecular beam scattering study, using velocity-map ion-imaging detection, of state-to-state rotational energy transfer for NO(A(2)Σ(+)) in collisions with the kinematically identical colliders He and D2. We report differential cross sections and angle-resolved rotational angular momentum polarization moments for transfer of NO(A, v = 0, N = 0, j = 0.5) to NO(A, v = 0, N' = 3, 5-12) in collisions with He and D2 at respective average collision energies of 670 cm(-1) and 663 cm(-1). Quantum scattering calculations on a literature ab initio potential energy surface for NO(A)-He [J. Kłos et al., J. Chem. Phys. 129, 244303 (2008)] yield near-quantitative agreement with the experimental differential scattering cross sections and good agreement with the rotational polarization moments. This confirms that the Kłos et al. potential is accurate within the experimental collisional energy range. Comparison of the experimental results for NO(A) + D2 and He collisions provides information on the hitherto unknown NO(A)-D2 potential energy surface. The similarities in the measured scattering dynamics of NO(A) imply that the general form of the NO(A)-D2 potential must be similar to that calculated for NO(A)-He. A consistent trend for the rotational rainbow maximum in the differential cross sections for NO(A) + D2 to peak at more forward angles than those for NO(A) + He is consistent with the NO(A)-D2 potential being more anisotropic with respect to NO(A) orientation. No evidence is found in the experimental measurements for coincident rotational excitation of the D2, consistent with the potential having low anisotropy with respect to D2. The NO(A) + He polarization moments deviate systematically from the predictions of a hard-shell, kinematic-apse scattering model, with larger deviations as N' increases, which we attribute to the shallow gradient of the anisotropic repulsive NO(A)-He potential energy surface.

Rotational and angular distributions of NO products from NO-Rg (Rg = He, Ne, Ar) complex photodissociation

We present the results of an investigation into the rotational and angular distributions of the NO à state fragment following photodissociation of the NO-He, NO-Ne, and NO-Ar van der Waals complexes excited via the à ← X̃ transition. For each complex, the dissociation is probed for several values of Ea, the available energy above the dissociation threshold. For NO-He, the Ea values probed were 59, 172, and 273 cm(-1); for NO-Ne they were 50 and 166 cm(-1); and for NO-Ar they were 44, 94, 194, and 423 cm(-1). The NO à state rotational distributions arising from NO-He are cold, with most products in low angular momentum states. NO-Ne leads to broader NO rotational distributions but they do not extend to the maximum possible given the energy available. In the case of NO-Ar, the distributions extend to the maximum allowed at that energy and show the unusual shapes associated with the rotational rainbow effect reported in previous studies. This is the only complex for which a rotational rainbow effect is observed at the chosen Ea values. Product angular distributions have also been measured for the NO à photodissociation product for the three complexes. NO-He produces nearly isotropic fragments, but the anisotropy parameter, β, for NO-Ne and NO-Ar photofragments shows a surprising change in sign from negative to positive as Ea increases within the unstructured excitation profile. Franck-Condon selection of a broader distribution of geometries including more linear geometries at lower excitation energies and more T-shaped geometries at higher energies can account for the changing recoil anisotropy. Two-dimensional wavepacket calculations are reported to model the rotational state distributions and the bound-continuum absorption spectra.

The modified quasi-quantum treatment of rotationally inelastic NO(X)-He scattering

A modified quasi-quantum treatment (MQQT) of molecular scattering has been developed to account for the softness of the repulsive part of the anisotropic atom-molecule PES. A contour of the PES is chosen such that the barrier height is just large enough to reflect the incoming kinetic energy, directed anti-parallel to the hard shell normal at the site of impact. The resulting rotationally inelastic quantum state resolved DCSs and ICSs of He + NO(X) at Ecol = 508 cm(-1) are compared to those obtained from regular QQT and from quantum mechanically exact calculations performed on the full highest quality ab initio Vsum PES. The MQQT parity changing DCSs for Δj ≤ 4 exhibit much better agreement with the QM DCSs than is obtained using regular QQT, particularly in the forward scattered direction. The improvements upon the remaining MQQT DCSs with respect to the regular QQT were minor, due to the near incompressible hard shell character of the n ≠ 1 or 3 anisotropic Legendre polynomial terms of the PES.

Rotationally inelastic scattering of methyl radicals with Ar and N2

The rotationally inelastic scattering of methyl radical with Ar and N2 is examined at collision energies of 330 ± 25 cm(-1) and 425 ± 50 cm(-1), respectively. Differential cross sections (DCSs) were measured for different final n' rotational levels (up to n' = 5) of the methyl radicals, averaged over k' sub-levels, using a crossed molecular beam machine with velocity map imaging. For Ar as a collision partner, we present a newly constructed ab initio potential energy surface and quantum mechanical scattering calculations of state-resolved DCSs. These computed DCSs agree well with the measurements. The DCSs for both Ar and N2 collision partners are strongly forward peaked for all spectroscopic lines measured. For scattering angles below 60°, the theoretical CD3-Ar DCSs show diffraction oscillations that become less pronounced as n' increases, but these oscillations are not resolved experimentally. Comparisons are drawn with our recently reported DCSs for scattering of methyl radicals with He atoms.

Collision dynamics of symmetric top molecules: a comparison of the rotationally inelastic scattering of CD3 and ND3 with He

We compare rotationally inelastic scattering of deuterated methyl radicals (CD3) and ammonia (ND3) in collisions with helium using close-coupling quantum-mechanical scattering calculations performed with ab initio potential energy surfaces (PESs). The theoretical methods have been rigorously tested against angle-resolved experimental measurements obtained using crossed molecular beam apparatuses in combination with velocity map imaging [O. Tkáč, A. G. Sage, S. J. Greaves, A. J. Orr-Ewing, P. J. Dagdigian, Q. Ma, and M. H. Alexander, Chem. Sci. 4, 4199 (2013); O. Tkáč, A. K. Saha, J. Onvlee, C.-H. Yang, G. Sarma, C. K. Bishwakarma, S. Y. T. van de Meerakker, A. van der Avoird, D. H. Parker, and A. J. Orr-Ewing, Phys. Chem. Chem. Phys. 16, 477 (2014)]. Common features of the scattering dynamics of these two symmetric top molecules, one closed-shell and the other an open-shell radical, are identified and discussed. Two types of anisotropies in the PES influence the interaction of an atom with a nonlinear polyatomic molecule. The effects of these anisotropies can be clearly seen in the state-to-state integral cross sections out of the lowest CD3 rotational levels of each nuclear spin symmetry at a collision energy of 440 cm(-1). Similarities and differences in the differential cross sections for the ND3-He and CD3-He systems can be linked to the coupling terms derived from the PESs which govern particular initial to final rotational level transitions.

Parity-dependent oscillations in collisional polarization transfer: CN(A²Π, v = 4) + Ar

We report the first systematic experimental and theoretical study of the state-to-state transfer of rotational angular momentum orientation in a (2)Π-rare gas system. CN(X(2)Σ(+)) was produced by pulsed 266 nm photolysis of ICN in a thermal bath (296 K) of Ar collider gas. A pulsed circularly polarized tunable dye laser prepared CN(A(2)Π, v = 4) in two fully state-selected initial levels, j = 6.5 F1e and j = 10.5 F2f, with a known laboratory-frame orientation. Both the prepared levels and a range of product levels, j' F1e and j' F2f, were monitored using the circular polarized output of a tunable diode laser via cw frequency-modulated (FM) spectroscopy in stimulated emission on the CN(A-X) (4,2) band. The FM Doppler lineshapes for co-rotating and counter-rotating pump-and-probe geometries reveal the time-dependence of the populations and orientations. Kinetic fitting was used to extract the state-to-state population transfer rate constants and orientation multipole transfer efficiencies (MTEs), which quantify the degree of conservation of initially prepared orientation in the product level. Complementary full quantum scattering (QS) calculations were carried out on recently computed ab initio potential energy surfaces. Collision-energy-dependent tensor cross sections for ranks K = 0 and 1 were computed for transitions from both initial levels to all final levels. These quantities were integrated over the thermal collision energy distribution to yield predictions of the experimentally observed state-to-state population transfer rate constants and MTEs. Excellent agreement between experiment and theory is observed for both measured quantities. Dramatic oscillations in the MTEs are observed, up to and including changes in the sign of the orientation, as a function of even/odd Δj within a particular spin-orbit and e/f manifold. These oscillations, along with those also observed in the state-to-state rate constants, reflect the rotational parity of the final level. In general, parity-conserving collisions conserve rotational orientation, while parity-changing collisions result in large changes in the orientation. The QS calculations show that the dynamics of the collisions leading to these different outcomes are fundamentally different. We propose that the origin of this behavior lies in interferences between collisions that sample the even and odd-λ terms in the angular expansions of the PESs.

A general scaling rule for the collision energy dependence of a rotationally inelastic differential cross-section and its application to NO(X) + He

The quasi-quantum treatment (QQT) (Gijsbertsen et al., J. Am. Chem. Soc., 2006, 128, 8777) provides a physically compelling framework for the evaluation of rotationally inelastic scattering, including the differential cross sections (DCS). In this work the QQT framework is extended to treat the DCS in the classically forbidden region as well as the classically allowed region. Most importantly, the QQT is applied to the collision energy dependence of the angular distributions of these DCSs. This leads to an analytical formalism that reveals a scaling relationship between the DCS calculated at a particular collision energy and the DCS at other collision energies. This scaling is shown to be exact for QM calculated or experimental DCSs if the magnitude of the (kinematic apse frame) underlying scattering amplitude depends solely on the projection of the incoming momentum vector onto the kinematic apse vector. The QM DCSs of the NO(X)-He collision system were found to obey this scaling law nearly perfectly for energies above 63 meV. The mathematical derivation is accompanied by a mechanistic description of the Feynman paths that contribute to the scattering amplitude in the classically allowed and forbidden regions, and the nature of the momentum transfer during the collision process. This scaling relationship highlights the nature of (and limits to) the information that is obtainable from the collision-energy dependence of the DCS, and allows a description of the relevant angular range of the DCSs that embodies this information.

Rotational alignment effects in NO(X) + Ar inelastic collisions: an experimental study

Rotational angular momentum alignment effects in the rotationally inelastic collisions of NO(X) with Ar have been investigated at a collision energy of 66 meV by means of hexapole electric field initial state selection coupled with velocity-map ion imaging final state detection. The fully quantum state resolved second rank renormalized polarization dependent differential cross sections determined experimentally are reported for a selection of spin-orbit conserving and changing transitions for the first time. The results are compared with the findings of previous theoretical investigations, and in particular with the results of exact quantum mechanical scattering calculations. The agreement between experiment and theory is generally found to be good throughout the entire scattering angle range. The results reveal that the hard shell nature of the interaction potential is predominantly responsible for the rotational alignment of the NO(X) upon collision with Ar.

New findings regarding the NO angular momentum orientation in Ar-NO(2Π(1/2)) collisions

This article reports a theoretical study of the stereodynamics of Ar + NO(X(2)Π, v = 0, j = 1/2, Ω = 1/2, ε = ±1) rotationally inelastic collisions. First, quantum scattering data are used to calculate all differential polarisation moments of the reagent and product molecules; this leads to the observation that the orientations of the reagent and product angular momenta are very strongly correlated. Next, canonical collision mechanisms theory [Aldegunde et al., Phys. Chem. Chem. Phys., 2008, 10, 1139] is used to separate and characterise the stereodynamics of the two independent collision mechanisms that contribute to the collision dynamics; this leads to the observation that the average product orientation is determined by the relative contributions of the two canonical mechanisms, which have comparable importance but are associated with starkly contrasting angular momentum orientations. These observations lead to a new and rigorous explanation of the experimental results reported a decade ago by Lorenz et al. [Science, 2001, 293, 2063]. The central fact of the new explanation is the incoherent, interference-free superposition of two independent collision mechanisms. This makes the new explanation radically different from the only one previously suggested, namely that the experimental observations might be due to quantum interference in a single collision mechanism.