Effects of conical intersections on hyperfine quenching of hydroxyl OH in collision with an ultracold Sr atom

The effect of conical intersections (CIs) on electronic relaxation, transitions from excited states to ground states, is well studied, but their influence on hyperfine quenching in a reactant molecule is not known. Here, we report on ultracold collision dynamics of the hydroxyl free-radical OH with Sr atoms leading to quenching of OH hyperfine states. Our quantum-mechanical calculations of this process reveal that quenching is efficient due to anomalous molecular dynamics in the vicinity of the conical intersection at collinear geometry. We observe wide scattering resonance features in both elastic and inelastic rate coefficients at collision energies below \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_{\text {B}}\times 10 \, \hbox {mK}$$\end{document}kB×10mK. They are identified as either p- or d-wave shape resonances. We also describe the electronic potentials relevant for these non-reactive collisions, their diabatization procedure, as well as the non-adiabatic coupling between the diabatic potentials near the CIs.

Scientific RepoRtS | (2020) 10:14130 | https://doi.org/10.1038/s41598-020-71068-w www.nature.com/scientificreports/ frequently than elastic collisions leading to overwhelming trap losses. A favorable elastic-to-inelastic ratio has been predicted for collisions of OH with atomic hydrogen with initial OH temperatures around 250 mK 27 . Sympathetic cooling of the internal states of neutral molecules with laser-cooled neutral atoms is far more challenging. Trap depths for neutral molecules are small and the energy release from vibrational, rotational, and even hyperfine relaxation rapidly decreases the number of cold molecules. On the other hand molecular ions, trapped in deep Penning or Paul traps, can survive sympathetic cooling of the internal states. The method was first demonstrated for the BaCl + molecular ion colliding with ultracold calcium 28 and further analyzed in Ref. 29 . A collision of this molecular ion with a polarizable ultracold atom encourages the two to thermalize via vibrational relaxation and elastic momentum-changing collisions.
Here, we examine the possibility of sympathetic cooling of external and internal degrees of freedom of OH molecules in their electronic X 2 � 3/2 ground state due to collisions with ultracold ground-state Sr atoms. We have developed a state-of-the-art quantum coupled-channel description of the OH + Sr system, thereby, allowing the calculations of collisional (elastic) and quenching (inelastic) rate coefficients. Our earlier study of the collisional complex SrOH 30 has proven that symmetry-required conical intersections (CIs) exist between its ground and excited 2 A ′ adiabatic electronic potential energy surfaces (PESs). Conical intersections are sets of degeneracy points between adiabatic PESs and are common features in the electronic structure of polyatomic molecules 31,32 . They play an important role in ultrafast radiation-less transitions from excited to ground electronic states found in photochemistry, molecular spectroscopy, and quantum reactive scattering [33][34][35][36] . Yet, there remain many fields of physics and chemistry, where the effects of CIs on the collisional dynamics of molecules is not fully understood. Additional studies are required. We focus on the role of CIs in the hyperfine quenching of OH molecules in collisions with ultracold Sr atoms. It is our goal to answer the natural question whether a conical intersection in the collisional complex can influence the quenching dynamics of the free radical.

Results
Potential Surfaces involved in collisional dynamics. The non-reactive collision physics of 88 Sr( 1 S) colliding with the tightly-bound v = 0, J = 3/2 ro-vibrational ground-state of 16 O 1 H(X 2 � 3/2 ) molecules is most conveniently described in three-dimensional Jacobi coordinates or vectors R and r , where R = |R| is the separation between Sr and the center of mass of OH and r = |r| is the separation between O and H. It is also useful to define angle θ between R and r . A schematic of the Jacobi coordinates is shown in Fig. 1b.
The relevant electronic structure of the tri-atomic system is well characterized by three non-relativisitic diabatic electronic potential surfaces: two shallow nearly-degenerate at large R van-der-Waals-bonded potentials labeled by 4 2 A ′ (F 2 �) and 4 2 A ′′ (F 2 �) that dissociate to ground-state Sr( 1 S ) and OH( 2 ) and one deep ionically-bound potential labeled by 1 2 A ′ (X 2 � + ) . These surfaces are shown in Figs. 1 and 2. Where possible, we have estimated the uncertainties of our calculations. For example, in table 3 of our previous publication 30 , in which we first presented the electronic states of SrOH, we compared the energies of three stretching and bending levels of the 1 2 A ′ (X 2 � + ) potential with the experimental values of Ref. 37 . The agreement is around 5 %. We also compared dissociation energies of our 1 2 A ′ (X 2 � + ) potential and its experimental value from 38 . Our dissociation energy of the (0,0,0) vibrational state is D 0 /hc = 33.3 × 10 3 cm −1 , which agrees well with the experimentally measured value of 32(1.4) × 10 3 cm −1 .
The potential surface of the ionically-bound 1 2 A ′ (X 2 � + ) state dissociates to an electronically-excited state of the Sr atom and the ground state OH(X 2 ) molecule. In our notation of states trimer symmetries 2 A ′ and 2 A ′′ are further specified by 2 ± labels in parenthesis. These describe the C ∞v symmetries of the electronic wavefunctions in co-linear geometries. It is convenient to denote trimer potentials by C ∞v labels as the equilibrium geometry of SrOH is linear with O in the center. In fact, the equilibrium separation between O and H is close to that of the X 2 ground-state potential of the OH dimer and the vibrational energy in the OH stretch is only 3 % Figure 1. Relevant two-dimensional potential energy surfaces of non-reactive Sr + OH( 2 ) as functions of Jacobi coordinates R. (a) Potential energies of the 1 2 A ′ (X 2 � + ) and 4 2 A ′ (F 2 �) states as functions of R at θ = 180 • (blue solid curves) and θ = 0 • (dashed-red curves), respectively. Red and blue cones indicate conical intersections between X and F curves. They are located at R = 7.4a 0 and 8.1a 0 , respectively. The arrow marks the entrance channel. (b) A schematic depiction of the Jacobi coordinates. (c) Jacobi coordinates of the SrOH molecule in optimized geometry.
Scientific RepoRtS | (2020) 10:14130 | https://doi.org/10.1038/s41598-020-71068-w www.nature.com/scientificreports/ larger than in the OH dimer. For our calculations it then also suffices to determine the three PESs for only a small range of OH separations r around the OH dimer equilibrium separation of 1.832a 0 , where a 0 is the Bohr radius. Finally, trimer states m 2 A ′ are labeled by integer m = 1, 2, . . . and to a lesser extent by characters X and F in the parenthesis. The value m follows the energy ordering of 2 A ′ PESs near the trimer equilibrium geometry. As the diabatic 4 2 A ′ and 4 2 A ′′ states are nearly degenerate at linear optimized geometry, the 2 A ′′ is also denoted by m = 4 even though it is not the fourth state with this symmetry. The excited 2 2 A ′ and 3 2 A ′ electronic states have been omitted in our description of the Sr + OH collision. These potentials cross the 4 2 A ′ (F 2 �) potential at much smaller separations R and the effects of their couplings are expected to be smaller. Figure 1a shows the two 2 A ′ electronic potentials at θ = 0 • and 180 • , co-linear C ∞v geometries, as functions of separation R with r = 1.832a 0 . The deeper and shallower of the two potentials correspond to the 1 2 A ′ (X 2 � + ) and 4 2 A ′ (F 2 �) states, respectively. In these co-linear geometries the two states have conical intersections near 8a 0 . The energy of the CI for θ = 180 • is lower than that of our entrance channel Sr( 1 S) + OH(X 2 �) . The corresponding energy for θ = 0 • is higher and thus classically forbidden. Figure 1b shows a schematic depiction of the Jacobi coordinates used in our model, whereas Fig. 1c describes Jacobi coordinates of the SrOH molecule in the optimized geometry.
We have determined these diabatic potentials and their couplings from ab initio non-relativistic coupledcluster electronic-structure calculation of the SrOH PESs with single, double, and perturbative triple excitations www.nature.com/scientificreports/ (CCSD(T)) and the equation-of-motion coupled cluster (EOM-CCSD(dT)) method described in detail in our previous study 30 . The procedures to diabatize the adiabatic potential energy surfaces, as determined by the nonrelativistic electronic-structure calculations, have also been described in this reference. Figure 2 shows a more quantitative view of the three diabatic potentials. In particular, the difference between the shallow potentials of the 4 2 A ′′ and 4 2 A ′ states is apparent in panels (a) and (b) with their depths are around hc × 400 cm −1 and hc × 730 cm −1 , respectively. In addition, the PES of the 4 2 A ′′ state is very shallow in the T-shape θ = 90 • region, exhibiting a saddle point between the two collinear minima. The PES of 4 2 A ′ is more attractive, allowing a closer approach of the Sr atom towards OH in a skewed geometry near θ = 120 • . The 4 2 A ′ state has two saddle points: one collinear with θ = 180 • and one around θ = 60 • . The 1 2 A ′ potential is very deep and shown in Fig. 2c. In all panels we have indicated the curve where the 1 2 A ′ and 4 2 A ′ states have the same energy. On this curve conical intersections occur at θ = 0 • and 180 • .
For heteronuclear tri-atomic molecules conical intersections are not isolated molecular geometries, but form a one-dimensional seam. The molecule can undergo nonadiabatic passage or transitions at any point of this seam. In SrOH the seam between the 1 2 A ′ and 4 2 A ′ potentials lies in the (R, r) plane with angle θ at either 0 • or 180 • . Figure 2d shows the 1 2 A ′ and 4 2 A ′ potentials in this plane for θ = 180 • as well as the seam located between R = 8.3a 0 and 8.7a 0 . The equal-energy contours of the 4 2 A ′ state are nearly independent of r. As this diabatic state dissociates to Sr( 1 S) + OH( 2 �) for large R and the OH( 2 ) state has a zero-point energy of nearly hc × 2000 cm −1 , only a limited region of r around 1.832a 0 is relevant. For the diabatic 1 2 A ′ state the contours are curved in the (R, r) plane with the largest R value very close to the OH equilibrium separation. In fact, the curvature at the equilibrium separation is almost independent of contour especially for contours with energy less than that of the Sr( 1 S) + OH( 2 �) limit. Equivalently, except for an energy off set, the 1 2 A ′ potential as function of r close to 1.832a 0 is nearly independent of R. In fact, as noted before the OH separation and zero-point energy of the OH stretch in the SrOH trimer at its equilibrium geometry (not shown in Fig. 2d) are close to that of the OH dimer. We will then assume that in the collision OH vibrational motion in the 1 2 A ′ state is limited to a small range of r around 1.832a 0 .
We require diabatic PESs for use in coupled-channels calculations of the Sr + OH collision. Only adiabatic PESs, however, are available from electronic structure calculations, here provided by the MOLPRO and Q-CHEM programs 39,40 . In addition, we computed the mixing angles β(R, r, θ) for the adiabatic 2 A ′ states by the DDR procedure within the MRCI method 30 . The corresponding adiabatic states and potentials were transformed into diabatic states and potentials described in the previous subsection, so that the problematic avoided crossings near and degenerate singular part at the CI seam are "removed" 41 . By construction diabatic states are independent of the Jacobi coordinates near the CI and coupled by coupling potentials H 1,2 (R, r, θ) between diabatic states 1 and 2. Specifically, near the CI seam between the 1 2 A ′ and 4 2 A ′ states we first computed a 2 × 2 orthogonal matrix O(R, r, θ) , parameterized by mixing angle β(R, r, θ) , to transfer between the adiabatic and diabatic states. From the orthogonal matrix the coupling function can be easily constructed. For details see Ref. 30 . Figure 3 shows the mixing angle β(R, r, θ) and coupling potential H 12 (R, r, θ) between 1 2 A ′ and 4 2 A ′ diabatic states as a function of Jacobi coordinates R and θ . At the CI the mixing angle changes rapidly from 90 • to 0 • . The value of the coupling function is zero at θ = 0 • and 180 • and has a maximal strength of approximately 3300 cm −1 near θ = 90 • and R = 8.5a 0 . For large R the coupling relatively fast decreasing.

Dynamics in the vicinity of a conical intersection.
To assess the collision of 88 Sr( 1 S) with the rovibrational ground state of 16 O 1 H(X 2 �) it is convenient to first introduce relevant energy correlation diagram in Fig. 4. It illustrates the reaction pathway of our system, which highlights stationary points (minima or saddle points) in the PESs. The pathway is based on ab-initio calculations described in Ref. 30 . The reaction to form SrO(X 1 � + ) + H( 2 S) is endothermic by hc × 2890 cm −1 when accounting for the significant difference in zero point energy of OH and SrO. The entrance-channel van-der-Waals interaction between Sr and OH creates a rela- The v = 0 level has 2 � �=1/2 and 2 � �=3/2 spin-orbit components separated by just over hc × 100 cm −1 . The energetically-lowest rotational level is a J = 3/2 state. The nuclear spin I of the hydrogen atom in 16 OH leads to hyperfine splittings of the ground state rotational levels into components labeled by quantum number F, where F = J + I is the total angular momentum of 16 OH . Finally, the � = 3/2 state is split into opposite parity states e and f due to a weak non-adiabatic coupling, called -doubling. The transition  We have developed a coupled-channels model to calculate atom-dimer quenching rate coefficients for 88 Sr + 16 O 1 H collision energies below hc × 10 −2 cm −1 or k B × 10 mK , where k B is the Boltzmann constant, in a rigid rotor approximation for OH. That is, under the assumption that the OH stretch in SrOH is not excited in the Sr + OH collision. The model includes the three diabatic non-relativistic 2 A ′ and 2 A ′′ trimer potential surfaces and their couplings as well as the spin-orbit, rotational, -doubling, and hyperfine interactions in the OH dimer. The vibrational wavefunction of the OH stretch is that of the v = 0 eigenstate of the OH dimer for both the 2 A ′ and 2 A ′′ symmetries. As discussed in the previous section this assumption is mainly justified by the observation that near the equilibrium geometry of the 1 2 A ′ (X 2 � + ) state the vibrational energy of the OH stretch is close to that in the dimer. Secondly, the depth of the potentials in the 4 2 A ′ (F 2 �) and 4 2 A ′′ (F 2 �) states is less than the ground vibrational spacing of the OH(X 2 �) dimer.
Hence, we expand that the six-dimensional scattering wavefunction as a superposition of products of spherical harmonics with relative orbital angular momentum ℓ in the orientation R of coordinate R and rigid-rotor v = 0 ro-vibrational states for OH in r , such that the total trimer angular momentum F tot = ℓ + F and parity are conserved. For this system negative (positive) parity correspond to states or channels with odd(even) value for partial wave ℓ . Details about the basis set are given in Methods.
The expansion leads to coupled Schrödinger equations in separation R with potential matrix element U ij (R) between basis set elements i and j. We have numerically solved the differential equations and for R → ∞ construct rate coefficients from the solutions. The coupled-channels calculations are computationally demanding as a large number of OH rotational channels J need to be included. The 2 A ′ and 2 A ′′ potentials have strong angular anisotropies that couple many OH spin-orbit, rotational and hyperfine states. In the remainder of this paper we will study elastic and inelastic rate coefficients of Sr and OH in states |f ; F = 1, 2� or |e; F = 1, 2� states, defined in Fig. 5. In addition, we intend to show how the conical intersection influences the outcome of the collision. This is achieved by switching the CI "off " by simply setting H 12 (R, r, θ) = 0. Figure 6 shows the F tot = 2 and even parity eigenvalues of matrix U ij (R) as functions of R at two energy and length scales. We have included the energetically-lowest 12 rotational states of the v = 0 X 2 � 1/2 and v = 0 X 2 � 3/2 states of 16 OH . These eigenvalues are also called adiabatic potentials, but now in the sense that only motion in the radial R direction is considered slow. Most importantly, we observe that the density of collisional channels is high with spacings much smaller than the coupling strength H 12 (R, r, θ) providing by the CI. In fact, near R = 8a 0 in both Figs. 6a, b a "bundle" of nearly-vertical curves avoids, in a complicated manner, a bundle of www.nature.com/scientificreports/ nearly-horizontal curves. The former bundle is due to the 1 2 A ′ (X 2 � + ) state. The latter is due to both 4 2 A ′ (F 2 �) and 4 2 A ′′ (F 2 �) states. Their avoided crossings are due to the CI. Figure 7 shows the total elastic and hyperfine quenching inelastic rate coefficients obtained from coupledchannels calculations as a function of collision energy E for OH in the hyperfine entrance channel |e; F = 2� , |f ; F = 2� , and |f ; F = 1� . Quenching leads to OH molecules in state |e; F ′ = 1� . For these calculations, we have included the energetically-lowest 12 rotational states of the v = 0 X 2 � 1/2 and v = 0 X 2 � 3/2 states of OH. Collisions with partial wave quantum numbers up to ℓ = 8 contribute to our entrance channels and collision energies E/k B < 10 mK . Rate coefficients calculated when the coupling near the CI is switched on and off are shown. Figure 7a,b show elastic and inelastic hyperfine quenching rate coefficients when OH is prepared in hyperfine entrance channel |e; F = 2� → |e; F ′ = 1� . Both rate coefficients have a resonance feature with a maximum value near E = k B × 1 mK but only when the coupling between the 1 2 A ′ and 4 2 A ′ states is switched on. This resonance is much weaker when the coupling is turned off. We observe that for both cases the elastic rate coefficients are about five time larger than the inelastic ones. Figure 7c,d show elastic and inelastic hyperfine quenching rate coefficients when OH is prepared in either hyperfine entrance channel |f ; F = 2� → |e; F ′ = 1� or |f ; F = 1� → |e; F ′ = 1� . The rate coefficients are indistinguishable on the scale of the figure.
Analysis, based on studying the contributions from individual F tot and parity channels, has shown that the resonances that appear in Fig. 7a,b are due to a shape resonance behind and tunneling through a ℓ = 1 , p-wave centrifugal barrier leading to quantum enhanced scattering. Similarly, the broader resonances near E = k B × 8 mK with CI coupling and near E = k B × 5 mK without CI coupling are found to be due to ℓ = 2 , d-wave shape resonances. Figure 8 explains the analysis of the resonances. The figure shows the relevant F tot = 1 adiabatic eigenvalues of matrix U ij (R) for large R close to the OH |e; F = 1� and |e; F = 2� dissociation limits. Panels (a) and (b) show potentials for even and odd partial wave channels, respectively. On these energy and length scales centrifugal barriers for the p and d wave channels are visible with approximate barrier heights of about k B × 1 mK and k B × 10 mK , respectively. Higher partial wave entrance channels have even higher barriers and do not significantly contribute to rate coefficients for collision energies below k B × 10 mK . These barriers are not shown in the figure even though the corresponding channels are included in our coupled channels calculations. For our |e; F = 2� entrance channel and F tot = 1 the ℓ = 0 state does not exist. (The s-wave channel for the Sr + OH |e; F = 2� collision only occurs when F tot = 2 .) The heights or tops of the p-and d-wave barriers are thus consistent with the two resonances when the coupling between 1 2 A ′ and 4 2 A ′ states is turned on. The location of the resonance at E = k B × 5 mK when the coupling is turned off can only be explained by the d-wave centrifugal barrier for the |e; F = 1� exit channel, whose dissociation energy is about k B × 2.5 mK lower in energy than that of the |e; F = 2� entrance channel.
Finally, we have to note that with current computing capabilities first-principle calculations can not be fully converged with respect to the number of OH rotational states J included in our coupled-channels calculations when 100% of the coupling between the 1 2 A ′ and 4 2 A ′ states is turned on. These uncertainties are indicated in Fig. 7 by colored bands and estimated from our results of two consecutive calculations of the elastic and inelastic rates with J = 25/2 and 27/2. The difference between the rate coefficients for J = 25/2 and 27/2 is taken as the one standard-deviation uncertainty. We have reached convergence when the coupling between the potential surfaces is turned off. Figure 7 also show that the inelastic rate coefficients obey the Wigner threshold law and approach a finite value in limit of zero collision energy.  , panels (a,c), and inelastic, panels (b,d), rate coefficients for 88 Sr + 16 O 1 H with v = 0, J = 3/2 16 OH(X 2 � 3/2 ) . Solid and dashed lines indicate results of coupled-channels calculations with and without conical intersection between 1 2 A ′ and 4 2 A ′ potentials, respectively. OH v = 0 rotational states up to J = 25/2 are included. The grey and orange bands reflect our estimate of the one-standard deviation uncertainty of the rate coefficient due to our inability to include all rotational states of OH when the CI is included in the calculations. Panels (a,b) show total elastic and partial inelastic hyperfine quenching rate coefficients for OH in its |e; F = 2� hyperfine state, whereas panels (c,d) show rate coefficients for OH in its |f ; F = 1� and |f ; F = 2� hyperfine states. Rate coefficients for |f ; F = 1� and |f ; F = 2� are indistinguishable on the scale of the figure.

Discussion
We have created computational tools to compute elastic and quenching rate coefficients of the ultracold 88 Sr + 16 OH collision in the presence of nonadiabatic coupling between potential surfaces. These tools allowed us to treat the electronic structure and the nonadiabatic nuclear dynamics, dominated by conical intersections, with quantum chemical and quantum dynamical methods. In particular, we have performed state-of-the-art calculations of non-reactive collisional dynamics of hydroxyl with Sr based on three electronic potential-energy surfaces using only a localized region of the tri-atomic nuclear coordinate space. An important aspect of the investigation has been to answer the question whether conical intersections affect the quenching of the hyperfine levels of the ultracold ground-state OH molecule. We therefore calculated collisional quenching rates under conditions where the conical intersection can be switched on and off. This was achieved by using diabatic electronic basis functions. The results have shown that at low collision energy the conical intersection has important consequences for the collisional properties of OH. Specifically, when OH is prepared in one of its energetically-lowest hyperfine states shape resonances can be observed, which change their location or are absent when the coupling is turned off.

Basis set:
The ultracold non-reactive 88 Sr + 16 OH scattering wave function | � is expanded in 16 OH molecular eigenstates using Jacobi coordinates R and r defined in Fig. 1. That is, where index i labels basis states or channels |ψ i � . The sum over channels is constrained to conserve total angular momentum of the trimer F tot and parity p tot = ±1.
Channel states |ψ i � are given by where C jm j 1 m 1 ,j 2 m 2 are Clebsch-Gordan coefficients, Y ℓm (R) are spherical harmonic functions of the orbital angular momentum of partial wave ℓ , and |ϕ τ ;FM;p � describe electronic and OH ro-vibrational states with combined angular momentum F and parity p. Label τ is used to further describe these states. Projection quantum numbers M tot , m, and M of F tot , ℓ , and F , respectively, are given with respect to a space-fixed laboratory axis. Here, F tot = ℓ + F , parity p tot = (−1) ℓ p.
In principle, states |ϕ τ ;FM;p � form a complete set of electronic and OH ro-vibrational states. In order to make the computations tractable, however, we must introduce various approximations. The first is that the electronic wavefunctions are restricted to be the non-relativistic diabatic states |1 2 A ′ (X 2 � + )� , |4 2 A ′ (F 2 �)� , and |4 2 A ′′ (F 2 �)� , defined in the main text. These diabatic states are assumed to be independent of R, r, and θ www.nature.com/scientificreports/ (an approximation already implicit in the notation used in Eqs. 1 and 2.) We further assume that these trimer electronic wavefunctions are given by simple products of Sr( 1 S) and either 2 and 2 + 16 OH dimer electronic wavefunctions. That is, |4 2 A ′ (F 2 �)� and |4 2 A ′′ (F 2 �)� are superpositions of |Sr( 1 S)�|OH(X 2 �)� with projection quantum number = ± or ±1 and |1 2 A ′ (X 2 � + )� → |Sr( 1 S)�|OH( 2 � + )� (The state has projection = or 0), re-enforcing the usefulness of labeling trimer electronic states with C ∞v symmetries at collinear geometries. Secondly, the vibrational motion of the OH stretch is limited to that of the v = 0 state of 16 OH(X 2 �) . This approximation relies on the observations that the 4 2 A ′ (F 2 �) and 4 2 A ′′ (F 2 �) potentials are less deep than the vibrational spacing of OH(X 2 �) and that near the conical intersections of the |1 2 A ′ (X 2 � + )� and |4 2 A ′ (F 2 �)� states the r dependence of the 1 2 A ′ (X 2 � + ) potential, except for a constant energy offset, is nearly independent of R. Near the equilibrium geometry of the 1 2 A ′ (X 2 � + ) potential the vibrational energy in the OH stretch is within 3 % of that of the OH(X 2 �) dimer.
Spin-orbit interactions and OH rotation are included based on our assumption that trimer electronic states are separable and determined by those in the OH dimer. Then spin-orbit interactions in the |1 2 A ′ (X 2 � + )� state are absent while those in the |4 2 A ′ (F 2 �)� and |4 2 A ′′ (F 2 �)� states correspond to those in the OH(X 2 �) dimer. States represented by 2 � + �=1/2 and 2 � �=1/2,3/2 with OH electronic angular momentum projection � = 1/2 and 3/2 on the internuclear axis of 16 OH are formed. The rotation, coriolis and hyperfine interactions, as well as -doubling in SrOH are similarly based on those in OH. In fact, these latter interactions weakly mix 2 � �=1/2 and 2 � �=3/2 states as well as mix electron-rotational angular momentum states J . See Refs. 43,44 for a discussion of symmetries of and the spin-orbit, hyperfine, -doublet interactions in OH.
In practice, the relevant states are where radial functions ϕ v=0,J� (r) are v = 0, J OH(X 2 � � ) rotational levels, obtained by numerical diagonalization of the OH Hamiltonian excluding the coriolis, hyperfine, -doubling interactions 43 , and OH electron-rotational states | 2 , JM J ; p� are with Here, operator J is the electron-rotational angular momentum of 16 OH with quantum number J and projections M J and along the laboratory axis and 16 OH axis r , respectively. Functions D J M,� (α, β, γ ) are Wigner rotation functions with Euler angles α , β , and γ that specify the orientation of OH and, thus, r in our space-fixed coordinate system. For the 16 OH electronic wavefunctions |OH( 2 � � )� we have = and (or 0 and 1). Finally, the ket |IM I � describes the nuclear spin wavefunction of H with M I = ±1/2 and total 16 OH angular momentum F = J + I . Consistent with our approximations we have ignored the electronic state dependence of ϕ v=0,J� (r) . Historically, e states are states with p = (−1) J−1/2 while f states are states with p = (−1) J+1/2 . In summary, index i in Eq. 1 represents labels and quantum numbers 2 � � ; p, [(JI)F, ℓ]F tot M tot ; p tot . Similarly, τ in Eq. 2 represents 2 ; JI . Reference 45 gives details of the required angular momentum algebra for a simpler case. Finally, the radial wave functions F i (R) are channel components and only depend on the separation R between the center of mass of the molecule and the atom. They are determined from numerically integrating the coupled radial Schrödinger equations for total energy E from R = 0 to R → ∞ . Here, vector F(R) contains all F i (R) and µ is the reduced mass of the atom-molecule system. The first term in curly brackets on the left hand side of this equation is a diagonal matrix for the radial kinetic energy (the same for each channel), while the second term U (R) is the potential matrix with elements U ij (R) = �ψ i |U|ψ j � between channels i and j and includes the three diabatic potentials V (R, r, θ) and their couplings, the rotation energy of Sr around OH, 2 ℓ/(2µR 2 ) , as well as the spin-orbit, rotational and hyperfine energies of the v = 0 vibrational level of OH. In fact, U (R) is anisotropic, orientation dependent as channels with different values for ℓ or m are coupled. We limit the number of channel components by only including 16 OH rotational states J ≤ 25/2.
At large separations U (R) is an "almost" diagonal matrix. For the basis in Eq. 3 weak couplings by OH coriolis, hyperfine and -doubling interactions persist and mix states with different values of and J. Hence, in order to compute scattering amplitudes and rate coefficients we first calculate the orthonormal matrix that diagonalizes U (R) at a large R and transform F(R) into the basis in which U (R) is diagonal.

Data availability
All data generated or analysed during this study are available upon request.