# Constraining P and T Violating Forces with Chiral Molecules

Chaja Baruch<EMAIL_ADDRESS>Physics Department, Technion –
Israel Institute of Technology, Haifa 3200003, Israel  P. Bryan Changala
<EMAIL_ADDRESS>Center for Astrophysics $|$ Harvard &
Smithsonian, Cambridge, MA 02138, USA  Yuval Shagam<EMAIL_ADDRESS>Schulich Faculty of Chemistry, The Helen Diller Quantum Center and the Solid
State Institute, Technion-Israel Institute of Technology, Haifa, 3200003,
Israel  Yotam Soreq<EMAIL_ADDRESS>Physics Department,
Technion – Israel Institute of Technology, Haifa 3200003, Israel

###### Abstract

New sources of parity and time reversal violation are predicted by well
motivated extensions of the Standard Model and can be effectively probed by
precision spectroscopy of atoms and molecules. Chiral molecules have
distinguished enantiomers which are related by parity transformation. Thus,
they are promising candidates to search for parity violation at molecular
scales, yet to be observed. In this work, we show that precision spectroscopy
of the hyperfine structure of chiral molecules is sensitive to new physics
sources of parity and time reversal violation. In particular, such a study can
be sensitive to regions unexplored by terrestial experiments of a new chiral
spin-1 particle that couples to nucleons. We explore the potential to hunt for
time reversal violation in chiral molecules and show that it can be a
complementary measurement to other probes. We assess the feasibility of such
hyperfine metrology and project the sensitivity in $\text{CHDBrI}^{+}$.

##  I Introduction

Discrete space-time symmetries such as parity (P), charge-conjugation (C) and
time-reversal (T) play an important role in fundamental physics. Well-known
examples are the discovery of P violation (PV) in Cobalt-60 $\beta$-decay and
of CP violation (CPV) in kaon decays, both of which were milestones in the
building of the Standard Model (SM). However, despite its unparalleled
success, the SM cannot be considered a complete description of Nature. Thus,
new degrees of freedom that extend it are required, see e.g. [1]. Moreover,
new physics sources of PV and CPV are predicted in many beyond SM (BSM)
scenarios and are, for instance, motivated by Baryogenesis.

Precision atomic and molecular spectroscopy is a powerful tool to probe PV,
CPV, and T violation (TV) effects from SM and new physics sources e.g., [2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12] and see [13] for a review. To date, PV is
observed at atomic scales and smaller, but it is yet to be observed in
molecular systems. Chiral molecules, which are polyatomic molecules with two
distinguishable enantiomers, left (L) and right (R), are promising candidates
to observe PV at molecular scales, induced by the electron-nuclei weak
interaction [14, 15]. Recent developments are expected to reach the SM PV for
the first time in vibrational spectroscopy e.g., [16, 17, 18, 19, 20, 21, 22]
as well as nuclear magnetic resonance spectroscopy [23]. In other attempts
using rotational spectroscopy [24, 25], the magnitude of the SM effect is
typically below the current achievable precision. Chiral molecules may be
utilized as a dark matter detectors by a comparison of the L and R vibrational
spectra [26]. In this work, we investigate the potential of precision
rotational and vibrational spectroscopy of chiral molecules to serve as a
probe of new physics sources of PV and CPV without relying on the dark matter
cosmological abundance.

New physics sources of PV can result from electron-electron, electron-nucleon
or nucleon-nucleon interactions. The first two can be probed in electronic
and/or atomic systems, e.g. [27], while the latter cannot. Thus, molecules are
uniquely sensitive to new nucleon-nucleon interactions with a typical range of
an angstrom. Since a parity transformation relates the two molecular
enantiomers, a comparison of them is a direct probe of PV, eliminating the
need for external electric or magnetic fields. Moreover, the SM weak
interaction contribution to the molecular rotational structure is suppressed
[14]. Therefore, the difference between the L and R hyperfine spectra of
chiral molecules is an excellent probe of new PV nucleon-nucleon interactions
with negligible SM contribution. Thus, there is no need for any sophisticated
SM inputs. As we show below, precision spectroscopy of chiral molecules can
potentially probe parameter space unexplored by terrestrial experiments of a
new PV spin-1 particle that couples to neutrons, or to neutrons and protons
with opposite coupling.

CPV or TV (we use them interchangeably as we assume CPT) in molecules
originates from several BSM sources, among them electron and nucleon
interaction, the electron EDM ($e$EDM), modification of the pion coupling and
others, see e.g., [28, 29, 30] for more details. Here we focus on the nucleon-
nucleon interaction and show that in order to hunt for this effect external
magnetic and electric fields are required. The current bounds from the
reinterpretation of the $e$EDM in diatomic molecules [31] are stronger than
the expected sensitivity for a new CPV spin-0 state. However, the proposed
measurement is complementary to the current $e$EDM searches.

The rest of the paper is organized as follows. We introduce our benchmark
models in Section II. In Section III we analyze the angular momentum hyperfine
structure of chiral molecules relevant to new physics searches. In Section IV,
we give the projection for the $\text{CHDBrI}^{+}$ molecule, which has been
discussed in the context of searches for SM PV via vibrational spectroscopy
[32, 18, 33]. We conclude in Section V. The Appendix contains technical
details which are relevant to this work.

##  II P- and CP-violating forces

In this section we present our benchmark models for P- and CP-violating
forces. For PV we consider a new spin-1 particle, $X_{\mu}$, with both vector-
and axial-vector couplings to the SM fermions. The resulting low energy
potential includes a PV part, denoted as $AV$, see e.g. [34, 35]. For CPV (or
TV), we consider a new spin-0 particle, $\phi$, with both scalar- and pseudo-
scalar couplings to the SM fermions, which can induce a CP-violating
potential. We denote the low energy CPV potential as $PS$, see e.g. [34, 35].

###  II.1 Benchmark parity violating spin-1 model

We consider a spin-1 particle $X_{\mu}$ with both vector and axial-vector
couplings to fermions, which induces a parity-violating effect at the
molecular scale. The effective couplings between $X_{\mu}$, the nucleons,
$N=n,p$, and the electrons are given by

$\displaystyle\mathcal{L}_{\rm
int}\subset\sum_{\psi=e,p,n}\\!\\!\\!X^{\mu}\left(g_{V}^{X}\overline{\psi}\gamma_{\mu}\psi+g_{A}^{X}\overline{\psi}\gamma_{\mu}\gamma_{5}\psi\right)\,.$
(1)

The effective couplings in Eq. (1) are constrained both by terrestrial and
astrophysical observations. At the molecular scale, the relevant constraints
are those for $X_{\mu}$ with $m_{X}\lesssim 10\,[k]$. Regarding terrestrial
constraints, the strongest bound on $g_{A}^{N}$ is set using the J-coupling
interaction in HD $g_{A}^{p}(g_{A}^{p}+g_{A}^{n})\lesssim 1.4\times 10^{-18}$
for $m_{X}\ll[k]$ [36]. The vector interactions $g_{\rm V}^{n}$ are
constrained by neutron scattering [37, 38, 39, 40] and molecular vibrational
modes in HD e.g., [41, 42] $g_{V}^{p}(g_{V}^{p}+g_{V}^{n})\lesssim 2.9\times
10^{-10}$ for $m_{X}\ll[k]$. (We do not consider bounds from anomalous
currents, e.g. [43, 44], which are model dependent.) The P-violating proton-
nucleon interaction is also constrained by the nuclear anapole momemt of 133Cs
$g_{p}^{A}g_{N}^{V}\leq 6.0\times 10^{-8}$ [45] , see [46] for a new method to
measure the TI anapole moment. For $m_{X}\lesssim 10\,[k]$, there are
stringent astrophysical bounds on $g_{V}^{N}$ obtained from stellar cooling
[47].

The effective P-violating potential between nuclei $i$ and $j$ due to the
interaction of Eq. (1) is given by [34, 48, 35]

$\displaystyle V_{AV}(r)=V^{(1)}_{AV}(r)+V^{(2)}_{AV}(r)\,,$ (2)

with

$\displaystyle V^{(1)}_{AV}(r)$
$\displaystyle=\alpha_{X}^{ij}\frac{\left(\vec{\sigma}_{i}\times\vec{\sigma}_{j}\right)\cdot\hat{r}_{ij}}{2\overline{m}_{N}}\left(\frac{1}{r}+m_{X}\right)\frac{e^{-rm_{X}}}{r}\,$
$\displaystyle V^{(2)}_{AV}(r)$
$\displaystyle=\alpha_{X}^{ij}\frac{\vec{\sigma}_{i}}{2}\cdot\left\\{\frac{\vec{p}_{i}}{m_{i}}-\frac{\vec{p}_{j}}{m_{j}},\frac{e^{-rm_{X}}}{r}\right\\}\,,$
(3)

where $\vec{\sigma}_{j}$ are the Pauli matrices that follow the spin of the
valence nucleon, $\overline{m}_{N}=939\,[M]$ is the average nucleon mass and
$\vec{r}$ is the inter-nuclear axis of the molecule. The new physics
interaction strength is defined as

$\displaystyle\alpha_{X}^{ij}\equiv-\frac{1}{4\pi}(Z^{i}g_{V}^{p}+N^{i}g_{V}^{n})(B^{j}_{p}g_{A}^{p}+B^{j}_{n}g_{A}^{n})\,,$
(4)

where $Z^{i}\,(N^{i})$ is the number of protons (neutrons) in nucleus $i$. The
relation between the nucleus and the nucleon spins, as well as the proton-
neutron mass difference, is encoded in $B^{j}_{n,p}$ [48, 49]. The electron-
electron and electron-nuclei potentials can be derived from Eqs. (2) and (4)
by taking $\alpha^{ee}_{X}=-g^{e}_{V}g^{e}_{A}/(4\pi)$ and
$\alpha^{ej}_{X}=-g^{e}_{V}(B^{j}_{p}g_{A}^{p}+B^{j}_{n}g_{A}^{n})/(4\pi)$ or
$\alpha^{je}_{X}=-(Z^{i}g_{V}^{p}+N^{i}g_{V}^{n})g_{A}^{e}/(4\pi)$ for $ee$
and $eN$, respectively. For nucleus-nucleus interactions the $V^{(1)}_{AV}(r)$
part of Eq. (2) is the dominant contribution, since $V^{(2)}_{AV}(r)$ scales
with the nuclei velocity, which is very small, see below.

###  II.2 Benchmark spin-0 model

We consider a spin-0 particle, $\phi$ with both scalar and pseudoscalar
couplings to fermions, which induces a CP-violating potential at the molecular
scale. The effective couplings between $\phi$, the nucleons and the electrons
are given by

$\displaystyle\mathcal{L}_{\rm int}\subset\sum_{\psi=e,p,n}\\!\\!\\!\phi$
$\displaystyle\left(g_{\rm S}^{\psi}\overline{\psi}\psi+ig_{\rm
P}^{\psi}\overline{\psi}\gamma_{5}\psi\right)\,.$ (5)

Such an interaction can be mapped onto UV models, such as e.g., the CPV axion
[50, 51, 52] and CPV light scalars that are a result of relaxion mixing with
the SM-Higgs [53, 54, 55]. For a summary of existing constraints, see [56]. As
for the P-violating case, the effective couplings of Eq. (5) are constrained
both by terrestrial and astrophysical observations. We probe interactions at
the molecular scale, i.e., $m_{\phi}\sim\mathcal{O}([k])$.

Regarding terrestrial observations, the strongest bounds set on $g_{\rm
P}^{n,p}$ come from diatomic eEDM $g_{\rm S}^{n,p}g_{\rm P}^{n,p}\leq
1.2\times 10^{-17}$ [31]. Additional bounds arise from the proton and neutron
EDMs, $|d_{p}|<2.1\times 10^{-25}e\,$ [57] and $|d_{n}|<1.8\times 10^{-26}e\,$
[58, 59], which can be translated following [52] to $g_{\rm S}^{p}g_{\rm
P}^{p}<8.4\times 10^{-10}$ and $g_{\rm S}^{n}g_{\rm P}^{n}<1.0\times
10^{-10}$. As for the spin-1 case, the scalar interactions are constrained by
neutron scattering [37, 38, 39, 40] and molecular vibrational modes in HD
e.g., [41, 42]. For additional bounds, see e.g. [60]. Additional bounds can be
set using the the same J-coupling interaction in HD and H2 molecules as for
the $g_{A}^{N}$ constraint, $g_{\rm P}^{p}(g_{\rm P}^{n}+g_{\rm
P}^{p})\lesssim 6.9\times 10^{-4}$ [36].

Astrophysical constraints for $m_{\phi}\lesssim 10\,$ are due to hot neutron
stars $(g_{\rm P}^{n},g_{\rm P}^{p})<(1.3,1.5)\times 10^{-9}$ [61], stellar
cooling $g_{\rm S}^{N}<6.5\times 10^{-13}$ [47, 62], and SN1987 $g_{\rm
P}^{N}<6.0\times 10^{-10}$ [63]. Additionally, the scalar and pseudoscalar
proton couplings contribute to the scalar-photon and pseudoscalar-photon
couplings at the one loop level. Translating these constraints as in [64] we
obtain a bound of $\mathcal{O}(10^{-16})$ on $g_{S}^{p}g_{P}^{p}$. It is worth
noting that both the SN1987 bound and globular cluster bounds [65] have been
suggested to be subject to substantial uncertainties, casting doubt on their
reliability [66]. All of these astrophysical bounds can be avoided in models
that are subject to environmental effects, see e.g., [67, 68, 69, 70, 71, 72]
and discussion in [56].

The effective T-violating (which is also PV and CPV) potential between two
nuclei $i$ and $j$ is given by [34, 48, 35]

$\displaystyle V_{PS}(r)$
$\displaystyle=\alpha_{\phi}^{ij}\frac{\vec{\sigma}_{j}\cdot\hat{r}_{ij}}{2\overline{m}_{N}}\left(\frac{1}{r}+m_{\phi}\right)\frac{e^{-rm_{\phi}}}{r}\,.$
(6)

The new physics interaction strength is defined as in Eq. (4) but we replace
$X\to\phi$, $V\to S$ and $A\to P$.

##  III Angular momentum structure in polyatomic molecules

Chiral molecules are polyatomic molecules that possess a handedness, i.e.,
their mirror images are distinguishable. This asymmetry probes parity-
violating energy differences caused by both SM interactions (e.g., $Z$ boson
exchange between electrons and nuclei [73]) and new physics interactions. In
this section, we identify hyperfine states of rotating chiral molecules
$|\Psi_{\mathrm{mol}}\rangle$ that satisfy two criteria: (i) the expectation
values of the new physics interaction potentials, Eq. (2) and/or (6), are non-
zero, and (ii) the sign of this expectation value can be experimentally
switched to enable a differential measurement of the new physics term. Meeting
these criteria requires an examination of the alignment and orientation of
various (rotational and spin) angular momenta and bond vectors within the
rotating molecule-fixed frame.

The key factor of the effective $AV$ potential, for example, is the term
$\langle\Psi_{\mathrm{mol}}|\left(\vec{\sigma}_{i}\times\vec{\sigma}_{j}\right)\cdot\hat{r}_{ij}|\Psi_{\mathrm{mol}}\rangle\equiv\langle\left(\vec{\sigma}_{i}\times\vec{\sigma}_{j}\right)\cdot\hat{r}_{ij}\rangle$.
As we discuss below, this operator has a non-zero expectation value if the
nuclear spins have a fixed relative orientation to each other, not necessarily
in the molecule-fixed frame, and can be switched by swapping L and R
enantiomers. The corresponding factor of the effective $PS$ potential,
$\langle\vec{\sigma}_{j}\cdot\hat{r}_{ij}\rangle$, requires the nuclear spins
to be oriented, not just aligned, in the molecule-fixed frame using external
magnetic and electric fields, but is otherwise also switchable with chiral
enantiomers.

###  III.1 The effective hyperfine Hamiltonian

The hyperfine eigenstates are characterized by a minimum of six quantum
numbers: $N$ the rotational angular momentum, $K$ the projection of $N$ on the
quantization axis of the molecule, $S$ the electronic spin, $J=N+S$ the total
electronic angular momentum, $I_{n}$ the nuclear spin of atom $n$, the
intermediate sums of electronic and nuclear angular momentum
$F_{n-1}=J+I_{1}+..+I_{n-1}$ and finally $F=N+S+\sum_{i}I_{i}$, the total
angular momentum. Additionally, we define the labels $K_{a},K_{c}$ for the
different projection states of an asymmetric top molecule [74, 75]. The
effective hyperfine Hamiltonian is

$\displaystyle H=H_{\rm rot}+H_{\rm SN}+H_{\rm HFS}\,.$ (7)

The rotational angular momentum is described by $H_{\rm rot}$, where we used a
rigid rotor approximation

$\displaystyle H_{\rm rot}=\frac{1}{2}\vec{N}\cdot\mu\cdot\vec{N}\,,$ (8)

where $\mu$ is the inverse moment-of-inertia tensor. We choose the molecule-
fixed frame to be the principal axis system $\\{x,y,z\\}$, in which $\mu$ is
diagonal.

The electronic spin of the molecule is coupled to the rotational angular
momentum via

$\displaystyle H_{\rm SN}=\vec{S}\cdot\epsilon\cdot\vec{N}\,,$ (9)

where $\epsilon$ is the electronic spin-rotation coupling tensor [76]. We
consider two hyperfine interactions, a nuclear electric quadrupole term and an
electron-nucleus spin-spin interaction $H_{\rm HFS}=H_{\rm Q}+H_{\rm SI}$,

$\displaystyle H_{\rm Q}$
$\displaystyle=\sum_{i}\vec{I}_{i}\cdot\chi_{i}\cdot\vec{I}_{i}$ (10)
$\displaystyle H_{\rm SI}$ $\displaystyle=\sum_{i}\vec{S}\cdot
A_{i}\cdot\vec{I}_{i}\,,$ (11)

where $\chi_{i}$ are the nuclear electric quadrupole coupling tensors and
$A_{i}$ are the spin-spin coupling tensors for the electronic and nuclear
spins. In this case $A_{i}$ is composed of the Fermi contact interaction
$a_{F}^{i}$ and $T_{i}$ which is the nuclear spin-dipole coupling.

###  III.2 Parity violating effect

For the $AV$ potential of Eq. (2) to induce a PV contribution to the molecular
energy levels, we require a nonzero
$\expectationvalue{(\vec{\sigma}_{i}\times\vec{\sigma}_{j})\cdot\hat{r}_{ij}}$.
To obtain this, the nuclear spins must only be mutually oriented in the
molecule-fixed frame. In other words, $H_{\rm rot}$ aligns the direction of
$\vec{N}$ in the molecule frame, $H_{\rm SN}$ transfers this alignment to
$\vec{S}$, and $H_{\mathrm{SI}}$ finally transfers it to each $\vec{I}_{i}$.
The nuclear electric quadrupoles $\chi_{i}$ slightly modify the effect.
Although the individual $\vec{I}_{n}$ are not oriented, their pairwise cross-
products are oriented, because they are coupled to a common electronic spin
$\vec{S}$. This means that PV nucleon-nucleon interaction can be probed in
chiral molecules without any external field, which is similar to the case of
SM PV via $Z$-exchange [73]. We consider eigenstates of $H$ (for which $F$ is
a good quantum number), and to have a nonzero contribution of the
$AV$-potential we require these states to be superposition states of different
$\sum_{i}I_{i}$ (or equivalently, different $J$).

For a given transition, we define the P switch as the difference between the
left and right enantiomer transition energies

$\displaystyle\Delta E_{\rm PV}\equiv\frac{E_{R}-E_{L}}{2}\approx E^{\rm
SM}_{\rm PV}+\sum_{ij}E^{ij}_{\rm PV}\,,$ (12)

where $E^{\rm SM}_{\rm PV}$ is the SM PV contribution, which can be neglected
in the transitions we are interested in, see below. The second term is the new
physics effect and the sum runs over all nuclei pairs. There are three general
new physics sources of PV

$\displaystyle E_{\rm
PV}^{ij}=W^{P}_{ee}\alpha_{X}^{ee}+W^{P,ie}_{eN}\alpha_{X}^{ie}+W^{P,ij}_{NN}\alpha_{X}^{ij}\,,$
(13)

i.e., due to electron-electron, electron-nucleon and nucleon-nucleon
interactions. Here we focus on $W^{P,ij}_{NN}$ and leave discussion of
$W^{P,ie}_{eN}$ and $W^{P}_{ee}$ to future work.

We obtain $W^{P,ij}_{NN}$ by first-order perturbation theory and the potential
of Eq. (2),

$\displaystyle W^{P,ij}_{NN}=\frac{\left\langle
V_{AV}\right\rangle}{\alpha_{X}^{ij}}\approx\frac{\expectationvalue{\vec{\sigma}_{i}\times\vec{\sigma}_{j}}\cdot\hat{r}_{\rm
eq}^{ij}}{2\overline{m}_{N}}\left(\frac{1}{r_{\rm
eq}^{ij}}+m_{\phi}\right)\frac{e^{-r_{\rm eq}^{ij}m_{\phi}}}{r_{\rm
eq}^{ij}}\,,$ (14)

where $r_{\rm eq}^{ij}$ is the equilibrium distance between the two nuclei. In
polyatomic molecules, many possible atom pairs contribute to $W^{P,ij}_{NN}$,
but their expectation values
$\expectationvalue{\vec{\sigma}_{i}\times\vec{\sigma}_{j}}$ vary drastically
between the different molecular eigenstates.

A full analysis of the contributions of the $eN$ and $ee$ interactions is
beyond the scope of this work. However, for $m_{X}\gg[M]$ the $VA$ potential
is a rescaling of the SM weak interaction. Thus, we estimate the sensitivity
to the $eN$ contributions as a function of the uncertainty in the SM
measurement. Assuming that the PV shift in the vibrational modes is consistent
with the SM prediction, we project an upper bound of

$\displaystyle\frac{\absolutevalue{g_{V}^{p}g_{A}^{e}}}{\sqrt{2}m_{X}^{2}}$
$\displaystyle\leq\Delta G_{\rm F}\left(\frac{1}{4}-s_{W}^{2}\right)\,,$
$\displaystyle\frac{\absolutevalue{g_{V}^{n}g_{A}^{e}}}{\sqrt{2}m_{X}^{2}}$
$\displaystyle\leq\Delta G_{\rm F}\left(\frac{1}{2}\right)\,,$ (15)

where $s_{W}$ is the sine of the Weinberg angle, $G_{\rm F}\simeq 1.17\times
10^{-5}\,[G]^{-2}$ is the Fermi constant and $\Delta$ is the relative
uncertainty in the measurement.

###  III.3 Time reversal violating effect

For the $PS$ potential of Eq. (6) to induce a T-violating contribution to the
molecular energy levels, we need a nonzero expectation value
$\expectationvalue{\vec{\sigma}\cdot\hat{r}}$. With zero external field, the
nuclear spins (or any other angular momentum) are aligned with respect to the
molecule frame, but not oriented, so that
$\expectationvalue{\vec{\sigma}\cdot\hat{r}}=0$. We obtain nonzero
$\expectationvalue{\vec{\sigma}\cdot\hat{r}}$ by applying both an external
electric and external magnetic field. The $E$-field couples the molecular
frame to the lab frame via the molecular electric dipole, and the $B$-field
lifts the remaining degeneracy between the orientations. Generally, to then
see any T-violating effects, the direction of both EM fields must be reversed.
This is not the case for chiral molecules, where we can reverse the sign of
just one of the EM fields, and the second switch is obtained by comparing the
transitions of the left- and right-handed enantiomer. Thus, we define the
double difference

$\displaystyle\Delta E_{\rm
TV}\equiv\frac{\left(E_{L}^{\uparrow}-E_{L}^{\downarrow}\right)-\left(E_{R}^{\uparrow}-E_{R}^{\downarrow}\right)}{4}\approx\sum_{ij}E_{\rm
TV}^{ij}\,,$ (16)

where the arrow indicates the reversal of the direction of the external
magnetic field in the lab frame. The SM TV contribution is negligibly small.

We consider four sources of T-violation in molecules (for other potential
sources see e.g. [2, 4, 5, 6, 8, 9]), which can be written as

$\displaystyle E_{\rm
TV}^{ij}=W^{T,ij}_{d_{e}}d_{e}+W^{T,ie}_{eN}\alpha_{\phi}^{ie}+W^{T}_{ee}\alpha_{\phi}^{ee}+W^{T,ij}_{NN}\alpha_{\phi}^{ij}\,.$
(17)

As for the $VA$ case, we focus on $W^{T,ij}_{NN}$ and leave the electron-
electron and electron-nucleon interactions for future work. Moreover, these
interactions can be probed in atomic systems e.g. [8]. To estimate the
contribution of a possible eEDM, we need to compute the effective electric
field exerted on an electron in the molecule, which is beyond the scope of
this work. The nucleon-nucleon contribution can be estimated by first-order
perturbation theory

$\displaystyle W^{T,ij}_{NN}=\frac{\left\langle V_{\rm
SP}\right\rangle}{\alpha_{\phi}^{ij}}\approx\frac{\langle\vec{\sigma}_{j}\cdot\hat{r}_{\rm
eq}^{ij}\rangle}{2\overline{m}_{N}}\left(\frac{1}{r_{\rm
eq}^{ij}}+m_{\phi}\right)\frac{e^{-r_{\rm eq}^{ij}m_{\phi}}}{r_{\rm
eq}^{ij}}\,.$ (18)

We briefly note that, in principle, the TV necessary for non-zero
$\expectationvalue{\vec{\sigma}\cdot\hat{r}}$ could be obtained instead by
preparing non-stationary rotational quantum states that have a non-zero signed
projection ($K$) of the rotational angular momentum $N$ in the molecule-fixed
frame. In such “gyroscopic” states, the molecule can rotate with a well-
defined orientation in both the molecule-fixed and lab frames. The spin-
rotation and hyperfine interactions would ultimately transfer this orientation
to individual nuclear spins. For potential eEDMmeasurement in molecules
without external fields see [77].

##  IV Projection for $\text{CHDBrI}^{+}$

###  IV.1 Parity violating projection

Parameter | CH279BrI+ | CHD79BrI+
---|---|---
$A$/MHz | 16888.0 | 14781.2
$B$/MHz | 1109.2 | 1106.7
$C$/MHz | 1048.3 | 1040.0
$\mu_{a}$/au |  $-$0.170 |  $-$0.168
$\mu_{b}$/au | 0.825 | 0.824
$\mu_{c}$/au | 0 |  $-$0.048
Table 1: The rotational constants and permanent dipole moments of CH2BrI+ and CHDBrI+. Tensor111For each tensor quantity, the rows and column are listed in axis order $abc$. | CH279BrI+ | CHD79BrI+
---|---|---
|  $-$20323.7 |  $-$1515.1 | 0.0 |  $-$17872.6 |  $-$1509.9 | 84.4
$\epsilon$222The spin-rotation tensor indices are ordered here as $\epsilon_{ij}(S_{i}N_{j}+N_{j}S_{i})/2$. The values for CHD79BrI+ are derived from those of the parent isotopologue assuming pure spin-orbit contributions [78]. |  $-$17887.0 |  $-$1740.7 | 0.0 |  $-$15640.1 |  $-$1724.1 | 100.3
| 0.0 | 0.0 | 74.4 | 935.5 | 107.1 | 67.8
| 419.1 | 321.0 | 0.0 | 420.6 | 319.1 |  $-$19.4
$T$(Br) | 321.0 |  $-$135.3 | 0.0 | 319.1 |  $-$137.4 |  $-$8.9
| 0.0 | 0.0 |  $-$283.8 |  $-$19.4 |  $-$8.9 |  $-$283.2
| 293.3 |  $-$302.8 | 0.0 | 291.8 |  $-$302.6 | 16.8
$\chi$(Br) |  $-$302.8 | 177.6 | 0.0 |  $-$302.6 | 176.9 |  $-$37.5
| 0.0 | 0.0 |  $-$470.9 | 16.8 |  $-$37.5 |  $-$468.7
| 565.2 |  $-$835.8 | 0.0 | 561.2 |  $-$835.7 | 47.6
$T$(I) |  $-$835.8 | 31.1 | 0.0 |  $-$835.7 | 33.0 |  $-$35.9
| 0.0 | 0.0 |  $-$596.2 | 47.6 |  $-$35.9 |  $-$594.2
|  $-$1381.1 |  $-$190.3 | 0.0 |  $-$1382.0 |  $-$187.6 | 14.6
$\chi$(I) |  $-$190.3 |  $-$478.7 | 0.0 |  $-$187.6 |  $-$469.8 | 136.4
| 0.0 | 0.0 | 1859.8 | 14.6 | 136.4 | 1851.8
| 872.4 | 925.8 |  $-$672.6 | 875.2 | 885.7 |  $-$734.2
$T$(H333The H values are listed only for the H nucleus that remains unsubstituted in the deuterated isotopologue.) $\times 10^{3}$ | 925.8 | 2220.7 |  $-$1805.7 | 885.7 | 1987.8 |  $-$2102.1
|  $-$672.6 |  $-$1805.7 |  $-$3093.0 |  $-$734.2 |  $-$2102.1 |  $-$2863.0
|  |  |  | 134.8 | 148.7 | 94.7
$T$(D) $\times 10^{3}$ |  |  |  | 148.7 | 369.7 | 227.4
|  |  |  | 94.7 | 227.4 |  $-$504.5
|  |  |  |  $-$97.6 | 4.0 | 6.5
$\chi$(D) $\times 10^{3}$ |  |  |  | 4.0 |  $-$4.3 | 142.0
|  |  |  | 6.5 | 142.0 | 101.9
Table 2: The spin-rotation and anisotropic hyperfine coupling constants of CH2BrI+ and CHDBrI+ (in MHz). Nucleus |  $a_{F}$ (MHz)
---|---
79Br | 167.0
I | 256.7
H |  $-$1.998
D |  $-$0.307
Table 3: The isotropic hyperfine coupling constants of CH2BrI+ and CHDBrI+. |  |  | state 1 | state 2
---|---|---|---|---
| $(i,j)$ |  $E_{1\rightarrow 2}$ [[G]] | $\bra{N,K_{a},K_{c},J,F_{1},F_{2},F_{3},F}$ | $\expectationvalue{\sigma_{i}\times\sigma_{j}}\cdot\hat{r}_{\rm eq}^{ij}$ | $\ket{N,K_{a},K_{c},J,F_{1},F_{2},F_{3},F}$ | $\expectationvalue{\sigma_{i}\times\sigma_{j}}\cdot\hat{r}_{\rm eq}^{ij}$
(a),(b),(c) | (D, I) | 3.34 | $\bra{1,1,0,\frac{3}{2},1,1,\frac{5}{2},0}$ | 0.49 | $\ket{1,0,1,\frac{1}{2},1,2,\frac{7}{2},1}$ | 0.12
(d) | (H, I) | 3.99 | $\bra{2,2,0,\frac{5}{2},3,3,\frac{5}{2},0}$ | 0.23 | $\ket{1,0,1,\frac{3}{2},1,2,\frac{3}{2},1}$ | 0
Table 4: Summary of the optimal transitions to use for probing the PV effect of Eq. (2) on the rovibrational energy levels of $\text{CHDBrI}^{+}$. We denote the atom pair that gives the leading contribution to the new physics effect $(i,j)$ for each of the benchmarks outlined above, the optimal transition $E_{1\to 2}$, the quantum numbers (with the nuclear spins organized as (H, D, Br, I)) and the expectation value of the polarization for both the initial and final state.  Atom | $a$ | $b$ | $c$
---|---|---|---
C | 0.51 | 1.27 | 0
I |  $-$1.20 |  $-$0.07 | 0
79Br | 1.84 |  $-$0.14 | 0
H | 0.54 | 1.85 |  $-$0.92
H | 0.54 | 1.85 | 0.92
Table 5: The Cartesian coordinates of CH2BrI+ and $\text{CHDBrI}^{+}$ in their
PAS, in angstrom.

Figure 1: Constraint on the V-A couplings for the different benchmark
scenarios, when assuming an accuracy of the measurement of $0.1$ (long dashes)
and $1[m]$ (short dashes). For (a)($g_{V}^{n}=g_{A}^{n}=0$), the existing
constraint is obtained by combining the bounds on $g_{V}^{p}$ from HD
vibrational modes [41, 42] and the $g_{A}^{p}$ bound from the J-coupling in HD
[36]. For (b)($g_{V}^{p}=g_{A}^{p}=0$) there are no terrestrial experiments
that constraint the coupling. For
(c)($g_{V}^{p}=g_{V}^{n},g_{A}^{p}=g_{A}^{n}$), the existing constraint in the
combined bound of neutron scattering [37, 38, 39] and J-coupling in HD [36].
The strongest terrestrial constraint for
(d)($g_{V}^{p}=-g_{V}^{n},g_{A}^{p}=-g_{A}^{n}$) is due to the nuclear anapole
moment of 133Cs [45].

In this section we estimate the sensitivity of precision hyperfine
spectroscopy of $\text{CHDBrI}^{+}$ to probe PV from new physics. The proposal
for producing the $\text{CHDBrI}^{+}$ in an internally cold state via
photoinization [32] makes it suitable for precision searches in rotational
spectroscopy in addition to the vibrational spectroscopy applications [33, 18,
32].

The rotational constants for $\text{CHDBrI}^{+}$ were calculated as part of
this work and are given in Tab. 1 and are in agreement with those computed in
Ref. [32]. The calculated hyperfine coupling matrices, including the nuclear
quadrupole-electric field gradient coupling tensor $\chi_{i}$, the electron
spin-rotation coupling tensor $\epsilon$ , and the electronic-nuclear spin-
dipole coupling tensor $T_{i}$, are given in Tab. 2. The Fermi contact
parameters ($a_{F}$) are presented in Tab. 3. The structural and electronic
properties were calculated using the CFOUR quantum chemistry code [79, 80].
Details about the methods used are given in Appendix A. Values in the tables
are given for both CH2BrI+ and the chiral isotopologue $\text{CHDBrI}^{+}$,
which is the molecule of interest.

For simplicity, we assume that there are only nucleon-nucleon new physics
interactions. Their dominant effect is given in Eq. (14), where we consider
only $V_{AV}^{(1)}(r)$ and neglect $V_{AV}^{(2)}(r)$. The effect of
$V_{AV}^{(2)}(r)$ is sub-leading since it is proportional to the momentum of
the nuclei. In $\text{CHDBrI}^{+}$, the most energetic vibrational mode is the
C-H stretch mode, which has a frequency of $91\,[T]$ [32] and leads to a
proportionality factor $p/\mu\sim\mathcal{O}(10^{-5})$ compared to
$V^{(1)}_{AV}(r)$, where $p$ is the momentum in the rest frame and $\mu$ the
reduced mass of the C-H system.

So far the precision in spectroscopy of $\text{CHDBrI}^{+}$ has only been
estimated for vibrational transitions at below $0.1\,$ for C-H stretch and
even better for the C-H wag mode due to the seconds long natural lifetime [18,
32, 33]. In the rotational spectroscopy needed for the BSM effects discussed
in this work, the natural lifetime of rotational transitions will be much
longer. Thus, the coherence time is expected to be limited by magnetic field
noise due to the non-zero electronic spin of the molecule and black body
radiation. In the proof of concept experiment in Satterthwaite et al. [24] a
statistically limited $\sim 0.7\,$ precision was achieved between R and L
1,2-propanediol neutral chiral molecules, while in HfF+ molecular ions a
precision of $\sim 20[\mu]$ was reached [10]. It is challenging to estimate
the precision on $\text{CHDBrI}^{+}$ due to the early stages of the
experiment. Therefore, in this work we consider two cases: a conservative case
of $0.1\,$ and an aggressive case of $1\,[m]$. As we progress in production
and trapping of $\text{CHDBrI}^{+}$ molecules, these estimates will become
more refined.

The SM PV shift is typically several orders of magnitude smaller for
rotational transitions than in the vibrational transitions e.g. [24]. However,
an interesting aspect of $\text{CHDBrI}^{+}$ is that is only isotopically
chiral, such that the two enantiomers have a nearly identical electronic
configuration. Despite this, the molecule exhibits surprisingly large PV
shifts in some of its vibrational transitions [33]. For rotational transitions
the SM PV likely scales in similar fashion to other molecules, but we leave
detailed calculations for future work.

In our analysis we consider four benchmark scenarios for $X$-nucleon
interactions: (a) only protons $g_{V}^{n}=g_{A}^{n}=0$; (b) only neutrons
$g_{V}^{p}=g_{A}^{p}=0$; (c) protons and neutrons equally
$g_{V}^{p}=g_{V}^{n},g_{A}^{p}=g_{A}^{n}$; and (d) protons and neutrons
oppositely $g_{V}^{p}=-g_{V}^{n},g_{A}^{p}=-g_{A}^{n}$. For each scenario, we
have computed which rovibrational transition is most sensitive to the effect
of Eq. (2). The relevant transitions and expectation values are detailed in
Tab. 4. The Z-matrix elements of $\text{CHDBrI}^{+}$ can be found in Tab. 5.
We choose the inital state of the transition used such that the new physics
effect is maximized, and determine the transition with the largest intensity
from that state using PGOPHER [81]. The quantum numbers defining the state are
those of the main contribution to the relevant eigenstate, but in reality, the
eigenstates are superpositions of different quantum numbers. We obtain the
expectation values $\expectationvalue{\sigma_{i}\times\sigma_{j}}$ by
diagonalizing the Hamiltonian for the case where only the nuclei $(i,j)$ have
spin. We verify that the correction from the inclusion all four nuclear spins
is small by checking the change in the energy state in PGOPHER. With this, we
obtain

$\displaystyle g_{V}^{p}g_{A}^{p}$ $\displaystyle\leq 4.8\times 10^{-13}$
$\displaystyle g_{V}^{n}g_{A}^{n}$ $\displaystyle\leq 3.4\times 10^{-13}$
$\displaystyle g_{V}^{p,n}g_{A}^{p,n}$ $\displaystyle\leq 1.0\times 10^{-13}$
$\displaystyle g_{V}^{p,-n}g_{A}^{p,-n}$ $\displaystyle\leq 2.0\times
10^{-12}\,,$ (19)

for $m_{X}\lesssim[k]$ and an accuracy of $0.1$. The mass-dependent
constraints are shown in Fig. 1. For (b), to the best of our knowledge there
exist no other terrestrial constraint on the couplings. This is because any
probes of these couplings done using HD molecules are not valid since they
probe $g^{p}\times(g^{p}+g^{n})$. For (a), in order to be competitive with the
existing bound an accuracy of $\sim 0.01\,$ is required. For (c), the existing
terrestrial constraints outperform the ones that can be set using chiral
molecules with the current precision. For (d), in order to be competitive with
the existing bound an accuracy of $\mathcal{O}([k])$ is required.

Regarding the electron-nucleon interaction, we assume that the vibrational
mode measurement is compatible with the SM expectation to the $\Delta\sim 0.1$
level, $\sim 0.1\,$. By using Eq. (III.2) we project the following limit of

$\displaystyle g_{V}^{p}g_{A}^{e}$ $\displaystyle\lesssim 5\times
10^{-12}\left(\frac{\Delta}{0.1}\right)\left(\frac{m_{X}}{10[M]}\right)^{2}\,,$
$\displaystyle g_{V}^{n}g_{A}^{e}$ $\displaystyle\lesssim 8\times
10^{-11}\left(\frac{\Delta}{0.1}\right)\left(\frac{m_{X}}{10[M]}\right)^{2}\,.$
(20)

This is comparable with the current bounds from APV [27, 45]. As a proof of
concept, we adopt the result of [15] for CHFClBr with is consistent with no PV
and put an upper limit of $13\,$ on the L and R difference. There is a large
uncertainty on the theoretical prediction of the SM PV, Ref. [82] predicts
$0.06\,$, while Ref. [83] predicts $1\,[m]$. Conservatively, we use the later
SM prediction, thus, $\Delta\sim 10^{4}$ and the resulting upper bounds are

$\displaystyle[g^{p}_{V}g^{e}_{A}]_{\rm CHFClBr}\lesssim 5\times
10^{-7}\left(\frac{m_{X}}{10[M]}\right)^{2}\,,$
$\displaystyle[g^{n}_{V}g^{e}_{A}]_{\rm CHFClBr}\lesssim 8\times
10^{-6}\left(\frac{m_{X}}{10[M]}\right)^{2}\,,$ (21)

which, to the best of our knowledge, are the first bounds on BSM from chiral
molecules.

###  IV.2 Time Violating projection

Figure 2: The constraints on $g^{p/n}_{S}g^{p/n}_{P}$ that can be set with a
measurement with aggressive precision of $0.001$ of rovibrational transitions
in $\text{CHDBrI}^{+}$. We show the bounds for the four benchmark scenarios
(a),(b),(c)and (d). We show the bounds that can be set by reinterpreting the
$e$EDM measurement in HfF$+$ as well as combined existing bound from neutron
scattering and vibrational spectroscopy in HD.

Here we do an order-of-magnitude estimation as to the sensitivity of precision
hyperfine measurements in $\text{CHDBrI}^{+}$ to probe TV new physics. We
consider the purely nucleon-nucleon interaction and leave an analysis of the
other T-violating sources to future work. The SM contribution to TV or CPV is
negligibly small. External electric and magnetic fields are required to orient
the molecule such that $\expectationvalue{\vec{\sigma}\cdot\hat{r}}\neq 0$. We
can choose a transition such that all nuclear spins are oriented along the
direction of the magnetic field. For simplicity, we take
$\vec{\sigma}=(0,0,1)$ for all nuclei. The bound obtained, for the four cases
(a), (b), (c) and (d) are

$\displaystyle g_{S}^{p}g_{P}^{p}$ $\displaystyle\leq 9.8\times 10^{-13}$
$\displaystyle g_{S}^{n}g_{P}^{n}$ $\displaystyle\leq 1.4\times 10^{-13}$
$\displaystyle g_{S}^{p,n}g_{P}^{p,n}$ $\displaystyle\leq 9.5\times 10^{-14}$
$\displaystyle g_{S}^{p,-n}g_{P}^{p,-n}$ $\displaystyle\leq 8.2\times
10^{-13}\,,$ (22)

for $m_{\phi}\lesssim[k]$ and assuming an accuracy of $0.1$. The mass
dependent bounds are shown in Fig. 2. For the bound to be competitive with
bounds set using diatomic eEDM bounds we need a precision of the measurement
comparable to the eEDM measurements, i.e., $10^{-5}$. This is because the TV
effect in both systems has a similar absolute value, the nuclear charges are
of the same order of magnitude and the polarization is comparable. We leave a
detailed analysis of the optimal transition to choose until experiment reaches
this limit. We comment that a measurement with such an accuracy may also be
able to probe the eEDM with a similar sensitivity as the current probes in
diatomic molecules [10, 12], but detailed calculations are required.

##  V Conclusion

In this work, we have shown that chiral molecules provide a unique environment
for probing PV new physics. For example, chiral molecules can be the most
sensitive terrestrial probe of PV in the case of a new spin-1 particle that
couples only to neutrons, or to neutrons and protons oppositely. The chirality
switch of chiral molecules provides a straightforward approach to remove the
majority of significant SM contributions, thus the proposed method is free of
SM background and does not require sophisticated SM calculations. For the case
of coupling only to the proton, for $[m]$ precision of the PV energy
difference measurement, we would improve current bounds. When coupling to
protons and neutrons evenly, our bound is less constraining than the combined
bound due to neutron scattering and HD precision measurements. We also show
that measurements of the SM PV energy difference can constrain the electron-
nucleon PV interaction, to the level of APV. The bound we extract from current
data is weaker than the APV bounds. Additionally, we have shown that chiral
molecules can probe TV new physics. They would present a competitive probe
only when the precision of the measurement becomes comparable to that of eEDM
measurements in diatomic molecules.

As development of state preparation techniques for $\text{CHDBrI}^{+}$
progresses, opportunities for precision metrology of properties other than the
vibrational degrees of freedom, which are well motivated, will arise. This
paper motivates the development of hyperfine and rotational precision
spectroscopy in $\text{CHDBrI}^{+}$ and chiral molecules in general.

###### Acknowledgements.

We thank Arie Landau, Inbar Savoray and Gilad Perez for collaboration at early
stages of this project. We would like to thank Joshua Baraban, Itay Bloch,
Yevgeny Stadnik and Stephan Schiller for useful discussions. We thank Dima
Budker and Inbar Savoray for comments on the manuscript. C.B. and Y.So. are
supported by grants from the NSF-BSF (grant No. 2021800), the ISF (grant No.
483/20), the BSF (grant No. 2020300) and by the Azrieli foundation. P.B.C. is
supported by the U.S. NSF (grant No. PHY-2110489). Y.Sh. gratefully
acknowledges funding by the European Union (ERC, 101116032 – Q-ChiMP). Views
and opinions expressed are however those of the authors only and do not
necessarily reflect those of the European Union or the European Research
Council Executive Agency. Neither the European Union nor the granting
authority can be held responsible for them. YSh. also acknowledges support
from the US-Israel Bi-national Science Foundation (BSF) [Grant no. 2022160]
and Israel Science Foundation (ISF) [Grant no. 1142/21].

##  Appendix A Quantum chemical calculations

The structural and electronic properties of CH2BrI+ and CHDBrI+ were computed
with coupled cluster theory including single, double, and perturbative triple
excitations [CCSD(T)] [84, 85] as implemented in the CFOUR quantum chemistry
code [79, 80]. The optimized geometry and first-order electronic properties
were calculated using analytic gradient techniques [86], while spin-rotation
parameters were calculated within the effective one-electron spin-orbit
operator approach of Ref. [87] based on analytic second-derivatives for open-
shell CCSD(T) [88]. Because this implementation of spin-rotation constants in
CFOUR is only available with non-relativistic CCSD(T) and without effective
core potentials, the all-electron 6-311G* and 6-311G** basis sets [89, 90, 91]
were chosen for all geometry and property calculations.

Preliminary calculations of fine and hyperfine properties were performed for
the planar CH2Br and CH2I radicals to assess the performance of the non-
relativistic CCSD(T) predictions for molecules containing the heavy atoms Br
and I. The calculated and experimental values for the spin-rotation
($\epsilon_{aa}$, $\epsilon_{bb}$, and $\epsilon_{cc}$), nuclear magnetic
hyperfine ($a_{F}$, $T_{aa}$ and $T_{bb}-T_{cc}$), and nuclear electric
quadrupole coupling constants ($\chi_{aa}$ and $\chi_{bb}-\chi_{cc}$) are
summarized in Table 6. The fractional agreement for each of the large in-plane
components of the spin-rotation tensor ($\epsilon_{aa}$ and $\epsilon_{bb}$)
is reasonably good (1–10%). The out-of-plane component ($\epsilon_{cc}$) has
much worse fractional accuracy (30–40%), but this behavior is typical of
planar $\pi$ radicals like CH2Br and CH2I for which the out-of-plane spin-
rotation parameter has a small absolute magnitude by symmetry arguments [92,
93]. The magnetic and electric quadrupole hyperfine coupling constants are
also in generally good agreement, with the largest fractional error for the
iodine Fermi contact parameter ($a_{F}$). This discrepancy is again not
surprising at this level of theory, which includes neither core-electron
correlation nor orbital-following zero-point vibrational effects, both of
which can be important for calculating accurate electronic spin densities in
planar $\pi$ radicals [94]. Nonetheless, this theoretical approach is shown to
provide qualitative to semi-quantitative accuracy for the fine and hyperfine
properties of Br and I-bearing radicals.

| CH279Br | CH2I
---|---|---
Parameter | Calc.111Analytic equilibrium values at the frozen-core UHF-CCSD(T) level of theory with the 6-311G* basis set. | Expt.222Experimental values from Refs. [92, 93] | Calc.111Analytic equilibrium values at the frozen-core UHF-CCSD(T) level of theory with the 6-311G* basis set. | Expt.222Experimental values from Refs. [92, 93]
$\epsilon_{aa}$ |  $-$11151.5 |  $-$12569.80(2) |  $-$29031.1 |  $-$29409.76(1)
$\epsilon_{bb}$ |  $-$633.8 |  $-$699.23(7) |  $-$910.5 |  $-$926.284(2)
$\epsilon_{cc}$ | 90.6 | 68.39(7) | 295.9 | 208.565(2)
$a_{F}$(Br/I) | 26.3 | 22.7904(9) | 5.4 |  $-$15.8381(5)
$T_{aa}$(Br/I) |  $-$151.1 |  $-$149.718(2) |  $-$165.1 |  $-$152.9644(7)
$T_{bb}-T_{cc}$(Br/I) |  $-$287.2 |  $-$365.68(2) |  $-$280.2 |  $-$430.966(2)
$\chi_{aa}$(Br/I) | 516.8 | 518.339(4) |  $-$1631.7 |  $-$1745.022(3)
$\chi_{bb}-\chi_{cc}$(Br/I) |  $-$107.7 |  $-$132.78(6) | 319.1 | 434.20(1)
$a_{F}$(H) |  $-$77.5 |  $-$60.208(1) |  $-$75.1 |  $-$57.6046(7)
$T_{aa}$(H) |  $-$20.5 |  $-$21.894(3) |  $-$19.3 |  $-$20.708(2)
$T_{bb}-T_{cc}$(H) | 29.6 | 28(3) | 28.6 | 19.3(1)
Table 6: The fine and hyperfine parameters (in MHz) of CH2Br and CH2I in their
2B1 electronic ground states.

The equilibrium geometry of the 2A′ ground electronic state of CH2BrI+
optimized at the UHF-CCSD(T)/6-311G** level of theory is listed in Table 5 in
its principal axis system (PAS). Table 1 lists the rotational constants and
permanent electric dipole moment components for CH2BrI+ and CHDBrI+, Table 2
lists the isotropic hyperfine coupling constants (which are the same for both
isotopologues), and Table 2 lists the spin-rotation tensor and anisotropic
magnetic and electric quadrupole hyperfine coupling constants. For the
magnetic and electric quadrupole hyperfine constants, it is straightforward to
transform the values calculated directly for CH2BrI+ to those for CHDBrI+ by
rotating the tensors into the new rotational principal axis system. The
transformation of the spin-rotation tensor requires a more elaborate procedure
because it is a second-order property that depends on the rotational kinetic
energy operator itself. Assuming that the spin-rotation values are dominated
by second-order spin-orbit contributions, we use well known isotopic scaling
relations [78] to derive the spin-rotation parameters of the deuterated
isotopologue.

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