# Universal stereodynamics of cold atom-molecule collisions in electric fields

Timur V. Tscherbul1 and Jacek Kłos2 1Department of Physics, University of
Nevada, Reno, NV, 89557, USA
2Department of Physics, Joint Quantum Institute, University of Maryland,
College Park, Maryland, 20742, USA

###### Abstract

We use numerically exact quantum dynamics calculations to demonstrate
universal stereoselectivity of cold collisions of ${}^{2}\Pi$ molecules with
1S-state atoms in an external electric field. We show that cold collisions of
OH molecules in their low-field-seeking $f$-states, whose dipole moments are
oriented against the field direction, are much more likely to lead to
inelastic scattering than those of molecules oriented along the field
direction, causing nearly perfect steric asymmetry in the inelastic collision
cross sections. The universal nature of this effect is due to the threshold
suppression of inelastic scattering between the degenerate $\pm M$ Stark
sublevels of the high-field-seeking $e$-state, where $M$ is the projection of
the total angular momentum of the molecule on the field axis. Above the
$\Lambda$-doublet threshold, the stereodynamics of inelastic atom-molecule
collisions can be tuned via electric-field-induced resonances, which enable
effective control of Ne + OH scattering over the range of collision energies
achievable in current merged beam experiments.

##  I Introduction

Modern experimental studies of ultracold molecular gases [1] have reached an
extraordinary level of control over molecular degrees of freedom using
external electromagnetic fields [2, 3, 4]. In particular, control over the
rotational motion [5] makes it possible to address a central question of
chemical physics concerning the role of the relative orientation of the
reactants in determining the outcome of molecular collisions and chemical
reactions [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 14, 17]. The steric effects
have been the subject of numerous experimental studies in crossed molecular
beams [7, 10, 9, 11, 12, 18, 13, 14], external field traps [19, 20], and, more
recently, in merged molecular beams [21, 22]. The latter experiments probed
the stereodynamics of cold HD + D2 collisions at 1 K [21, 22] and observed a
dramatic preference for the perpendicular alignment of collision products. The
single partial-wave regime accessed in these experiments is optimal for
studying and controlling collision stereodynamics [23] due to the absence of
detrimental averaging over many partial waves (or impact parameters), which
tends to obscure steric effects [6].

Recent quantum scattering calculations revealed the important role of single
scattering resonances in determining the stereodynamics of cold HD$(v=1,j=2)$
\+ H2 collisions [24] and suggested the possibility of tuning shape resonances
in cold HD$(v=1,j=2)$ \+ H2 collisions by aligning the rotational angular
momentum of HD with respect to the initial relative velocity vector [25].
Additional calculations explored the stereodynamics of cold rotationally
inelastic He + HD [26] and HCl + H2 collisions [27] in the presence of
overlapping resonances and identified a universal trend in the stereodynamic
preference of state-to-state integral cross sections.

Previous theoretical work on steric effects in cold molecular collisions [24,
25, 26, 27] has focused on molecules in nondegenerate electronic states of
$\Sigma$ symmetry in the absence of external fields. Open-shell molecular
radicals such as OH$(^{2}\Pi)$ and NO$(^{2}\Pi)$ are readily controllable by
external fields due to their quasi-degenerate $\Lambda$-doublet levels of the
opposite parity [28]. The OH radical was among the first molecules cooled and
trapped at low temperatures [29, 30] and its cold collisional properties with
rare-gas atoms have been extensively studied [29, 30, 31, 32, 33, 34].
External fields orient or align the molecules along a laboratory-fixed
quantization axis [35, 3, 36, 37, 38], providing an extra spatial direction
for observing novel stereodynamical effects. In addition, external fields are
commonly used to tune the scattering properties of cold atoms and molecules
via Feshbach resonances [39]. A combination of steric and external field
control may thus lead to new and powerful ways to engineer the quantum
dynamics of molecular collisions at ultralow temperatures.

Here, we explore the stereodynamics of cold atom-molecule collisions in an
electric field using Ne + OH as a representative example (rare gas - OH
collisions serve as prototype systems for studying steric effects in molecular
collisions [7, 10, 9, 11, 12]). Using rigorous quantum scattering
calculations, we uncover a universal stereodynamical trend: Collisions of
${}^{2}\Pi$ molecules initially oriented against the field direction are much
more likely to lead to inelastic scattering than those of molecules oriented
along the field direction. This is the first theoretical study of cold atom-
molecule collision stereodynamics in the presence of external fields using
numerically exact quantum dynamics calculations. We also show that the
stereodynamics of cold atom-molecule collisions can be controlled by an
external electric field, and find that such control can be extensive even in
the multiple partial wave regime, which can be reached experimentally in
merged molecular beams [40, 41]. Our predictions can thus be verified in
current molecular beam scattering experiments.

##  II Theory

To explore the stereodynamics of cold atom-molecule collisions in an external
electric field, we carry out rigorous quantum scattering calculations for the
benchmark collision system Ne + OH parametrized by accurate ab initio
interaction potentials as described in the Appendix. The stereodynamical
observables of interest are encoded in the atom-molecule scattering amplitude
[42, 43]

$q_{\alpha\to\alpha^{\prime}}(\hat{k}_{\alpha},\hat{R})=2\pi\sum_{\begin{subarray}{c}\ell,m_{\ell},\\\
\ell^{\prime},m_{\ell}^{\prime}\end{subarray}}{i}^{\ell-\ell^{\prime}}Y^{*}_{\ell
m_{\ell}}(\hat{k}_{\alpha})Y_{\ell^{\prime}m_{\ell}^{\prime}}(\hat{R})T_{\alpha\ell
m_{\ell};\alpha^{\prime}\ell^{\prime}m_{\ell}^{\prime}}$ (1)

where $\mathbf{k}_{\alpha}=k_{\alpha}\hat{k}_{\alpha}$ the incident
wavevector, $\hat{k}_{\alpha}$ gives the direction of the incident flux with
respect to the space-fixed (SF) $Z$-axis defined by the direction of the
external electric field, $\hat{R}$ specifies the direction of the scattered
flux, $Y_{lm}(\hat{R})$ are the spherical harmonics, $\ell$ and $m_{\ell}$ are
the quantum numbers for the orbital angular momentum and its SF projection,
$\alpha$ refers to the internal states of the molecule, and $T_{\gamma\ell
m_{\ell};\gamma^{\prime}\ell^{\prime}m_{\ell}^{\prime}}$ are the transition
$T$-matrix elements.

We will assume that the external field is collinear with the incident relative
velocity vector ($Z\parallel\hat{k}_{\alpha}$) [24, 25, 26, 27] which allows
us to set $\hat{k}_{\alpha}=0$ in Eq. (1) to yield [43]

$q_{\alpha\to\alpha^{\prime}}^{(0)}(\hat{R})=\sqrt{\pi}\sum_{\ell,\ell^{\prime},m_{\ell}^{\prime}}{i}^{\ell-\ell^{\prime}}(2\ell+1)^{1/2}Y_{\ell^{\prime}m_{\ell}^{\prime}}(\hat{R})T_{\alpha\ell
0;\alpha^{\prime}\ell^{\prime}m_{\ell}^{\prime}}$ (2)

The integral cross section (ICS) corresponding to a fixed orientation of the
incident flux ($\hat{k}_{\alpha}=\hat{0}$) may be obtained by integrating the
differential cross section
${d\sigma_{\alpha\to\alpha^{\prime}}(\hat{R})}/{d\Omega}={k_{\alpha}^{-2}}|q^{(0)}_{\alpha\to\alpha^{\prime}}(\hat{R})|^{2}$
over all angles. Substituting the scattering amplitude from Eq. (2) and
performing the integration, we obtain [43]

$\sigma_{\alpha\to\alpha^{\prime}}^{(0)}=\frac{\pi}{k_{\alpha}^{2}}\biggl{[}\sum_{\ell}\sum_{\ell^{\prime}m_{\ell}^{\prime}}(2\ell+1)|T_{\alpha\ell
0;\alpha^{\prime}\ell^{\prime}m_{\ell}^{\prime}}|^{2}\\\
+\sum_{\begin{subarray}{c}\ell_{1}\neq\ell_{2}\\\
\ell^{\prime},m_{\ell}^{\prime}\end{subarray}}[(2\ell_{1}+1)(2\ell_{2}+1)]^{1/2}{i}^{\ell_{2}-\ell_{1}}T^{*}_{\alpha\ell_{1}0;\alpha^{\prime}\ell^{\prime}m_{\ell}^{\prime}}T_{\alpha\ell_{2}0;\alpha^{\prime}\ell^{\prime}m_{\ell}^{\prime}}\biggr{]}.$
(3)

The first term on the right represents the incoherent contribution to the ICS,
which does not depend on the phases of $T$-matrix elements. The second term is
an interference term, which originates from fixing the direction of the
incident collision flux in Eq. (2), as required for the description of
molecular beam stereodynamics experiments [24, 25, 26, 27]. The “steric” ICS
defined by Eq. (3) is notably different from the conventional state-to-state
ICS $\sigma_{\alpha\to\alpha^{\prime}}=\pi
k_{\alpha}^{-2}\sum_{\ell,m_{\ell}}\sum_{\ell^{\prime},m_{\ell}^{\prime}}|T_{\alpha\ell
m_{\ell};\alpha^{\prime}\ell^{\prime}m_{\ell}^{\prime}}|^{2}$ obtained by
averaging the absolute square of the scattering amplitude (1) over
$\hat{k}_{\alpha}$ and integrating over $\hat{R}$ [42] leading to the
disappearance of interference terms. We note that the steric ICS (3) becomes
identical to the conventional ICS in the $s$-wave limit, where
$\ell_{1}=\ell_{2}=0$ [43] and both types of ICS obey the same threshold laws
for $s$-wave scattering [44]. We will omit the prefix “steric” when referring
to the ICS (3) unless necessary to avoid confusion.

The $T$-matrix elements in Eqs. (2)–(3) are obtained using numerically exact
quantum scattering methodology [45, 46] based on the accurate ab initio Ne-OH
PESs calculated as described in the Appendix. Unlike previous theoretical
studies [7, 10, 9, 11, 12], our calculations explicitly account for the
effects of external electric fields on quantum dynamics [45, 46] as required
for the proper theoretical description of cold atom-molecule collisions [47,
48].

##  III Results and discussion

Figure 1(a) shows the Stark energy levels of OH($X^{2}\Pi$) in its ground
vibronic state. The ground-state $J=3/2$ level is split by the spin-orbit
interaction between the ground and excited electronic states of OH into a
$\Lambda$-doublet consisting of two closely lying levels of opposite parity
[28, 49]. The low-field seeking $M$-components of the upper $f$-state
($|M|=1/2$ and $|M|=3/2$) increase in energy with increasing electric field.
The OH molecules residing in the $f$ states are oriented against the direction
of the applied electric field [7]. In contrast, the energy of the high-field-
seeking Stark sublevels of the lower $e$-state decreases with increasing field
as their dipole moments are oriented along the field direction [7].

In Fig. 1(b) we show the total inelastic ICS
$\sigma^{\text{inel}}_{i}=\sum_{k\neq i}\sigma_{i\to k}^{(0)}$ (hereafter
referred to simply as the ICS, with the ICS type explicitly indicated only
when necessary to avoid confusion) for the highest-energy low-field seeking
Stark state $|i\rangle=|f,M=3/2\rangle$ of OH($J=3/2$) as a function of
collision energy and electric field. We observe two pronounced resonance peaks
in the collision energy ($E_{\text{coll}}$) dependence of the ICS near $E=1$
kV/cm. These resonances are due to the trapping of the collision partners
behind centrifugal barriers in either the incoming or outgoing collision
channels [46]. At $E_{\text{coll}}>0.1$ cm-1, Ne + OH collisions occur in the
multiple partial-wave regime, where destructive interference between different
partial wave contributions washes out the resonance structure in the ICS
apparent at lower collision energies.

Figure 1: (a) Stark energy levels of OH(${}^{2}\Pi$) in the $J=3/2$ ground-
state manifold as a function of electric field [panel (a) is the inset in
panel (b)]. The initial states used in steric asymmetry computations are shown
by dashed lines. Panels (b) and (c) show the ICSs (in units of $a_{0}^{2}$)
for Ne + OH plotted vs. collision energy and electric field for the
$|f,M=3/2\rangle$ (b) and $|e,M=3/2\rangle$ (c) initial states of OH. The
zero-field $\Lambda$-doublet energy is marked by the vertical dashed line and
the dependence $\Delta E_{\Lambda}(E)$ is marked by the slanted line. The area
labeled “UR” indicates the universal threshold regime, where the inelastic ICS
$\sigma^{\text{inel}}_{f,M}$ is suppressed.

Figure 1(c) shows the ICS for the ground high-field-seeking state of OH
$|e,M=3/2\rangle$ colliding with Ne. At low collision energies
($E_{\text{coll}}<\Delta E_{\Lambda}$, where $\Delta E_{\Lambda}=0.055$ cm-1
is the $\Lambda$-doublet splitting energy) inelastic scattering of OH
molecules oriented along the electric field direction is strongly suppressed,
leading to the universal steric preference phenomenon considered below. The
origin of the suppression is that only three inelastic channels
($|e,M^{\prime}=-3/2\rangle$ and $|e,M^{\prime}=\pm 1/2\rangle$) remain open
at zero collision energy in the low field limit. As these channels have
$M^{\prime}\neq M$ inelastic scattering must be accompanied by a change in
$M$. According to the Krems-Dalgarno threshold laws [44] transitions with
nonzero $\Delta M$ scale with the collision energy as $E_{\text{coll}}^{\Delta
M}$ (for even $\Delta M$) and $E_{\text{coll}}^{\Delta M+1}$ (for odd $\Delta
M$). Thus, transitions with $\Delta M\neq 0$ will be suppressed in the
$s$-wave limit by centrifugal barriers in the outgoing collision channel.

At higher collision energies above the $\Lambda$-doublet threshold, inelastic
channels of opposite parity and $\Delta M=0$ (such as
$|f,M^{\prime}=3/2\rangle$) open up, causing a substantial increase of the ICS
and the appearance of near-threshold resonances. A total of 8 isolated
resonances occur over the range of collision energies 0.06-0.2 cm-1 below 9
kV/cm. Remarkably, the resonances survive not only in the few partial wave
regime ($E_{\text{coll}}<0.05$ cm-1) but also at higher collision energies.
This is due to the limited number of inelastic channels available for the
$|e,M=3/2\rangle$ initial state to decay into, resulting in a more pronounced
$S$-matrix pole structure compared to the $|f,M=3/2\rangle$ initial state
[50].

The resonance structure shown in Fig. 1(c) displays an interesting pattern,
shifting to higher collision energies with increasing field. This occurs due
to the field-induced repulsion between the opposite parity $e$ and $f$ states.
As a result of widening $\Lambda$-doublet energy gap, the minimum collision
energy required to access the inelastic channels in the $f$-manifold increases
with the $E$-field, shifting the onset of the resonance pattern to higher
collision energies.

Additionally, as seen in Fig. 1(c), increasing the electric field suppresses
inelastic scattering from the $|e,M=3/2\rangle$ state of OH: the resonance
maxima of the ICS become less pronounced at higher electric fields. This is
caused by the energy gap between the $|M|=1/2$ and $|M|=3/2$ states in the
$e$-manifold growing linearly with increasing field [see Fig. 1(a)] until the
$|M|=1/2$ channels become closed. The ICS for the single energetically allowed
transition $|e,M=3/2\rangle\to|e,M^{\prime}=-3/2\rangle$ scales as
$\sigma^{\text{inel}}\simeq E_{\text{coll}}^{\Delta M+1}=E_{\text{coll}}^{4}$
[44] and is thus strongly suppressed at ultralow collision energies. As shown
below, the suppression of the ICS is a universal trend, which manifests itself
in ultracold collisions of ${}^{2}\Pi$ molecules in the $|e,M=3/2\rangle$
initial state with spherically symmetric atoms. In contrast, no such trend
exists for the $|f,M=3/2\rangle$ initial state.

To gain additional insight into cold Ne + OH collision stereodynamics, we
calculate the steric asymmetry parameter [7, 10, 9, 11, 12]

$\mathcal{S}^{\text{inel}}=\frac{\sigma_{e,M}^{\text{inel}}-\sigma_{f,M}^{\text{inel}}}{\sigma_{e,M}^{\text{inel}}+\sigma_{f,M}^{\text{inel}}}$
(4)

where $\sigma_{i}^{\text{inel}}$ is the ICS for the $i$-th initial state of OH
aligned along and against the field axis (see above), $M=3/2$, and the $k$ sum
runs over all energetically accessible final channels. The steric asymmetry
measures the difference between the collisional properties of OH molecules
oriented along vs. against the field direction (with coincides with the
incident atom-molecule velocity vector). A value of
$\mathcal{S}^{\text{inel}}$ close to -1 (+1) indicates a strong stereodynamic
preference for Ne + OH collisions with OH oriented against (along) the field
axis. Experimental measurements and theoretical calculations of the steric
asymmetry have provided a wealth of valuable information about stereodynamic
effects in collisions of OH and NO molecules with rare-gas atoms at collision
energies of 300 K and above [7, 10, 9, 11, 12].  In particular, Schreel and
Meulen [8] and Beek et al. [9] measured and calculated the steric asymmetry of
rotationally inelastic collisions of OH($J=3/2,f$) molecules with He, Ar, and
H2 at collision energies $E_{\text{coll}}=394{-}746$ cm-1. They observed
steric asymmetries ranging from $\mathcal{S}^{\text{inel}}=-0.01$ to $-0.27$
depending on the collision energy and the rotational transition (the largest
asymmetry was observed for the $J=3/2,f\to 7/2,e$ transition in He + OH
collisions at $E_{\text{coll}}=394$ cm-1 [8].) Larger steric asymmetries (up
to $|\mathcal{S}^{\text{inel}}|=0.78$) were reported for rotationally
inelastic Ar + OH collisions at $E_{\text{coll}}=746$ cm-1 [9]. Interestingly,
the preference for the negative steric asymmetry observed in high-temperature
He + OH collisions [8] is also apparent in our Ne + OH calculations (see Fig.
2).

It should be noted, however, that previous experimental and theoretical work
on He + OH and Ar + OH collisions [8, 9] was performed at much higher
collision energies ($E_{\text{coll}}=394{-}746$ cm-1) and for a different type
of inelastic process (rotationally inelastic collisions) than explored here.
Under the conditions of Refs. [8, 9] the stereodynamical properties of atom-
molecule collisions are determined by contributions from many partial waves.
Because of these multiple competing contributions, the physical mechanisms
responsible for, e.g., the negative sign of the steric asymmetry in high-
temperature He + OH collisions [8, 9] are significantly more challenging to
identify and interpret than the low-temperature trends explored here.

Figure 2: (a) Steric asymmetry (4) for inelastic Ne + OH collisions as a
function of collision energy and electric field computed using the present Ne-
OH PESs (a) and the PESs from Ref. [51] (b) [see the Appendix for details].
The $\Lambda$-doublet splitting energy of OH is marked by the vertical dashed
line ($E=0$) and by the sloped dashed line (as a function of $E$).

Figure 2(a) is a two-dimensional map of the steric asymmetry for inelastic Ne
+ OH collisions plotted as a function of collision energy and electric field.
Two distinct regions may be observed in the map, which we will refer to as the
universal region and the resonance region. The universal region corresponds to
the regime $E_{\text{coll}}<\Delta E_{\Lambda}$, where the upper components of
the $\Lambda$-doublet are closed, and inelastic scattering from the lowest
low-field-seeking state $|e,J=3/2,M=3/2\rangle$ of OH is strongly suppressed
by the threshold laws for $M$-changing collisions as noted above. Thus, in the
universal region, $\sigma_{f,M}^{\text{inel}}\gg\sigma_{e,M}^{\text{inel}}$
and $\mathcal{S}^{\text{inel}}\simeq-1$. This remarkable trend in only
apparent in the inelastic ICSs as further discussed in the Appendix.

Significantly, the universal suppression of inelastic scattering persists at
nonzero electric fields because the $M=3/2$ and $M=-3/2$ magnetic sublevels
remain degenerate, and thus zero-field Krems-Dalgarno threshold laws [44]
remain applicable to the $|e,M=3/2\rangle\to|e,M=-3/2\rangle$ transition. The
degeneracy can be lifted by an external magnetic field, which is expected to
break the universal behavior of the steric asymmetry, leading to a rapid
increase of $\mathcal{S}^{\text{inel}}$ as the $s$-wave threshold scaling of
the $M$-changing ICS changes from $~{}E_{\text{coll}}^{4}$ to
$E_{\text{coll}}^{-1/2}$.

In the resonance regime defined by the condition $E_{\text{coll}}>\Delta
E_{\Lambda}$ excitation transitions occur from the initial $e$-state, such as
$|e,M=3/2\to|e,M=\pm 1/2\rangle$ and $|e,M=3/2\to|f,M=\pm 1/2\rangle$. While
these are also $M$-changing transitions, they are not so strongly suppressed
compared to the $|e,M=3/2\rangle\to|e,M=-3/2\rangle$ transition due to their
smaller $\Delta M$. As s result, the background value of the ICS increases and
so does the steric asymmetry. In addition, scattering resonances begin to
appear near and above the excitation thresholds leading to distinct spikes in
$\sigma_{e,M}^{\text{inel}}$ [see Fig. 1(c)]. As noted above and seen in Fig.
1(b), $\sigma_{f,M}^{\text{inel}}$ does not vary strongly with either
collision energy or electric field in the resonant regime. Taken together,
these factors cause the appearance of the resonance peaks in the steric
asymmetry in Fig. 2(a). As the details of the resonance structure are
sensitive to the underlying PESs (a well-documented phenomenon in cold
molecular collisions [50, 52, 53]), the resonance regime can also be regarded
as nonuniversal. The significance of the nonuniversal behavior is that it
allows one to control the stereodynamics of cold atom-molecule collisions by
applying an external electric field. For example, as shown in Fig. 2, by
varying the external electric field between zero and 10 kV/cm (an interval
well within the reach of modern experimental capabilities), it is possible to
tune the steric asymmetry over a substantial range from $-1$ to 0.4,
essentially reversing the stereodynamical preference of inelastic Ne + OH
collisions. The nonuniversal behavior could be observed as a function of
collision energy and external fields in merged beam experiments with state-
selected OH molecules and Ne atoms as was done for cold He∗ \+ H2 and Ne∗ \+
Ar collisions [40, 41].

Figure 3: (a) Electric field dependence of state-to-state ICSs
$\sigma_{f,M=3/2\to k}$ [panels (a), (b)] and $\sigma_{e,M=3/2\to k}$ [panels
(c), (d)] for Ne + OH collisions. The collision energy is 0.02 cm-1 [panels
(a), (c)] and 0.2 cm-1 [panels (b), (d)]. The ICSs summed over all final $k$
are shown by the top traces.

To illustrate the distinction between the universal vs. nonuniversal regimes,
we plot in Fig. 2(b) the steric asymmetry calculated using a different set of
Ne-OH interaction PESs [51]. While the differences between the PESs are small
(see the Appendix), they have a dramatic effect on the resonance structure at
$E_{\text{coll}}\geq\Delta E_{\Lambda}$, as expected in the nonuniversal
regime [50, 52, 53].

Remarkably, by comparing Fig. 2(a) and Fig. 2(b) we observe that the steric
asymmetries calculated using the different PESs are in nearly perfect
agreement with each other at $E_{\text{coll}}<\Delta E_{\Lambda}$
($\mathcal{S}^{\text{inel}}\simeq-1$). This independence is a clear signature
of the universal regime, where inelastic scattering of OH($|e,M=3/2\rangle$)
is strongly suppressed due to the threshold effects [44] (see above).

In Figs. 2(a) and 2(b) we observe small deviations from the universal behavior
at $E_{\text{coll}}\simeq\Delta E_{\Lambda}$. To understand the origin of
these deviations, we plot in Fig. 3 the state-to-state ICSs
$\sigma_{f,M=3/2\to k}$ and $\sigma_{e,M=3/2\to k}$, which define the steric
asymmetry (4). Below the $\Lambda$-doublet threshold the electric field
dependence of $\sigma_{f,M=3/2\to k}$ is determined by isolated shape
resonances with the dominant contribution due to the $|e,M=3/2\rangle$ final
state. At higher electric fields and/or collision energies, the resonances
broaden and begin to overlap, leading to the disappearance of distinct peaks
in the total ICS [46]. We note that the resonance structure in the state-to-
state ICS can survive at collision energies as high as 0.2 cm-1, as
illustrated in Fig. 3(b) for the final state $|e,M=1/2\rangle$.

As shown in Figs. 3(c) and 3(d), the ICSs $\sigma_{e,M=3/2\to k}$ increase by
2-4 orders of magnitude with increasing collision energy by a factor of 10,
which is consistent with their $E_{\text{coll}}^{\Delta M}$ threshold scaling
discussed above [44]. In contrast, the ICSs $\sigma_{f,M=3/2\to k}$ decrease
due to their different threshold scaling $\simeq E_{\text{coll}}^{-1/2}$. We
verified that the deviations from the perfect universal scaling
($\mathcal{S}^{\text{inel}}=-1$) occur due to the small contributions of the
$\ell\geq 1$ partial waves to the ICS $\sigma_{e,M=3/2\to k}$ at
$E_{\text{coll}}<\Delta E_{\Lambda}$ [see Fig. 3(c)]. At lower collision
energies, these contributions freeze out as $E_{\text{coll}}^{4}$ and the
universal relation $\mathcal{S}^{\text{inel}}=-1$ holds to an increasingly
better accuracy.

##  IV Summary and outlook

In summary, we have established a universal stereodynamical trend in cold
collisions of ${}^{2}\Pi$ molecular radicals with ${}^{1}\text{S}_{0}$-state
atoms in an external electric field. Using rigorous quantum scattering
calculations based on highly accurate ab initio interaction potentials, we
show that the steric anisotropy of Ne + OH collisions approaches $-1$ in the
limit of zero collision energy due to a suppression of $M$-changing
transitions from the $|e,M=3/2\rangle$ initial state, in which the dipole
moment of OH is oriented along the field direction. The suppression occurs
universally in the $s$-wave threshold regime, where the $M$-changing cross
sections vanish [44], and it persists at collision energies below the
$\Lambda$-doublet energy $\Delta E_{\Lambda}$ regardless of the magnitude of
the applied electric field [see Fig. 2]. Above the $\Lambda$-doublet energy,
nearly perfect stereoselectivity is lost and scattering occurs in the
nonuniversal resonant regime, where extensive control is possible over
collision stereodynamics via electric field-induced resonances. Remarkably,
the resonances survive at collision energies as high as 0.2 cm-1 [see Fig. 2],
which can be realized in merged molecular beams [40, 41], making our
predictions verifiable in current cold molecule experiments.

## Acknowledgements

We are grateful to Jun Ye, Gerrit Groenenboom, Hao Wu, David Reens, and Piotr
Wcisło for stimulating discussions, and to Yoshihiro Sumiyoshi for sharing the
Ne-OH PESs [51]. This work was supported by the NSF EPSCoR RII Track-4
Fellowship (Award No. 1929190).

## Appendix A: Ab initio calculations of Potential Energy Surfaces

In this Appendix we provide the details of our ab initio calculations of Ne-
OH(${}^{2}\Pi$) interaction potentials. Our quantum scattering calculations
based on these potentials are described in Sec. B below.

We performed electronic structure calculations of the adiabatic potential
energy surfaces (PESs) for the Ne-OH(X${}^{2}\Pi$) van der Waals complex at
the explicitly correlated coupled cluster level of theory. Due to the fact
that the OH molecule is a spin doublet radical, we used the spin-restricted
version of the coupled cluster method with single, double and perturbative
triple corrections (RCCSD(T)-F12a) using the F12a ansatz with scaled triples
corrections as implemented in the MOLPRO program [54]. The singly occupied
molecular orbital of OH(X${}^{2}\Pi$) splits its degeneracy upon approach of
the Ne atom to form the electronic states of the Ne-OH complex. The electronic
wavefunctions of the complex transform according to either $A^{\prime}$ or
$A^{\prime\prime}$ irreducible representations of the Cs symmetry group, which
are either symmetric or antisymmetric with respect to reflection in the
triatomic plane (for non-collinear geometries) and according to the $\Pi$
representation of the C∞v point group in collinear geometries.

The initial single reference wavefunctions are obtained from spin-restricted
Hartree-Fock (RHF) calculations, which control the corresponding $A^{\prime}$
or $A^{\prime\prime}$ symmetries of the wavefunctions of the Ne-OH dimer and
OH monomer for subsequent RCCSD(T)-F12 calculations. We use augmented
correlation-consistent triple-zeta atomic basis sets (aug-cc-pVTZ) [55] with
automatically generated auxiliary density fitting sets for the explicitly-
correlated part of the calculations. We additionally augment the basis with
mid-bond functions that improve the description of the dispersion-bound
complexes such as Ne-OH. The mid-bond functions are placed at the half-
distance between the Ne and the center of mass of OH and composed of
3s3p2d2f2g functions with the following exponents: $sp$ 0.9, 0.3, 0.1 and $df$
0.6, 0.2. The threshold for convergence of the total energies was set to 10-10
Eh.

We calculated the $A^{\prime}$ and $A^{\prime\prime}$ adiabatic PESs on a
discrete grid of Jacobi coordinates $(R,\theta)$ with the OH distance being
frozen at $r_{0}=1.8509$ $a_{0}$, the average internuclear distance of the
ground vibrational state. The Jacobi radius $R$ is the distance of the Ne atom
from the center of mass of OH calculated with the masses of the most stable
isotopes of O and H. The Jacobi angle $\theta$ describes the anisotropy of the
PESs originating from the rotation of the Ne atom around the OH molecule (with
$\theta=0^{\circ}$ corresponding to the collinear Ne–H-O arrangement). The
discrete grid was composed of 52 radial points for the $R$ variable and 19
angular $\theta$ points for the total of 988 points for each of the
$A^{\prime}$ and $A^{\prime}$ PESs. The radial grid covered distances from 2.5
$a_{0}$ to 500 $a_{0}$ with variable step sizes $\Delta R=$ 0.1, 0.25, 1.0,
2.0, and larger). The $\theta$ grid was set up from $0^{\circ}$ to
$180^{\circ}$ every 10∘. The largest radial distance of 500 $a_{0}$ was used
to correct the RCCSD(T)-F12a interaction energies for the size consistency
error, which arises due to the fact that the triple (T) corrections are not
explicitly correlated.

In order to perform scattering calculations we need to represent the discrete
set of interaction PESs in the form of analytical functions of the Jacobi
coordinates $R$ and $\theta$. To this end, we fit the angular dependence of
the diabatic half-sum and half-difference PESs $V_{\text{sum}}$ and
$V_{\text{diff}}$ given by Eq. (Appendix A: Ab initio calculations of
Potential Energy Surfaces) to a series of reduced Wigner functions $d_{\lambda
0}(\cos\theta)$ and $d_{\lambda 2}(\cos\theta)$, respectively. The resulting
radial coefficients $V_{\lambda\mu}(R)$, $\mu=0,2$ were fitted using the
reproducing kernel Hilbert space method of Rabitz and coworkers [56] with the
kernel describing dispersion-like terms for extrapolation. The resulting fit
was smoothly joined with a long-range analytical expression based on the
$C_{n}$ dispersion coefficients ($n=6{-}9$) obtained in a separate fit of the
long-range part of the potential.

Figures 4(a) and 4(b) show the half-difference and half-sum diabatic PES of
the Ne-OH complex,
$V_{\text{diff}}(R,\theta)=\frac{1}{2}(V_{A^{\prime\prime}}-V_{A^{\prime}})$
and
$V_{\text{sum}}(R,\theta)=\frac{1}{2}(V_{A^{\prime\prime}}+V_{A^{\prime}})$ as
a function of $R$ and $\theta$. The global minimum of the PESs is found in the
collinear Ne–H-O geometry at $\theta_{e}=0^{\circ}$, $R_{e}=6.57$ $a_{0}$ and
$D_{e}=60.026$ cm-1. There is additionally a local minimum on the $A^{\prime}$
PES, which occurs in a slightly skewed geometry at $\theta=66^{\circ}$,
$R=5.85$ $a_{0}$, and $D_{e}=59.804$ cm-1. The collinear Ne–O-H minimum at
$\theta=180^{\circ}$ is shallower with a well depth of 45.156 cm-1 at $R=6.14$
$a_{0}$. This critical point is a minimum on the half-sum PES $V_{\text{sum}}$
and on the $A^{\prime\prime}$ adiabatic PES, but it is a saddle point on the
$A^{\prime}$ adiabatic PES. The T-shape region of the half-sum PES is
characterized by a saddle point at $R=6.22$ $a_{0}$, $\theta=102^{\circ}$ and
$D_{e}=-36.88$ cm-1. This suggests that the Ne-OH complex has a fairly floppy
structure with the Ne atom easily delocalized between the two collinear
minima. The half-difference PES $V_{\text{diff}}$ is mostly repulsive with a
shallow ($D_{e}\simeq 1$ cm-1) long-range minimum, as shown in Fig 4(b).

Figure 4: Contour plots of the half-difference (a) and half-sum (b) Ne-OH PESs
calculated in this work as a function of the Jacobi coordinates $R$ and
$\theta$ (in units of cm-1). (c)-(d) Radial dependence of the half-difference
and half-sum PESs $V_{\text{diff}}(R,\theta)$ and $V_{\text{sum}}(R,\theta)$
calculated in this work (full lines) and computed by Sumiyoshi et al. [51]
(dashed lines). The corresponding values of $\theta$ are indicated in degrees
next to each curve.

Figures 4(c) and (d) compare the radial dependence of our ab initio Ne-OH PESs
with those computed by Sumiyoshi et al. [51]. While the PESs are overall very
similar, small differences remain in the well depth and its location. In
particular, the global minimum of our half-sum PES ($D_{e}=60.026$ cm-1) is
slightly deeper that that of the Ne-OH PES ($D_{e}=59.4$ cm-1) determined by
Sumiyoshi et al. [51]. This difference may originate from the slightly
different values of the OH interatomic distance $r$ used in our calculations
vs those of Ref. [51]. The latter used the equilibrium distance of OH
($r=r_{e}$), whereas we use the expectation value of the internuclear distance
$r_{0}=\langle{v=0}|r|{v=0}\rangle$ in the ground rovibrational state of OH.
Other numerical uncertainty factors resulting from the fitting procedures, and
using the spin-unrestricted coupled-cluster approach by Sumiyoshi et al.,
could also contribute to the discrepancies. The most sensitive to the level of
ab initio calculations is the half-difference PES shown in comparison to the
Sumiyoshi et al.’s PES in Fig 4(c). The long-range shallow minimum in our
half-difference PES is more attractive in the T-shaped Ne-OH geometry. These
differences, albeit small, have a dramatic effect on cold Ne + OH scattering
in the resonant regime as discussed in the main text.

To represent the half-sum and half-difference PESs in a form suitable for
quantum scattering calculations, we expand them in angular basis functions
[57, 45]

$\displaystyle V_{\text{sum}}(R,\theta)$
$\displaystyle=\frac{1}{2}(V_{A^{\prime\prime}}+V_{A^{\prime}})=\sum_{\lambda=0}^{\lambda_{\text{max}}}V_{\lambda
0}(R)P_{\lambda}(\cos\theta),$ $\displaystyle V_{\text{diff}}(R,\theta)$
$\displaystyle=\frac{1}{2}(V_{A^{\prime\prime}}-V_{A^{\prime}})=\sum_{\lambda=2}^{\lambda_{\text{max}}}V_{\lambda
2}(R)d_{02}^{\lambda}(\cos\theta),$ (5)

where $P_{\lambda}(\cos\theta)$ are the Legendre polynomials, and
$d_{02}^{\lambda}(\cos\theta)$ are the reduced Wigner D-functions.

The radial coefficients $V_{\lambda 0}(R)$ and $V_{\lambda 2}(R)$ are
evaluated using a 30-point Gauss-Legendre quadrature, and the expansions
(Appendix A: Ab initio calculations of Potential Energy Surfaces) are
truncated at $\lambda_{\text{max}}=17$ to provide converged matrix elements of
the interaction potential [45].

## Appendix B: Quantum scattering calculations

The Hamiltonian of the Ne-OH collision complex is (in atomic units) [57, 45]

$\hat{H}=-\frac{1}{2\mu R}\frac{\partial^{2}}{\partial
R^{2}}R+\frac{\hat{L}^{2}}{2\mu
R^{2}}+\hat{V}(R,r,\theta)+\hat{H}_{\text{mol}},$ (6)

where $\mathbf{R}$ is the vector pointing from Ne to the center of mass of OH,
$\mathbf{r}$ is the internuclear separation vector in OH, and $\theta$ is the
angle between $\mathbf{R}$ and $\mathbf{r}$. Further, $\mu$ and $\hat{L}^{2}$
are the reduced mass and the orbital angular momentum for the collision, and

$\hat{H}_{\text{mol}}=\hat{H}_{\text{rot}}+\hat{H}_{\text{SO}}+\hat{H}_{\Lambda}+\hat{H}_{\text{E}}+\hat{H}_{\text{B}}$
(7)

is the molecular Hamiltonian consisting of the rotational, spin-orbit,
$\Lambda$-doubling, Stark, and Zeeman terms [45]. To solve the Schrödinger
equation $\hat{H}|\Psi\rangle=E|\Psi\rangle$, we use the coupled-channel (CC)
expansion of the wavefunction of the collision complex

$|\Psi\rangle=\frac{1}{R}\sum_{\beta,\ell m_{\ell}}F_{\beta\ell
m_{\ell}}(R)|\beta\rangle|\ell m_{\ell}\rangle,$ (8)

where $|\ell m_{\ell}\rangle$ are the eigenstates of $\hat{L}^{2}$ and
$\hat{L}_{Z}$, and $\beta=|JM\Omega\Lambda\Sigma\rangle$ are Hund’s case (a)
basis functions for the internal degrees of freedom of OH [57]. These
functions depend on the total angular momentum of the molecule $J$ and its
projections on the space-fixed ($M$) and molecule-fixed ($\Omega$)
quantization axes, and well as on the electronic basis functions
$|\Lambda\Sigma\rangle$. The CC expansion leads to a system of CC equations
[45] for the radial solutions $F_{\beta\ell m_{\ell}}(R)$

$\left[\frac{d^{2}}{dR^{2}}+2\mu E\right]F_{\beta\ell
m_{\ell}}(R)=2\mu\sum_{\beta^{\prime}\ell^{\prime}m_{\ell}^{\prime}}\langle\beta\ell
m_{\ell}|\hat{V}(R,\theta,r)\\\ +\frac{\hat{L}^{2}}{2\mu
R^{2}}+\hat{H}_{\text{mol}}|\beta^{\prime}\ell^{\prime}m_{\ell}^{\prime}\rangle
F_{\beta^{\prime}\ell^{\prime}m_{\ell}^{\prime}}(R).$ (9)

which is integrated numerically using the log-derivative algorithm [58] to
produce the radial solutions $F_{\beta\ell m_{\ell}}(R)$. The application of
the scattering boundary conditions to the $F_{\beta\ell m_{\ell}}(R)$ yields
the $S$ and $T$-matrix elements as a function of collision energy and electric
field [45, 46]. The $T$-matrix elements are used to calculate the scattering
amplitude [Eq. (2) of the main text], from which the ICS are computed by
integration over the spherical polar angles of vector $\hat{R}$ ($\theta_{R}$
and $\phi_{R}$) using two-dimensional Gauss-Legendre quadratures. We verified
that the resulting ICSs are the same as given by Eq. (3) of the main text.

The matrix elements of the Hamiltonian in Eq. (9) are evaluated as described
in our previous work [45]. The matrix elements of the molecular Hamiltonian
$\hat{H}_{\text{mol}}$ are parametrized by the spectroscopic constants of
OH$(^{2}\Pi)$ [28, 49]. The Ne-OH interaction is described by two adiabatic
potential energy surfaces (PES) of $A^{\prime}$ and $A^{\prime\prime}$
symmetries [57, 45] (see above).

The CC equations (9) are integrated numerically using the modified log-
derivative method [58] on the radial grid extending from
$R_{\text{min}}=2.5\,a_{0}$ to $R_{\text{max}}=75.02\,a_{0}$ with a grid step
of $\Delta R=0.02,a_{0}$ (for collision energies below 0.1 cm-1). At higher
collision energies, we use $R_{\text{max}}=50.0\,a_{0}$ and a grid step of
$\Delta R=0.05\,a_{0}$. The CC basis set included all total angular momentum
states of OH with $J\leq 4.5$ and partial waves with $\ell\leq 7{-}9$
depending on the collision energy, and the following masses of collision
partners were used: $m_{\text{Ne}}=19.9924401754$ and
$m_{\text{OH}}=17.002739$ amu. The spectroscopic constants of OH used to
parametrize the molecular Hamiltonian (7) are the same as in our previous work
on cold He + OH collisions [45, 46]. These references also contain further
details regarding the evaluation of the matrix elements of the interaction
potential starting from Eq. (Appendix A: Ab initio calculations of Potential
Energy Surfaces).

Figure 5: ICSs for the $f\to e$ transition in Ne + OH($f,M=3/2)$ collisions
summed over all $M^{\prime}$ states in the $e$ manifold vs. collision energy
calculated using the HIBRIDON suite of scattering codes by Alexander et al.
[59] and the code developed in Ref. [45] (full line) at zero electric field.

At $R=R_{\text{max}}$ the log-derivative matrix is transformed to a basis,
which diagonalizes the asymptotic Hamiltonian [45] and then matched to the
Riccati-Bessel functions and their derivatives [60, 45] to yield the
$S$-matrix elements, from which the scattering cross sections are obtained
using standard expressions [60]. The calculated scattering cross sections are
converged to $<10\%$ with respect to the basis set and radial grid parameters.

To benchmark our quantum scattering calculations, we compared the ICSs
calculated using an in-house 1S atom + OH quantum scattering code developed in
Ref. [45] with those computed using the publicly available general-purpose
scattering code HIBRIDON [59]. Figure 5 shows the ICSs for Ne +
OH$(f,J=3/2,M=3/2)$ collisions as a function of collision energy at zero
electric field. We observe excellent overall agreement, including near the
low-energy scattering resonances, which serves as a verification of our
quantum scattering calculations.

## Appendix C: Steric asymmetry for elastic scattering

Figure 6 shows that elastic steric asymmetry defined as

$\mathcal{S}^{\text{el}}=\frac{\sigma_{e,M}^{\text{el}}-\sigma_{f,M}^{\text{el}}}{\sigma_{e,M}^{\text{el}}+\sigma_{f,M}^{\text{el}}}$
(10)

for $M=3/2$. Similarly to the inelastic steric asymmetry, this quantity
measures the difference between the magnitude of the elastic ICSs for OH
molecules colliding along vs. against the field direction.

We observe that unlike its inelastic counterpart [see Eq. (4) of the main
text] the elastic steric asymmetry is highly sensitive to the details of the
PESs used in scattering calculations over the whole range of collision
energies and electric fields. This occurs due to the absence of the threshold
suppression effects in elastic scattering. As a result, no universal regime
exists for the stereodynamics of elastic atom-molecule collisions at low
temperatures.

Figure 6: Steric asymmetry calculated from the elastic ICSs using the present
Ne-OH PESs (left panel) and the PES of Ref. [51] (right panel).

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