# Multichannel quantum defect theory with a frame transformation
for ultracold molecular collisions in magnetic fields

Masato Morita1, Paul Brumer1, and Timur V. Tscherbul2 1Chemical Physics Theory
Group, Department of Chemistry, and Center for Quantum Information and Quantum
Control, University of Toronto, Toronto, Ontario, M5S 3H6, Canada  2Department
of Physics, University of Nevada, Reno, Nevada, 89557, USA  1Chemical Physics
Theory Group, Department of Chemistry, and Center for Quantum Information and
Quantum Control, University of Toronto, Toronto, Ontario, M5S 3H6, Canada
2Department of Physics, University of Nevada, Reno, NV, 89557, USA

###### Abstract

We extend the powerful formalism of multichannel quantum defect theory
combined with a frame transformation (MQDT-FT) to ultracold molecular
collisions in magnetic fields. By solving the coupled-channel equations with
hyperfine and Zeeman interactions omitted at short range, MQDT-FT enables a
drastically simplified description of the intricate quantum dynamics of
ultracold molecular collisions in terms of a small number of short-range
parameters. We apply the formalism to ultracold Mg + NH collisions in a
magnetic field, achieving a 104-fold reduction in computational effort.

Recent advances in cooling and trapping diatomic and polyatomic molecules have
established ultracold molecular gases as an emerging platform for quantum
information science, ultracold chemistry, and precision searches for new
physics beyond Standard Model [1, 2, 3, 4]. The exquisite control over
molecular degrees of freedom achieved in these experiments enables the
exploration of novel regimes of ultracold chemical dynamics tunable by
external electromagnetic fields [5, 6, 7]. However, the use of molecules for
these applications has been hindered by rapid losses observed in ultracold
molecular gases due to the formation of intermediate complexes in two-body
collisions [8, 9, 10, 11, 12, 13, 14, 15]. Therefore, understanding and
controlling ultracold molecular collisions have long been recognized as major
goals in the field [1, 3]. This, however, has proven to be an elusive task due
to the enormous number of molecular states (rotational, vibrational, fine, and
hyperfine) strongly coupled by highly anisotropic interactions and external
fields, which results in skyrocketing computational costs of rigorous coupled
channel (CC) calculations on ultracold molecular collisions [16, 17]. While
efficient basis sets have been developed to mitigate this problem [18, 19, 20,
17, 21], fully converged CC calculations including all degrees of freedom are
yet to be reported on ultracold K + NaK [22, 23], Na + NaLi [24, 25], and Rb +
KRb [14] collisions explored in recent experiments.

A promising avenue toward resolving these difficulties could be based on
multichannel quantum defect theory (MQDT), an elegant technique for solving
the time-independent Schrödinger equation based on the separation of distance
and energy scales in ultracold collisions [26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37]. MQDT allows one to avoid the costly numerical procedure of
solving CC equations over extended ranges of the radial coordinate $R$,
collision energy, and magnetic field, leading to a substantial reduction of
computational cost [33, 34, 35, 36, 37, 38, 39, 40]. However, in conventional
MQDT, it is still necessary to solve CC equations in the short-range collision
complex region, which is forbiddingly difficult for the atom-molecule systems
interacting via deep and strongly anisotropic potentials, such as those
studied in recent experiments [22, 23, 24, 25, 14].

MQDT becomes especially powerful when combined with a frame transformation
(FT) approach, in which the hyperfine-Zeeman structure of colliding atoms is
neglected at short range. This results in a large reduction of computational
effort and provides a physically meaningful description of ultracold atomic
collisions in terms of a few short-range parameters [38, 39, 40, 41, 42, 43,
44]. An extension of MQDT-FT to the much more computationally intensive (and
less well understood) ultracold molecular collisions would thus be highly
desirable. Previous theoretical work has shown that MQDT can be successfully
applied to describe ultracold atom-molecule collisions [33, 34, 35, 36] and
chemical reactions [37], but these calculations did not consider the essential
FT aspect of MQDT, which makes it such an indispensable tool in modern
ultracold atomic collision theory [38, 39, 40, 41, 42, 43, 44].

Here, we show that the MQDT-FT approach can be extended to ultracold molecular
collisions in a magnetic field, enabling one to drastically simplify their
rigorous theoretical description. This is achieved by using compact CC basis
sets at short range, which exclude the hyperfine and Zeeman interactions.
These interactions are incorporated at long range via MQDT-FT boundary
conditions, resulting in a complete description of ultracold atom-molecule
collision dynamics across large collision energy and magnetic field ranges in
terms of a small number of short-range parameters. These parameters can be
used to, e.g., fit experimental observations and to obtain insight into
complex molecular collision dynamics without performing expensive CC
calculations. Our results show the potential of MQDT-FT to significantly
extend the scope of ultracold atom-molecule collisions and chemical reactions
amenable to rigorous quantum dynamical studies, which would facilitate the
accurate characterization of atom-molecule Feshbach resonances observed in
recent pioneering experiments [22, 23, 24, 25], as well as new insights into
quantum chaotic behavior and microscopic interactions in atom-molecule
collision complexes [45].

Theory. We begin by briefly reviewing the key concepts of MQDT as they apply
to ultracold atom-molecule collisions in a magnetic field. The Hamiltonian of
the atom-molecule collision complex is (in atomic units) [46]

${\hat{H}=-\frac{1}{2\mu R}\frac{\partial^{2}}{\partial
R^{2}}R+\frac{\hat{\mathbf{l}}^{2}}{2\mu
R^{2}}+\hat{H}_{\text{as}}+\hat{V}(R,\theta)}.$ (1)

Below we will use ultracold Mg(1S) + NH($\tilde{X}^{3}\Sigma^{-}$) collisions
as a representative example, which served as a testbed for applying MQDT to
ultracold atom-molecule collisions [33]. We note, however, that our treatment
can be readily generalized to collisions of atoms and molecules with a more
complex internal structure, and to molecule-molecule collisions. In Eq. (1),
$\mu$ and $\hat{\mathbf{l}}$ are the reduced mass and the orbital angular
momentum for the collision, $R$ is the atom-molecule distance, and
$\hat{H}_{\mathrm{as}}$ is the asymptotic Hamiltonian

$\hat{H}_{\text{as}}=B_{e}\,\hat{\mathbf{N}}^{2}+\hat{H}_{\text{fs}}+\hat{H}_{\text{hfs}}+\hat{H}_{\text{Z}},$
(2)

where $B_{e}$ is the rotational constant, $\hat{\mathbf{N}}$ is the rotational
angular momentum of the molecule,
$\hat{H}_{\text{fs}}=\gamma_{\mathrm{sr}}\,\hat{\mathbf{N}}\cdot\hat{\mathbf{S}}+\frac{2}{3}\lambda_{\mathrm{ss}}\sqrt{\frac{24\pi}{5}}\sum_{q}(-1)^{q}\,Y_{2\,-q}(\hat{\bm{r}})[\hat{\mathbf{S}}\otimes\hat{\mathbf{S}}]_{q}^{(2)}$
is the intramolecular fine-structure Hamiltonian, which depends on the
electron spin $\hat{\mathbf{S}}$ and the orientation of the molecular axis
$\bm{r}$, and $\gamma_{\text{sr}}$ and $\lambda_{\text{ss}}$ are the spin-
rotation and spin-spin interaction constants [47, 48]. The Zeeman interaction
$\hat{H}_{\text{Z}}=g_{S}\mu_{\mathrm{B}}\mathbf{B}\cdot\hat{\mathbf{S}}$,
where $g_{S}\simeq 2.002$ is the electron $g$-factor, $\mu_{\mathrm{B}}$ is
the Bohr magneton, the magnetic field vector $\mathbf{B}$ defines the space-
fixed quantization axis $z$, and the hyperfine interaction
$\hat{H}_{\text{hfs}}$ is considered below. We use the accurate ab initio
interaction potential $V(R,\theta)$ for Mg-NH [49], which depends on $R$ and
the atom-molecule Jacobi angle $\theta$ (see the Supplemental Material for
details of CC and MQDT-FT calculations [50]).

In MQDT, the matrix solution $\bm{\Psi}$ of the Schrödinger equation
$\hat{H}\bm{\Psi}=E\bm{\Psi}$ in the basis of eigenvectors of the asymptotic
Hamiltonian (2) is given by [31, 33]

$\bm{\Psi}=R_{m}^{-1}\,[\bm{f}(R_{m})+\bm{g}(R_{m})\,\bm{K}^{\text{sr}}\,],$
(3)

where the matching radius $R=R_{m}$ marks the boundary between the short-range
and long-range regions, $E$ is the total energy, $\bm{K}^{\text{sr}}$ is the
short-range $K$-matrix, which includes open and weakly closed channels, and
$\bm{f}(R)$ and $\bm{g}(R)$ are diagonal matrices of regular and irregular
solutions of the one-dimensional radial Schrödinger equations

$\left[\,-\frac{1}{2\mu}\frac{\partial^{2}}{\partial
R^{2}}+U_{i}^{\text{ref}}(R)\,\right]\,\psi_{i}(R)\,=\,E\,\psi_{i}(R).$ (4)

Here, $U_{i}^{\text{ref}}(R)\,=\,V_{0}(R)+{l_{i}(l_{i}+1)}/{(2\mu
R^{2})}+E_{i}^{\infty}$ is the MQDT reference potential [33, 50], for the
$i$-th reference channel with threshold energy $E_{i}^{\infty}$ and orbital
angular momentum $l_{i}$, and $V_{0}(R)$ is the isotropic part of the
interaction potential.

Having defined the short-range $K$-matrix, we take into account the coupling
between the open and weakly closed channels by forming the matrix

$\bar{\bm{K}}^{\text{sr}}=\bm{K}_{\mathrm{o,o}}^{\text{sr}}-\bm{K}_{\mathrm{o,wc}}^{\text{sr}}\,[\,\tan{\bm{\nu}}+\bm{K}_{\mathrm{wc,wc}}^{\text{sr}}\,]^{-1}\,\bm{K}_{\mathrm{wc,o}}^{\text{sr}},$
(5)

where $\tan{\bm{\nu}}$ is a diagonal matrix with elements $\tan\nu_{i}$ for
weakly closed channels [27, 31, 33]. The zeros of the bound-state phase
($\tan\nu_{i}=0$) locate the bound state energy in the $i$-th reference
potential and $\bm{K}_{\mathrm{wc,wc}}^{\text{sr}}$ introduces shifts in the
resonance positions [31]. We neglect the strongly closed channels, whose
coupling with open and weakly closed channels has no effect on scattering
dynamics [33, 34, 50]. Here, we define strongly closed channels as those, for
which $U_{i}^{\text{ref}}(R)>E$ for all $R$ [50].

Finally, we take into account the threshold effects using the diagonal
matrices of MQDT parameters $\bm{C}$ and $\tan{\bm{\lambda}}$ for open
channels to obtain the $\bar{\bm{R}}$ matrix

$\bar{\bm{R}}=\bm{C}^{-1}\,[\,({\bar{\bm{K}}^{\text{sr}}})^{-1}-\tan{\bm{\lambda}}\,]^{-1}\,\bm{C}^{-1},$
(6)

where $C_{i}$ determines the amplitude of the reference functions and
$\lambda_{i}$ is the phase difference between the long-range and short-range
irregular solutions [32]. Finally, the transition $T$-matrix may be written as

$\bm{T}=\bm{I}-e^{i\bm{\xi}}\,[\,\bm{I}+i\bar{\bm{R}}\,]\,[\,\bm{I}-i\bar{\bm{R}}\,]^{-1}\,e^{i\bm{\xi}},$
(7)

where $\bm{I}$ is the unit matrix and $\bm{\xi}$ is the diagonal matrix of
phase shifts $\xi_{i}$ associated with the energy-normalized reference
functions [27, 31, 32, 33, 34]. The state-to-state integral cross sections are
obtained from the transition probability $|T_{i,f}|^{2}$ as $\sigma_{i\to
f}=(\pi/k_{i}^{2})|T_{i,f}|^{2}$, where $i$ and $f$ label the initial and
final internal molecular states, $k_{i}=\sqrt{2\mu E_{\mathrm{c}}}$ is the
incident wavevector, and $E_{\mathrm{c}}$ is the collision energy.

In the spirit of MQDT-FT for atomic collisions [38, 39, 40, 41, 42, 43, 44],
we seek to describe short-range quantum dynamics with a simplified Hamiltonian
$\hat{H}_{\mathrm{0}}$ obtained by omitting the terms, which are small
compared to $\hat{V}$, from $\hat{H}_{\text{as}}$ in Eq. 2. To this end, the
exact short-range $K$-matrix ($\bm{K}^{\text{sr}}$) corresponding to the
original Hamiltonian $\hat{H}$ is obtained from the approximate $K$-matrix
($\bm{K}_{\mathrm{0}}^{\text{sr}}$), which corresponds to the simplified
Hamiltonian $\hat{H}_{0}$:

$\bm{K}^{\text{sr}}\simeq\bm{U}^{\dagger}\bm{K}_{\mathrm{0}}^{\text{sr}}\bm{U},$
(8)

where $\bm{U}$ is the unitary FT matrix composed of the eigenvectors of
$\hat{H}_{\text{as}}$ in Eq. 2 (see the Supplemental Material [50] for
details). Note that $\bm{K}_{\mathrm{0}}^{\text{sr}}$ is much easier to
compute than the exact $\bm{K}^{\text{sr}}$ because $\hat{H}_{0}$ can be
chosen in such a way as to exclude the small hyperfine and Zeeman interactions
(see below), thus eliminating the need to use enormous CC basis sets including
the electron and/or nuclear spin basis functions of the collision partners
[21]. In addition, since $\hat{H}_{0}$ is independent of external fields,
$\bm{K}_{\mathrm{0}}^{\text{sr}}$ can be efficiently computed using the total
angular momentum (TAM) basis as shown below, and it is independent of $M$, the
$z$ projection of $J$.

Once the FT approximation for $\bm{K}^{\text{sr}}$ is implemented, the
standard MQDT procedure [27, 31, 32, 33] can be applied to obtain scattering
observables as a function of collision energy and magnetic field. This
procedure is computationally efficient, scaling as $O(N)$ with the total
number of channels $N$. The only additional modification is that the threshold
energies $E_{i}^{\infty}$ should be replaced by exact threshold energies
[eigenvalues of $\hat{H}_{\text{as}}$ in Eq. 2].

We now apply MQDT-FT to describe the quantum dynamics of ultracold Mg + NH
collisions in a magnetic field, focusing on the inelastic transitions between
the different Zeeman sublevels of NH($v=0,N=0$) labeled by the $z$-projection
of the molecule’s electron spin $M_{S}=-1,0,1$. These transitions cause trap
loss and thus limit the efficiency of sympathetic cooling of NH molecules by
Mg atoms [47]. For now, we neglect the hyperfine interaction
($\hat{H}_{\text{as}}=B_{e}\hat{\bm{\mathrm{N}}}^{2}+\hat{H}_{\text{fs}}+\hat{H}_{\text{Z}}$)
and choose the simplified short-range Hamiltonian $\hat{H_{0}}$ by neglecting
the Zeeman interaction as $\hat{H}_{0}=\hat{H}-\hat{H}_{\text{Z}}$. Since
$\hat{H}_{0}$ is field-independent, we obtain
$\bm{K}_{\mathrm{0}}^{\text{sr}}$ by solving CC equations in the TAM basis
$|(NS)jlJM\rangle$, where $\hat{\mathbf{J}}=\hat{\mathbf{l}}+\hat{\mathbf{j}}$
is the total angular momentum of the atom-molecule system and
$\hat{\mathbf{j}}=\hat{\mathbf{N}}+\hat{\mathbf{S}}$ is that of the diatomic
molecule. Because $\hat{H}_{0}$ is block diagonal in $J$ and independent of
$M$, the computational cost of MQDT-FT is much smaller than that of
conventional MQDT [33].

Figure 1:  Probabilities for the spin relaxation transition $M_{S}=1$ ($l=0$)
$\to$ $M^{\prime}_{S}=0$ ($l^{\prime}=2$) in cold Mg +14NH($N=0$) collisions
plotted as a function of collision energy (a) and magnetic field (b). Solid
lines – MQDT-FT calculations, open circles – exact CC results.

Figure 1 compares the probabilities $|T_{i\to f}|^{2}$ for the spin relaxation
transition $M_{S}=1\to 0$ between the Zeeman levels of NH ($N=0$) in cold
Mg+NH collisions calculated using MQDT-FT with exact CC results.
Encouragingly, MQDT-FT provides an essentially exact description of the
transition probability as a function of collision energy and external magnetic
field with only 139 channels as compared to 954 channels required in standard
MQDT [34], a 103-fold increase in computational efficiency. The excellent
performance of MQDT-FT illustrated in Fig. 1 validates the assumption that
$\hat{H}_{\text{Z}}$ plays a negligible role in short-range dynamics. The
strong magnetic field dependence of the inelastic probability at 1 $\mu$K thus
arises entirely from long-range physics. Specifically, inelastic scattering is
suppressed by the centrifugal barrier in the outgoing $d$-wave channel at low
magnetic fields [51, 52]. This is an example of physically meaningful insight
into ultracold atom-molecule collisions gained from MQDT-FT simulations.

Figure 2: Hyperfine energy levels of 15NH ($N=0$) at zero field (a) and as a
function of magnetic field (b). State-to-state transition probabilities
$|T_{il,fl^{\prime}}|^{2}$ plotted as a function of collision energy at
$B=100$ G (c) and $B=1000$ G (d) for Mg + NH($N=0$) collisions for $l=0$ and
$l^{\prime}=2$. Solid lines – MQDT-FT results, symbols – exact CC
calculations. The initial and final states of NH are indicated next to each
curve as $i\to f$ [see panel (b)].

We next address the question of whether MQDT-FT can be used to describe
hyperfine interactions, which play a crucial role in ultracold molecular
collisions [2, 15], yet are notoriously difficult to describe using standard
CC methods, requiring enormous basis sets [21]. As depicted in Fig. 2(a) the
hyperfine structure of 15NH arises from the nuclear spins of 15N ($I_{1}=1/2$)
and H ($I_{2}=1/2$) which couple to $\hat{\mathbf{j}}$ (see above) to produce
the total angular momentum $\hat{\mathbf{F}}$ of NH, which is a good quantum
number at low $B$-field. The hyperfine structure is described by the
Hamiltonian [53, 54, 50]

$\hat{H}_{\text{hf}}=\sum_{i=1,2}a_{i}\hat{\mathbf{I}}_{i}\cdot\hat{\mathbf{S}}+\hat{H}_{\text{ahf}},$
(9)

where $a_{i}$ and $c_{i}$ are the isotropic and anisotropic hyperfine
constants for the $i$-th nucleus, and
$\hat{H}_{\text{ahf}}=\sum_{i}\frac{c_{i}\sqrt{6}}{3}\sqrt{\frac{4\pi}{5}}\sum_{q=-2}^{2}(-1)^{q}Y_{2\,-q}(\bm{r})[\hat{\mathbf{I}}_{i}\otimes\hat{\mathbf{S}}]_{q}^{(2)}$
is the anisotropic hyperfine interaction. In the high $B$-field limit, the 12
hyperfine levels of 15NH are arranged in three groups, with 4 states per
group, according to the value of $M_{S}=-1,0,$ and $1$, as shown in Fig. 2(b).

To account for the hyperfine structure, we perform MQDT-FT calculations using
the same simplified short-range Hamiltonian $\hat{H}_{\mathrm{0}}$ and the TAM
basis set as described above without extra computational costs. It is only
necessary to augment $\bm{K}_{\mathrm{0}}^{\text{sr}}$ by the nuclear spin
basis states $\ket{I_{1}M_{I_{1}}}\ket{I_{2}M_{I_{2}}}$ before applying the FT
[50].

Figures 2(c) and (d) show transition probabilities between the different
hyperfine levels of NH in ultracold collisions with Mg atoms. We observe very
good agreement between MQDT-FT and exact CC results, which indicates that the
effects of hyperfine structure on ultracold atom-molecule collisions can be
accurately described by MQDT-FT. Importantly, only 147 channels need to be
coupled in our MQDT-FT calculations, whereas full CC computations involve as
many as 3854 channels [48]. Given the $O(N^{3})$ scaling of the computational
effort with the number of channels $N$, MQDT-FT affords a massive reduction of
computational cost by a factor of 104.

Figure 3: Probabilities for the transition $M_{S}=1$, $l=0$ $\to$
$M^{\prime}_{S}=0$, $l^{\prime}=2$ in cold Mg + 14NH$(N=0)$ collisions as a
function of collision energy. Solid lines – MQDT-FT results based on a
$2\times 2$ short-range K-matrix ($K^{\text{sr}}$:$2\times 2$), open circles –
exact CC calculations.

An attractive feature of the MQDT-FT approach is its capability to describe
the complex quantum dynamics of ultracold collisions in terms of a small
number of ”universal” parameters [39, 40]. While this capability has provided
invaluable physical insight into ultracold atomic collisions [39, 40], it has
been unclear whether it can be extended to more complex molecular collisions.
A minimal MQDT-FT model of ultracold Mg + NH collision is based on a $2\times
2$ short-range $K$-matrix obtained using only the elements of
$\bm{K}^{\text{sr}}$ corresponding to the initial and final channels
($K_{ii}^{\text{sr}}$, $K_{if}^{\text{sr}}=K_{fi}^{\text{sr}}$, and
$K_{ff}^{\text{sr}}$).

As shown in Fig. 3, the probability for the $M_{S}=1$ $\to$ $M^{\prime}_{S}=0$
transition obtained with the two-channel model is in excellent agreement with
the full 954-channel results. Remarkably, the quantum dynamics of inelastic
spin relaxation in ultracold Mg + NH collisions in a magnetic field can be
described nearly exactly using just three short-range parameters
($K_{ii}^{\text{sr}}$, $K_{if}^{\text{sr}}$, and $K_{ff}^{\text{sr}}$) over a
wide range of collision energies. This result highlights the universal nature
of complex ultracold atom-molecule collisions in a dc magnetic field, and
suggests MQDT-FT as a possible alternative to the approaches based on a
single-channel universal model (UM) [55, 56]. While the phenomenological $y$
parameter of the UM can be related to the results of CC calculations only in
some special cases [56], the MQDT-FT parameters (${K}^{\text{sr}}_{ij}$) are
readily available from modest CC calculations, which only need to be performed
in the short-range region, and do not need to include the Zeeman and hyperfine
interactions.

Figure 4: Collision energy dependence of transition probabilities between the
different fine-structure levels of 14NH in ultracold collisions with Mg atoms.
Symbols – exact CC calculations, lines – MQDT-FT results obtained using the
recoupling FT. The quantum numbers for the initial and final states involved
in each transition are indicated next to the corresponding curves. All results
are for a single incident $s$-wave component ($l=0$) and summed over all final
$l^{\prime}$.

Finally, we consider the possibility of MQDT-FT based on an even more
drastically simplified short-range Hamiltonian $\hat{H}_{0}$, which includes
only rovibrational degrees of freedom. Because $\hat{H}_{0}$ excludes all
spin-dependent interactions, it cannot describe inelastic transitions between
the different hyperfine-Zeeman sublevels of NH($N=0$), which, for collisions
with structureless atoms such as Mg, are mediated by intramolecular spin-
dependent interactions [46, 52]. We thus focus on inelastic transitions
between the different fine-structure components of excited rotational states
of NH. We use a modified recoupling technique, which has been successfully
applied to such transitions at high collision energies [57, 58, 59, 60, 61].
To extend this technique to the ultracold domain, we apply it to the short-
range $K$-matrix rather than the $T$-matrix, as was done previously [57, 58,
59, 60, 61].

Specifically, we obtain $\bm{K}^{\mathrm{sr}}_{0}$ by solving CC equations for
the simplified Hamiltonian
$\hat{H}_{0}=\hat{H}-\hat{H}_{\text{fs}}-\hat{H}_{\text{hfs}}-\hat{H}_{\text{Z}}$
(see Eqs. 1 and 2) using the basis set $|(Nl)J_{r}M_{r}\rangle$, where
$\hat{\mathbf{J}}_{r}=\hat{\mathbf{N}}+\hat{\mathbf{l}}$ is the total
rotational angular momentum of the collision complex [21]. The basis functions
$|(Nl)J_{r}M_{r}\rangle$ are the eigenstates of $\hat{H}_{\mathrm{0}}$ in the
limit $R\to\infty$. The eigenstates of
$\hat{H}_{\mathrm{0}}+\hat{H}_{\text{fs}}$ are well approximated by the TAM
basis functions $|(NS)jlJM\rangle$ for ${}^{2S+1}\Sigma$ molecules [57]
provided the weak spin-spin interaction between the different $N$ states can
be neglected. To obtain the desired short-range $K$-matrix in the
$|(NS)jlJM\rangle$ basis from that in the $|(Nl)J_{r}M_{r}\rangle$ basis, we
use an analytical recoupling transformation [57, 50].

Figure 4 shows that the probabilities for rotationally inelastic fine-
structure transitions in ultracold Mg + NH collisions are well described by
MQDT-FT. Thus, using a spin-independent $\hat{H}_{0}$ in short-range CC
calculations in the framework of MQDT-FT enables one to describe a wide range
of transitions between rotational and fine-structure levels of open-shell
molecules in ultracold collisions with atoms.

In summary, we have generalized the powerful MQDT-FT approach to ultracold
atom-molecule collisions in external magnetic fields, providing a robust
conceptual and numerical framework for their theoretical description. We have
applied the approach to a realistic atom-molecule collision system (Mg + NH)
using a variety of short-range Hamiltonians, obtaining encouraging agreement
with exact CC calculations in all cases. Our calculations show that MQDT-FT
can provide a dramatic ($10^{4}$-fold) reduction of computational effort of CC
calculations of atom-molecule collisions compared to standard MQDT [34] due
the fact that there is no need to explicitly account for the hyperfine-Zeeman
structure of the collision partners at short range. In many cases of practical
interest, it will only be necessary to solve CC equations in the strong
interaction region employing a rovibrational (TAM) basis in the absence of
external fields, which can be efficiently accomplished using currently
available computational techniques [62, 63, 64, 45, 65, 36]. This opens up the
possibility of performing rigorous quantum scattering calculations on a wide
array of previously intractable ultracold atom-molecule collisions and
chemical reactions in external fields, including those probed in recent
experiments [22, 23, 24, 25, 14].

From a conceptual viewpoint, our results show that it is possible to describe
the intricate quantum dynamics of ultracold atom-molecule collisions over wide
ranges of collision energies and magnetic field in terms of a small number of
short-range parameters (three for Mg+NH). This provides a solid basis for the
development of few-parameter MQDT-FT models of ultracold atom-molecule (and
possibly even molecule-molecule [66]) collisions. Such models, which have been
so fruitful in the field of ultracold atomic collisions [39, 40], may allow
for novel insights into fascinating quantum dynamics of ultracold molecular
collisions.

We thank Chris Greene, John Bohn, James Croft, and Brian Kendrick for
stimulating discussions. This work was supported by the NSF CAREER program
(PHY-2045681) and by the U.S. AFOSR under Contract FA9550-22-1-0361 to P.B.
and T.V.T. Computations were performed on the Niagara supercomputer at the
SciNet HPC Consortium.

Supplemental Material

Supplemental Material for
“Multichannel quantum defect theory with a frame transformation
for ultracold molecular collisions in magnetic fields”

Masato Morita1, Paul Brumer1 and Timur V. Tscherbul2

###### Contents

  1. I  Computational details
    1. I.1 Coupled channel (CC) calculations
    2. I.2 Physical constants
    3. I.3 Interaction potential
    4. I.4 Multi-channel quantum defect theory (MQDT)
  2. II  Frame transformation (FT)
  3. III  Electron spin transitions
  4. IV  Hyperfine transitions
    1. V  Short-range K-matrix
  5. VI  Fine structure transitions

In this Supplemental Material (SM), we provide technical details of our MQDT-
FT calculations as well as some additional results, which complement the
figures and discussions in the main text. We describe the computational
details in Section I, and present the cross sections for Zeeman and hyperfine
transitions in cold Mg+NH($N=0$) collisions in Sections III and IV. Results
for the short-range K-matrix as a function of collision energy are presented
in Section V. In Section VI, the capabilities of the recoupling FT approach to
calculate the short-range K-matrix are illustrated by MQDT-FT calculations of
fine structure transitions from the $N=3$, $j=3$ initial state of NH.

##  I  Computational details

###  I.1 Coupled channel (CC) calculations

We performed coupled-channel (CC) calculations to obtain the reference results
for cold Mg + NH collisions in a magnetic field, against which we compare our
MQDT-FT calculations in the main text. The Hamiltonian of the atom-molecule
collision complex is given by Eq. (1) of the main text. We use converged
uncoupled basis sets,
$|NM_{N}\rangle|SM_{S}\rangle|I_{1}M_{I_{1}}\rangle|I_{2}M_{I_{2}}\rangle|lm_{l}\rangle$
for Fig. 2 and $|NM_{N}\rangle|SM_{S}\rangle|lm_{l}\rangle$ for Figs. 1 and 3,
with $N_{\text{max}}=6$ and $l_{\text{max}}=8$ [47]. On the other hand, for
Fig. 4, we employ a coupled total angular momentum (TAM) basis set
$|(NS)jlJM\rangle$ with $N_{\text{max}}=6$ and $J_{\text{max}}=5$. The CC
equations are solved with the log-derivative propagation method [67] between
$R_{\text{min}}=4\,a_{0}$ and $R_{\text{max}}=500\,a_{0}$. We use the
propagation interval of $\Delta\,R=0.05\,a_{0}$ for $R\leq 30\,a_{0}$ and
$\Delta\,R=0.1\,a_{0}$ for $R>30\,a_{0}$.

###  I.2 Physical constants

We use the same reduced mass of Mg+14NH ($\mu=9.232679959$ amu), and
spectroscopic constants of 14NH ($B_{e}=16.343$ cm-1,
$\gamma_{\mathrm{sr}}=-0.055$ cm-1, and $\lambda_{\mathrm{ss}}=0.92$ cm-1) as
in Refs. [47, 49]. For the calculations shown in Fig. 2 of the main text, we
use the reduced mass of $\mu=9.60046$ amu for Mg+15NH, the spectroscopic
constants of 15NH ($B_{e}=16.2712$ cm-1, $\gamma_{\mathrm{sr}}=-0.0546$ cm-1,
and $\lambda_{\mathrm{ss}}=0.9199$ cm-1), and the hyperfine constants of 15NH
($b_{1}=-1.9406\times 10^{-3}$ cm-1, $c_{1}=3.1783\times 10^{-3}$ cm-1,
$b_{2}=-3.2108\times 10^{-3}$ cm-1 and $c_{2}=3.0203\times 10^{-3}$ cm-1)
[54].

###  I.3 Interaction potential

The interaction potential between Mg (1S) and NH ($\tilde{X}\,^{3}\Sigma^{-}$)
used in our calculations is published as part of the MOLSCAT program suite
[49] as pot-Mg_NH.data and vstar-Mg_NH.f. This potential is slightly different
from that used in Refs. [47, 48, 33].

###  I.4 Multi-channel quantum defect theory (MQDT)

Except for the frame transformation step to obtain the short-range K-matrix
($\bm{K}^{\text{sr}}$), the required calculation procedures in MQDT-FT are
essentially the same as those of conventional atom-molecule MQDT [33, 34].

The total number of reference channels is
$N_{\mathrm{ref}}=N_{\mathrm{o}}+N_{\mathrm{wc}}$ [33], where $N_{\mathrm{o}}$
and $N_{\mathrm{wc}}$ are the numbers of open and weakly closed channels,
respectively. As stated in the main text, we use only weakly closed channels
when calculating the short-range $K$-matrix (i.e., we exclude strongly closed
channels as defined in the main text). The value of the matching radius
$R_{m}$ should be determined carefully by exploring the $R_{m}$ dependence of
the results.

We use $R_{m}=30\,a_{0}$ for all MQDT-FT calculations presented here and in
the main text. No sensitivity to $R_{m}$ was observed in the range
$15\,a_{0}\leq R_{m}\leq 40\,a_{0}$ for the results shown in Fig. 1 of the
main text and in the range $25\,a_{0}\leq R_{m}\leq 35\,a_{0}$ for those shown
in Fig. 2 of the main text.

The reference potentials, $U_{i}^{\text{ref}}(R)\,(i=1,..,N_{\text{ref}})$,
describe the long-range part of the scattering problem and dissociate into
accurate channel energies including Zeeman and/or hyperfine interactions. We
employed the isotropic part, $V_{0}(R)$, of the interaction potential to
construct the reference potentials. The reference potentials contain a hard
wall at $R=R_{\text{wall}}$ in the short range region
($U_{i}^{\text{ref}}(R_{\text{wall}})=+\infty$) restricting the amplitude of
the regular reference function at $R=R_{\text{wall}}$. We employ a common wall
position $R_{\text{wall}}=5\,a_{\text{0}}$ for all reference channels although
the final results are not affected by the choice of $R_{\mathrm{wall}}$. We
obtain the reference functions for the $i$-th reference channel by numerically
solving the following 1D radial equation [27, 33]

$\left[\,\frac{d^{2}}{dR^{2}}+K_{i}^{2}(R)\,\right]\,\psi_{i}(R)\,=\,0,$ (1)

where $K_{i}(R)=\sqrt{2\mu[\,E-U_{i}^{\mathrm{ref}}(R)\,]}$ is the local
wavevector and $E$ is the total energy. In the following, we refer to the
regular solution as $f_{i}(R)$ ($f_{i}(R)\to 0$ as $R\to 0$) and to the
irregular solution as $g_{i}(R)$. Here, we use the reference potentials with
minima at a finite $R$ in the range $0<R<R_{m}$ (in Eq. (5) in the main text)
and the WKB normalizations are taken at the minimum of the reference
potentials following Mies et al. [27].

Once we obtain the values of the regular and irregular reference functions at
the matching point $(R=R_{m})$, we calculate the $N_{\mathrm{ref}}\times
N_{\mathrm{ref}}$ short-range K matrix $\bm{K}_{\mathrm{0}}^{\text{sr}}$
following the matching boundary condition at $R=R_{m}$:

$\bm{\Psi}(R=R_{m})=R_{m}^{-1}[\bm{f}(R_{m})+\bm{g}(R_{m})\bm{K}_{\mathrm{0}}^{\text{sr}}],$
(2)

where $\bm{f}(R_{m})$ and $\bm{g}(R_{m})$ are the diagonal matrices in the
asymptotic basis with their diagonal elements given by the values of the
reference functions $f_{i}(R_{m})$ and $g_{i}(R_{m})$ [35].

##  II  Frame transformation (FT)

Here, as in the main text, $\hat{H}$ denotes the Hamiltonian of the atom-
molecule collision complex, and $\hat{H}_{0}$ denotes the simplified
Hamiltonian used in short-range CC calculations of $\bm{K}_{\mathrm{0}}$ in
MQDT-FT. We approximate $\bm{K}^{\text{sr}}$ by the matrix obtained via the
frame transformation (FT) as
$\bm{K}^{\text{sr}}\simeq\bm{U}^{\dagger}\bm{K}_{\mathrm{0}}^{\text{sr}}\bm{U}$,
where $\bm{U}$ is a unitary matrix composed of the eigenstates of the
asymptotic Hamiltonian $\hat{H}_{\text{as}}=\lim_{R\to\infty}\hat{H}$. Since
$\bm{K}_{\mathrm{0}}^{\text{sr}}$ is initially expressed in the basis of
eigenstates of the simplified asymptotic Hamiltonian
$\hat{H}_{0,\text{as}}=\lim_{R\to\infty}\hat{H}_{\mathrm{0}}$, it is necessary
to establish a relationship between the eigenstates of $\hat{H}_{\text{as}}$
and $\hat{H}_{0,\text{as}}$ to represent $\bm{K}^{\text{sr}}$ in the basis of
eigenstates of $\hat{H}_{\text{as}}$.

Once we represent $\hat{H}_{0}$ and $\hat{H}_{\text{as}}$ in the same basis
set, it is straightforward to apply the FT. However, it can be challenging to
obtain the eigenvectors of $\hat{H}_{\text{as}}$ in the basis set used in
short-range CC calculations. The latter are best performed using the TAM basis
set, and it is not straightforward to derive and/or implement the matrix
elements of $\hat{H}_{\text{as}}$ in this basis. To avoid these difficulties,
we propose to use the fully uncoupled basis set, in which the matrix elements
of $\hat{H}_{\text{as}}$ are readily available [46].

For example, to obtain the results shown in Fig. 1 of the main text, the
following sequence of steps is used

  1. 1.

The matrix of $\hat{H}_{\text{as}}$ is constructed and diagonalized in the
fully uncoupled basis $|NM_{N}\rangle|SM_{S}\rangle|lm_{l}\rangle$. The
resulting eigenstates are expressed as linear combinations of the fully
uncoupled basis functions

$|\,\alpha\,\rangle\,=\,\sum_{i}\,c^{\alpha}_{i}|\,i\,\rangle,$ (3)

where $|\,i\,\rangle$ is a short-hand notation for the uncoupled basis
functions $|NM_{N}\rangle|SM_{S}\rangle|lm_{l}\rangle$, which includes the
quantum numbers $N$, $M_{N}$, $M_{S}$, $l$, and $m_{l}$ (note that $S=1$ is
conserved), and the expansion coefficients {$c_{i}^{\alpha}$} correspond to
the $\alpha$-th eigenvector of $\hat{H}_{\text{as}}$. The calculated
eigenvectors are stored for subsequent use in step 4.

  2. 2.

The CC equations for the simplified Hamiltonian $\hat{H}_{0}$ expressed in the
primitive TAM basis $|(NS)jlJM\rangle$ are integrated numerically out to
$R=R_{m}$. The log-derivative matrix is then transformed from the primitive
TAM basis set to the basis of eigenvectors of $\hat{H}_{0,\text{as}}$ to
obtain the short-range $K$ matrix $\bm{K}_{\mathrm{0}}^{\text{sr},J}$ using
the matching condition given by Eq. 2 for each $J$ ($J_{\text{max}}=5$). Thus,
$\bm{K}_{\mathrm{0}}^{\text{sr}}=\bm{K}_{\mathrm{0}}^{\text{sr},J=0}\oplus\bm{K}_{\mathrm{0}}^{\text{sr},J=1}\oplus\cdots\oplus\bm{K}_{\mathrm{0}}^{\text{sr},J=5}$.

  3. 3.

For Mg + NH collisions, the eigenstates of $\hat{H}_{0,\text{as}}$ are well
approximated by the individual TAM basis functions due to the small spin-spin
interaction in NH (in other words, the matrix of $\hat{H}_{0,\text{as}}$ is
nearly diagonal in the TAM basis). Therefore, in the following, we assume that
$\bm{K}_{\mathrm{0}}^{\text{sr}}$ is expressed in the TAM basis. If
$\hat{H}_{0,\text{as}}$ is not diagonal in this basis, $\bm{K}^{\text{sr}}$
would need to be transformed to the basis, in which $\hat{H}_{0,\text{as}}$ is
diagonal (see the discussion after Step 4).

The matrix elements of $\bm{K}^{\text{sr}}$ in the basis of eigenstates
$\alpha$ and $\beta$ of $\hat{H}_{\text{as}}$ is obtained by applying the FT

$\langle\,\alpha\,|\,\bm{K}^{\text{sr}}\,|\,\beta\,\rangle=\sum_{p,q}\,{d_{p}^{\alpha}}^{\ast}d_{q}^{\beta}\langle\,p\,|\,\bm{K}_{0}^{\text{sr}}\,|\,q\,\rangle,$
(4)

where $|p\rangle$ and $|q\rangle$ are the TAM basis functions,
{$d_{p}^{\alpha}$} and {$d_{q}^{\beta}$} are the eigenvectors of
$\hat{H}_{\text{as}}$ in the TAM basis. Instead of calculating these
eigenvectors directly, we express them via the eigenvectors of
$\hat{H}_{\text{as}}$ in the fully uncoupled basis (see Step 1) as described
below.

  4. 4.

As noted above, we wish to calculate $\bm{K}^{\text{sr}}$ in the eigenbasis of
$\hat{H}_{\text{as}}$ with the latter operator is expressed in the fully
uncoupled basis. This is motivated by the ease of evaluating the matrix
elements of $\hat{H}_{\text{as}}$ in the fully uncoupled basis [46]. It
follows from Eq. (3) that

$\langle\,\alpha\,|\,\bm{K}^{\text{sr}}\,|\,\beta\,\rangle=\sum_{i,j}\,{c_{i}^{\alpha}}^{\ast}c_{j}^{\beta}\langle\,i\,|\,\bm{K}_{0}^{\text{sr}}\,|\,j\,\rangle.$
(5)

To express the matrix elements on the right-hand side in terms of the known
quantities $\langle\,p\,|\bm{K}_{0}^{\text{sr}}\,|\,q\,\rangle$, we use the
completeness relation $\hat{1}=\sum_{p}|p\rangle\langle p|$:

$\langle\,\alpha\,|\,\bm{K}^{\text{sr}}\,|\,\beta\,\rangle=\sum_{i,j}\,{c_{i}^{\alpha}}^{\ast}c_{j}^{\beta}\langle\,i\,|\,\left[\,\sum_{p,q}|\,p\,\rangle\,\langle\,p\,|\bm{K}_{0}^{\text{sr}}\,|\,q\,\rangle\,\langle\,q\,|\,\right]\
|\,j\,\rangle\\\
=\sum_{i,j,p,q}\,{c_{i}^{\alpha}}^{\ast}c_{j}^{\beta}\langle\,i\,|\,p\,\rangle\,\langle\,q\,|\,j\,\rangle\,\langle\,p\,|\bm{K}_{0}^{\text{sr}}\,|\,q\,\rangle\,,$
(6)

where the overlaps between the fully uncoupled basis state $|\,i\,\rangle$ and
the TAM basis state $|\,p\,\rangle$ are given analytically by Clebsh-Gordan
coefficients. We note that the expression in Eq. 6 is more generally
applicable to the cases, where the overlaps $\langle\,i\,|\,p\,\rangle$ are
only available from a numerical computation (which could rely on
approximations). Additionally, it is not necessary to explicitly construct the
matrix $\bm{K}_{0}^{\text{sr}}$ in the fully uncoupled basis set in this
calculation.

For completeness, we also consider the situation, in which the eigenstates of
$\hat{H}_{0,\text{as}}$ are strongly different from the individual TAM basis
functions (this is not the case in the present work). In this situation, the
assumption discussed in Step 3 above does not hold, and additional summations
over the TAM basis functions and the associated overlap coefficients
(eigenvectors) appear in Eq. 6. Let $|\phi\rangle$ be an eigenstate of
$\hat{H}_{0,\text{as}}$, which is expanded in the TAM basis set as

$|\,\phi\,\rangle\,=\,\sum_{p}\,h_{p}^{\phi}|\,p\,\rangle.$ (7)

From Eq. 6, we obtain

$\langle\,\alpha\,|\,\bm{K}^{\text{sr}}\,|\,\beta\,\rangle=\sum_{i,j,\phi,\chi}\,{c_{i}^{\alpha}}^{\ast}c_{j}^{\beta}\langle\,i\,|\,\phi\,\rangle\,\langle\,\chi\,|\,j\,\rangle\,\langle\,\phi\,|\bm{K}_{0}^{\text{sr}}\,|\,\chi\,\rangle\,.$
(8)

The matrix elements $\langle\,\phi\,|\bm{K}_{0}^{\text{sr}}\,|\,\chi\,\rangle$
and the expansion coefficients {$h_{p}$} Eq. 7 are available from short-range
CC calculations in the TAM basis subject to the matching condition Eq. 2.

The most general FT is then obtained using Eq. 7 as

$\langle\,\alpha\,|\,\bm{K}^{\text{sr}}\,|\,\beta\,\rangle=\sum_{i,j,\phi,\chi}\,{c_{i}^{\alpha}}^{\ast}c_{j}^{\beta}\langle\,\phi\,|\bm{K}_{0}^{\text{sr}}\,|\,\chi\,\rangle\
\sum_{p,q}\,{h_{p}^{\phi}}{h_{q}^{\chi}}^{\ast}\langle\,i\,|\,p\,\rangle\,\langle\,q\,|\,j\,\rangle\,.$
(9)

To produce the results shown in Fig. 2 of the main text, we implemented the
following sequence of MQDT-FT steps

  1. 1.

Build and diagonalize the matrix of $\hat{H}_{\text{as}}$ (including the
hyperfine interactions) in the fully uncoupled basis set
$|NM_{N}\rangle|SM_{S}\rangle|lm_{l}\rangle|I_{1}M_{I_{1}}\rangle|I_{2}M_{I_{2}}\rangle$.

  2. 2.

This step is the same as Step 2 above. The only difference is that the number
of $J$-blocks is increased to $J_{\text{max}}=7$.

  3. 3.

As in Step 3 above, we assume that the individual TAM basis functions
$|(NS)jlJM\rangle$ are the eigenstates of $\hat{H}_{0,\text{as}}$. In this
case, $\hat{H}_{\text{as}}$ includes the nuclear spin degrees of freedom for
the two nuclei of NH but $\hat{H}_{0,\text{as}}$ is identical to that used
above (see the steps for Fig. 1).

Because the short-range $K$-matrix is computed with the nuclear spin degrees
of freedom omitted, we need to reintroduce them before applying MQDT boundary
conditions (which do include the nuclear spin degrees of freedom). To this
end, we augment
$\bm{K}_{\mathrm{0}}^{\text{sr}}\to\bm{K}_{\mathrm{0}}^{\text{sr}}\otimes\bm{I}$,
where $\bm{I}$ is a $4\times 4$ unit matrix in the basis of 4 possible nuclear
spin states of 15NH, $|M_{I_{1}}=\pm 1/2,M_{I_{2}}=\pm 1/2\rangle$. The new
basis for $\bm{K}_{\mathrm{0}}^{\text{sr}}\otimes\bm{I}$ is given by the
direct product of the TAM basis and
$|I_{1}M_{I_{1}}\rangle|I_{2}M_{I_{2}}\rangle$, namely
$|(NS)jlJM\rangle|I_{1}M_{I_{1}}\rangle|I_{2}M_{I_{2}}\rangle$.

  4. 4.

Finally, we identify
$|i\rangle\leftrightarrow|NM_{N}\rangle|SM_{S}\rangle|lm_{l}\rangle|I_{1}M_{I_{1}}\rangle|I_{2}M_{I_{2}}\rangle$
and
$|p\rangle\leftrightarrow|(NS)jlJM\rangle|I_{1}M_{I_{1}}\rangle|I_{2}M_{I_{2}}\rangle$
and use Eq. 6 to obtain the desired matrix elements of $\bm{K}^{\text{sr}}$ in
the basis of eigenstates of $\hat{H}_{\text{as}}$.

##  III  Electron spin transitions

Figure 1: Probabilities for the spin relaxation transitions (a) $M_{S}=1$
($l=1$) $\to$ $M^{\prime}_{S}=0$ ($l^{\prime}=1$) and (b) $M_{S}=1$ ($l=0$)
$\to$ $M^{\prime}_{S}=-1$ ($l^{\prime}=2$) in Mg + 14NH ($N=0$) collisions.
Open circles – exact CC calculations, solid lines – MQDT-FT results. The
magnitude of the external magnetic field is indicated next to each curve.
Figure 2:  Probabilities for the state-to-state transitions ($l=0$ $\to$
$l^{\prime}=2$) at the magnetic field of (a) $B=100$ G (b) and $B=1000$ in
Mg+15NH ($N=0$) collisions. Open circles – exact CC calculations, solid lines
– MQDT-FT results. The initial and final states of 15NH ($N=0$) are indicated
next to each curve.

To further illustrate the good agreement between our MQDT-FT results and exact
CC calculations (see Fig. 1 of the main text), we show in Fig. 1 a sampling of
state-to-state transitions in ultracold Mg + NH collisions. While our main
interest here is in ultracold $s$-wave collisions, we also show the results
for the $M_{S}=1$ ($l=1$) $\to$ $M^{\prime}_{S}=0$ ($l^{\prime}=1$) transition
in panel (a) to demonstrate the capability of MQDT-FT to describe $p$-wave
collisions. In panel (b), we show the results for the $M_{S}=1$ ($l=0$) $\to$
$M^{\prime}_{S}=-1$ ($l^{\prime}=2$) transition, which is similar to the
$M_{S}=1$ ($l=0$) $\to$ $M^{\prime}_{S}=0$ ($l^{\prime}=2$) transition (see
Fig. 1 of the main text).

##  IV  Hyperfine transitions

In Fig. 2, we show the cross sections for the hyperfine transitions to the
final states of 15NH ($N=0$) corresponding to $M^{\prime}_{S}=-1$ (see the
main text) in $s$-wave collisions of Mg atoms with 15NH molecules initially in
the highest energy state (12). We observe excellent agreement between MQDT-FT
and exact CC results for most transitions. The agreement improves at higher
magnetic fields.

Figure 3: Elements of the $2\times 2$ short-range K-matrix plotted as
functions of collision energy at $B=1000$ G for the spin relaxation transition
$M_{S}=1$ ($l=0$) $\to$ $M^{\prime}_{S}=0$ ($l^{\prime}=2$) in Mg+14NH ($N=0$)
collisions.

##  V  Short-range K-matrix

In Fig. 3 of the main text, we show MQDT-FT results for spin relaxation cross
sections in Mg + NH collisions obtained using a reduced $2\times 2$ short-
range $K$-matrix. Figure 3 shows the variation of the matrix elements
$K_{ij}^{\text{sr}}$ with collision energy. Both diagonal (blue and red) and
off-diagonal (green) elements are constant throughout the low energy region
($E_{\text{c}}<10$mK), indicating that 3 parameters are sufficient to describe
spin relaxation in cold Mg+NH collisions over a wide range of collision
energies spanning 4 orders of magnitude.

We also observed no magnetic field dependence of $K_{ij}^{\text{sr}}$. In
particular, the off-diagonal matrix element
$K_{M_{S}=1(l=0),M_{S}=0(l=2)}^{\text{sr}}$ is conserved to within 6
significant digits in the $B=1-1000$ G range at $E_{\text{c}}=10^{-6}$K.

##  VI  Fine structure transitions

Figure 4:  Fine structure transition probabilities in ultracold Mg +
14NH($N=3$, $j=3$) collisions plotted as functions of collision energy for all
possible final states and $l=0$.

To calculate the cross sections for fine-structure transitions via the MQDT-FT
procedure outlined in the main text, it is necessary to express the short-
range $K$-matrix in the $|(NS)jlJM\rangle$ basis. However, our short-range
$K$-matrix is defined in the spin-free $|(Nl)J_{r}M_{r}\rangle$ basis. These
basis sets are related by an analytical recoupling transformation given by Eq.
(2.27) of Ref. [57]

$\begin{split}K^{\text{sr},J}_{N^{\prime}Sj^{\prime}l^{\prime},NSjl}=\sum_{J_{r}=|N-l|}^{N+l}[J_{r}][j^{\prime}j]^{1/2}(-1)^{-l^{\prime}+l+j^{\prime}-j}\begin{Bmatrix}S&N^{\prime}&j^{\prime}\\\
l^{\prime}&J&J_{r}\end{Bmatrix}\begin{Bmatrix}S&N&j\\\
l&J&J_{r}\end{Bmatrix}K^{\text{sr},J_{r}}_{0,N^{\prime}l^{\prime},Nl},\end{split}$
(10)

where $\\{\\}$ denotes a 6-$j$ symbol, $[J_{r}]=(2J_{r}+1)$, and
$[j,j^{\prime}]=(2j+1)(2j^{\prime}+1)$.

In Fig. 4(b) of the main text, we show the cross sections for fine structure
transitions from the $N=3$, $j=3$ ($l=0$) initial state of NH in cold
collisions with Mg atoms. Figure 4 shows an extended sample of the results for
all possible final states of NH using the recoupling FT technique (see the
main text). All the transitions accompanied by rotational de-excitation ($N=3$
$\to$ $N^{\prime}=0,1,$ and $2$) are seen to be accurately described by MQDT-
FT. For the $N$-conserving, $j$-changing transitions within the $N=3$
manifold, we observe a slight discrepancy between the CC and MQDT results for
$j^{\prime}=2$. The overall collision energy dependence of this transition is
well reproduced by MQDT-FT. This can be explained by noting that the
recoupling transformation (Eq. (10) of the main text) relies on the electron
spin playing a spectator role and neglects the intramolecular spin-spin
interaction in NH, which plays an important role in $N$-conserving fine
structure transitions.

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