Probing Fundamental Symmetries of Deformed Nuclei in Symmetric Top Molecules
Phelan Yu
∗
and Nicholas R. Hutzler
†
Division of Physics, Mathematics, and Astronomy,
California Institute of Technology, Pasadena, California 91125, USA
(Dated: August 21, 2020)
Precision measurements of Schiff moments in heavy, deformed nuclei are sensitive probes of beyond
Standard Model
T,P
-violation in the hadronic sector. While the most sensitive limits on Schiff
moments to date are set with diamagnetic atoms, polar polyatomic molecules can offer higher
sensitivities with unique experimental advantages. In particular, symmetric top molecular ions
possess
K
-doublets of opposite parity with especially small splittings, leading to full polarization at
low fields, internal co-magnetometer states useful for rejection of systematic effects, and the ability to
perform sensitive searches for
T,P
-violation using a small number of trapped ions containing heavy
exotic nuclei. We consider the symmetric top cation
225
RaOCH
+
3
as a prototypical and candidate
platform for performing sensitive nuclear Schiff measurements and characterize in detail its internal
structure using relativistic
ab initio
methods. The combination of enhancements from a deformed
nucleus, large polarizability, and unique molecular structure make this molecule a promising platform
to search for fundamental symmetry violation even with a single trapped ion.
Searches for permanent electric dipole moments
(EDMs) in atoms and molecules are powerful probes
of time reversal and parity (
T
,
P
-) violating physics
posited by beyond Standard Model (BSM) theories [1, 2].
Unsuppressed
T
,
P
-violation, and by extension charge
conjugation-parity (
CP
-) violation, are needed to explain
the observed lack of free antimatter in the universe [3]. In
the Standard Model, however,
CP
-violation only weakly
manifests in quark and neutrino mixing phases and is
apparently absent for strong interactions (the strong CP
puzzle) [4]. The hadronic sector thus provides a natu-
ral venue for introducing many new
CP
-violating BSM
interactions to resolve this discrepancy [5–7].
New
T
,
P
-violating nuclear effects, including nucleon-
nucleon interactions mediated by QCD, are understood
to induce a collective EDM in atomic nuclei of non-zero
spin known as a Schiff moment [8–10]. This effect, which
scales with atomic mass
Z
, is particularly pronounced in
heavy, octopole-deformed nuclei, such as
225
Ra [11, 12],
where low-lying nuclear states couple strongly to the
opposite-parity ground state [13]. The resulting Schiff
moment, and corresponding sensitivity to BSM physics,
is a factor of
&
100 larger [14–16] when compared to
heavy spherical nuclei, such as
129
Xe [17, 18] and
199
Hg
[19], the latter of which is used in the current most sen-
sitive Schiff moment experiment.
Heavy, octopole-deformed isotopes, however, are typ-
ically short-lived and difficult to produce in large quan-
tities [11, 13, 20]. Maximizing experimental sensitiv-
ity and coherence time is thus paramount to overcom-
ing a limited count rate. One demonstrated method
for increasing experimental sensitivity is to use a polar
molecule, whose internal fields can be easily oriented to
provide an enhancement of
&
100 over atoms in EDM
measurements [21–23]. TlF, for instance, is sensitive to
the Schiff moment of
205
Tl nuclei [24, 25], and theoret-
ical proposals have identified a wide variety of diatomic
(ThO
+
, ThF
+
, AcF, AcO
+
, AcN, EuO
+
, EuN, RaO,
RaF) [26–31] and triatomic molecules (RaOH
+
, TlOH,
ThOH
+
, TlCN) [27, 32, 33] suitable for Schiff moment
measurements. Combining enhancements due to nuclear
deformation and the polarizability of molecules results
in
&
10
5
sensitivity increase relative to atomic Schiff mo-
ment measurements with spherical nuclei.
Molecular ions have proven to be a powerful platform
for very sensitive measurements of symmetry violation
[22] due to long trapping and coherence times [34]. This
enables the ability to perform measurements with small
quantities of the target molecule, for example those con-
taining scarce or unstable nuclei. However, many BSM-
sensitive species, including radium, do not have the pre-
requisite electronic structure to make diatomic molecular
ions with opposite-parity (Ω) doublets, which are needed
to fully realize the advantages of this approach. Poly-
atomic molecules, by contrast, possess rovibrational par-
ity doublets [32], and thus provide a generic approach
to conducting molecular ion measurements with a broad
range of useful, and possibly rare, species.
In this manuscript, we consider a symmetric top
molecule (STM), the radium monomethoxide cation
(RaOCH
+
3
), as a platform to combine nuclear and molec-
ular enhancements with the advantages of a polyatomic
structure and extended coherence time achievable with
an ion trap. This molecule, which was recently produced
and co-trapped [35] with laser-cooled Ra
+
[36], has ax-
ial symmetry that gives rise to near-degenerate opposite
parity
K
-doublets, thereby enabling full polarization in
small fields and the co-magnetometer states necessary
for sensitive measurements in an ion trap. The ground
electronic state (
̃
X
1
A
1
) is diamagnetic, suppressing sen-
sitivity to magnetic noise. Due to this combination of en-
hancements and features, even a single trapped RaOCH
+
3
ion could be used to explore interesting parameter space
for new physics.
arXiv:2008.08803v1 [physics.atom-ph] 20 Aug 2020
2
K
=
−
1
K
=+1
m
F
=
−
2
m
F
=+2
E
-field
K
×
m
N
> 0
(energy < 0)
K
×
m
N
< 0
(energy > 0)
K
=
−
1
K
=+1
K
= 0
ortho-NSI
para-NSI
Ra
O
C
+
−
Figure 1. Labeled
225
RaOCH
+
3
STMs in stretched states,
grouped by energy and total angular momentum projection
m
F
. The internuclear axis runs from the negatively charged
end (methyl group) to the positively charged metal (radium).
K
is the molecule-frame projection of angular momentum
without spin
N
, while
m
N
is its projection onto the lab frame.
Nuclear spins for each spin-1/2 nucleus (
1
H,
225
Ra) are indi-
cated. The
|
K
|
= 1 states correspond to the mixed para
nuclear spin isomer (NSI) of the hydrogens, while the
K
= 0
state coincides with the stretched ortho-NSI.
There are several advantages to using a more complex
STM ion, as opposed to a triatomic analog (e.g. RaOH
+
)
[27, 32, 37]. First, the increased rovibrational complexity
of an STM, which makes laser cooling of neutral species
more difficult (though indeed possible [38, 39]), does not
pose challenges for the control of STM ions, as trapped
ions do not require photon cycling to achieve high preci-
sion [22, 34]. Furthermore,
K
-doublets, which arise from
rotational degrees of freedom, can arise in any vibrational
state, such as the ground state considered here. They
are therefore are low-lying (
ν
∼
100 GHz), have vastly
longer radiative lifetimes than excited vibrational modes,
and possess smaller splittings than the
`
-doublets of tri-
atomics.
Our theoretical analysis focuses on
225
RaOCH
+
3
, which
contains the short-lived (
τ
1
/
2
≈
15 d) spin-1
/
2 radium
isotope. We examine in detail the ground state hyper-
fine structure, as well as the various contributions to
the degeneracy-breaking of the
K
-states. We further-
more identify states suitable for measurement of a Schiff
moment, including co-magnetometer states, and examine
the Stark and Zeeman effects in the molecule.
Internal Structure and
K
-doubling.
The internal struc-
ture of the electronic ground state (
̃
X
1
A
1
) is analyzed
with explicit diagonalization of the effective molecular
Hamiltonian
H
total
=
H
rot
+
H
stark
+
H
zeeman
+
H
ss
+
H
nsr
+
H
sm
.
(1)
We have included the rotational (rot), Stark, Zeeman,
nuclear spin dipolar (ss), and nuclear spin-rotation terms
(nsr), all of which are generic to STMs. The Schiff mo-
ment (sm) term arises from the
225
Ra(
I
= 1
/
2) nucleus,
and is
T
,
P
-violating. Similar to the closed-shell alkali
monomethyls [40], electron spin terms are omitted. We
obtain molecular parameters (see table I) using a va-
riety of relativistic
ab initio
methods. Geometries are
optimized at the level of CCSD(T) with an ANO-RCC-
VQZ basis [41–46] via CFOUR [47–49], and scalar rela-
tivistic effects are modeled using the one-electron vari-
ant of the spin-free X2C Hamiltonian [50–52]. Nuclear
spin-rotation and rotational Zeeman parameters are com-
puted via a four-component relativistic linear response
approach [53–55] in the DIRAC19 code [56] using the
dyall.v4z basis [57], and electron correlation is treated at
the DFT level with a B3LYP functional [58]. Additional
details on the derivation of the Hamiltonian and
ab initio
parameters can be found in the Supplemental Material.
The rotational structure of symmetric tops is parame-
terized by three quantum numbers: the electronic angu-
lar momentum apart from spin (
N
), its molecule-frame
projection (
K
), and its lab-frame projection (
m
N
). For
|
K
|
>
0, which corresponds to rotation about the symme-
try axis, the cylindrical symmetry of the molecule gives
rise to a pair of degenerate +
K
and
−
K
states within
each
|
N,
|
K
|〉
rotational manifold (see fig. 1). These
degeneracies can be lifted by hyperfine and centrifugal
terms that couple states of different
K
, leading to the
formation of near-degenerate opposite parity
K
-doublets,
|±〉
= (
|
N,
+
K
〉±|
N,
−
K
〉
)
/
√
2.
We propose to use the
N
=
|
K
|
= 1 manifold for the
Schiff moment search. This state is
∼
160 GHz above
the absolute ground state, and accordingly has a much
longer lifetime than any vibrationally excited mode (with
frequencies
&
1 THz) such as those in triatomics. The
spontaneous decay rate is further suppressed as the tran-
sition to the lower
K
= 0 ground state is spin-forbidden,
making the radiative lifetime much longer than 1 hour or
any other relevant experimental timescale.
Similar to Ω and
`
-doublets, opposite parity
K
-
doublets can be mixed in electric fields where the Stark
energy exceeds the zero-field
K
-doublet splitting. In this
regime, the molecule is polarized and its internal fields
are oriented in the lab frame (see. fig 2). This gives
insensitivity to electric field fluctuations by largely satu-
rating the Schiff moment sensitivity, as well as enabling
co-magnetometer states. In open-shell species, such as
CaOCH
3
, the splitting between the ground state
K
-
doublets is
∼
0
.
3 MHz [59], dominated by an anisotropic
hyperfine interaction between the proton spins and the
metal-centered electron. In contrast, the absence of un-
paired electron spin in the
225
RaOCH
+
3
ground state im-
plies that the dominant hyperfine contributions to
K
-
doubling are from nuclear spin interactions, which are
suppressed generically by at least an order of magnitude
due to the minute size of nuclear magnetic moments com-
pared to electronic magnetic moments [60, 61].
For the
N
=
|
K
|
= 1 manifold, we calculate that
anisotropic nuclear spin-spin and spin-rotation contri-
3
K
×
m
N
0
−
1
+1
−
2
−
1
0
1
2
Electric Field (mV/cm)
Dipole Moment (D)
0
10
20
30
40
50
Figure 2. Lab frame dipole moment of hyperfine states of the
|
N
= 1
,
|
K
|
= 1
〉
manifold in the limit where the
K
doublets
are fully mixed yet rotational mixing is negligible. High, low,
and no field seekers correspond to states with negative, pos-
itive, and zero dipole moment (
K
×
m
N
= +1, 0, and
−
1).
The jumps indicate avoided crossings.
butions from the hydrogen nuclei generate sub-kHz
K
-
doublings. Combined with the calculated dipole moment
of
≈
5 D, this results in an extremely low threshold for
reaching the high-field limit and polarizing the molecule.
Indeed, we find that states in this manifold are
>
90%
polarized in external electric fields of 50 mV cm
−
1
and
>
99
.
9% polarized in fields of 250 mV cm
−
1
. This thresh-
old is even lower for stretched states with maximal pro-
jection of total angular momentum (
m
F
), which reach
full polarization (
>
99
.
9%) in fields as low as
.
mV cm
−
1
(see fig. 2), small enough that the molecules could be
polarized in
∼
mK deep optical traps [62]. Rotational
mixing can be neglected at these small fields, which we
assume is the case for the remainder of the manuscript.
The absence of unpaired electron spin in the
̃
X
1
A
1
ground state also has implications for the magnetic level
structure, as only the nuclear and rotational moments
contribute to the Zeeman energy of the ground state, and
are of order
∼
μ
N
, the nuclear magneton. This results
in suppressed sensitivity to magnetic fields and effective
g
-factors that are a factor of
∼
10
3
smaller than a Bohr
magneton
μ
B
(see fig 3).
Hyperfine Structure.
The large number of internal de-
grees of freedom in
225
RaOCH
+
3
necessitates a detailed
treatment of the hyperfine structure to both identify re-
solvable states for the Schiff moment measurement as
well as elucidate the sources of
K
-doubling. We use a
fully decoupled basis,
|
N,K,m
N
〉|
Γ
,I
H
,m
IH
〉|
I
M
,m
IM
〉
,
to describe the hyperfine structure of the
̃
X
1
A
1
electronic
state. The spin of the metal
225
Ra nucleus is denoted
I
M
, and
m
IM
is its corresponding lab frame projection.
Similarly, the total nuclear spin of the three hydrogen
atoms and its lab frame projection are denoted
I
H
and
m
IH
, while Γ denotes the symmetry character of the hy-
drogen spin wavefunction under
C
3
v
transformations. In
concrete terms, these hydrogen spin wavefunctions cor-
respond to either of two nuclear spin isomers (NSIs): the
“ortho” stretched states (Γ =
A,I
H
= 3
/
2) and the
“para” mixed states (Γ =
E,I
H
= 1
/
2).
In each
|
N,
|
K
|〉
manifold, symmetry arguments re-
strict the allowed NSIs [63]. In particular, ortho states
are only allowed with
K
= 3
n
rotational states (for inte-
ger
n
), while the para states are associated with
K
6
= 3
n
states. All other combinations are forbidden by quantum
statistics and
C
3
v
symmetries. (See Supplemental Mate-
rial for details.) Accounting for these restrictions results
in a total of 24 hyperfine states in the
N
= 1
,
|
K
|
= 1
manifold, which are all resolvable in high fields.
Two types of hyperfine terms are present in the
̃
X
1
A
1
state: dipolar couplings between the spins of different nu-
clei, and couplings between the nuclear spin and molec-
ular rotation. The Hamiltonian for dipolar nuclear spin-
spin interaction between two spins
I
1
and
I
2
can be ex-
pressed in terms of spherical tensors [64],
H
ss
=
−
√
6
μ
0
γ
1
γ
2
~
2
4
π
T
2
(
C
dip
)
·
T
2
(
I
1
,
I
2
)
.
(2)
where
μ
0
is the vacuum permeability,
γ
1
,
γ
2
are the gyro-
magnetic ratios, and
C
dip
is a spin-spin coupling tensor.
For the
N
=
|
K
|
= 1 manifold, we only need to consider
spin-spin interactions between the ortho-NSI hydrogens
and the
225
Ra atoms, as the inter-hydrogen matrix ele-
ments vanish between para-NSI states [65, 66]. The dipo-
lar spin coupling tensor
T
2
(
C
dip
), which can be directly
evaluated as a sum of spherical harmonics, gives a diago-
nal shift of
∼
40 kHz for
I
H
·
I
R
interactions. Anistropic
couplings of
∼
400 Hz mix states differing by ∆
K
= 2,
which contributes to the
K
-doubling.
Nuclear spin-rotation is the interaction between a nu-
clear magnetic moment associated with a spin
I
and the
magnetic field created by the rotational angular momen-
tum
N
[67],
H
nsr
=
1
2
2
∑
k
=0
[
T
k
(
C
nsr
)
·
T
k
(
N
,
I
) +
T
k
(
N
,
I
)
·
T
k
(
C
nsr
)
]
,
(3)
where
C
nsr
is a spin-rotation coupling tensor. Both
225
Ra
and the hydrogen nuclei in the ortho-NSI contribute to
the nuclear spin-rotation interaction. The former pro-
duces a diagonal shift of
∼
4 kHz for
N
·
I
R
interactions,
while the latter produces both a diagonal shift of
∼
15
kHz for
N
·
I
H
interactions and
∼
300 Hz off-diagonal
couplings between states differing by ∆
K
= 2, which
contributes to the
K
-doubling.
Measurement.
Since the energy shift from the Schiff
moment is proportional to the projection of the radium
spin onto the molecular axis (
I
M
·
ˆn
), different hyperfine
states have different Schiff moment sensitivities. In the
fully decoupled limit (
≥
1 V/cm), the value of
〈
I
M
·
ˆn
〉
is
K
·
m
N
·
m
IM
/
2, where the prefactor arises because
we do not mix rotational states beyond
N
= 1 and thus
4
−
1
0
+1
K
×
m
N
+1.2 MHz
−
1.2 MHz
~6.7 kHz
~6.7 kHz
~1.8 kHz
H
NSR
+
H
SS
−
2
+2
+1
−
1
0
m
F
+0.871
+0.871
−
0.871
−
1.017
−
0.871
+0.311
+1.017
−
0.311
−
0.458
+0.458
−
0.312
−
1.017
−
0.458
+0.458
+1.017
+0.312
Figure 3. Level structure and Schiff moment sensitivities for
the 24 hyperfine states of the
|
N
= 1
,
|
K
|
= 1
〉
manifold in
the decoupled regime (1 V/cm), grouped by the projection of
total angular momentum
m
F
=
m
N
+
m
IH
+
m
IM
and their
Stark manifold (
K
×
m
N
). Gold states have +1/4 effective
Schiff sensitivity, while blue states have
−
1
/
4 effective Schiff
sensitivity. Dashed lines denote zero Schiff sensitivity. Labels
above/below the states indicate the effective
g
-factor at zero
magnetic field, in terms of nuclear magnetons (
μ
N
). Table S4
lists the admixtures for each state.
the maximum projection of the internuclear axis on the
lab frame is 1
/
2. (see fig. 3 and table S4). There are
therefore three distinct classes of states, with positive,
zero, and negative Schiff energy shifts. Even in the in-
termediate regime when a molecule is polarized, but the
spins are not decoupled (
∼
50 mV cm
−
1
), we still find
stretched states that maximally project the radium spin
onto the molecular axis.
The measurement manifold is naturally populated at
cold temperatures, with
∼
1% of molecules occupying
the
N
=
|
K
|
= 1 manifold at 4 K. This yield could
be increased via state-controlled reactions of Ra
+
and
Table I. Molecular parameters for
225
RaOCH
+
3
ground state
(
̃
X
1
A
1
). See Supplemental Material for details.
Hyperfine
T
zz
|
T
xx
−
T
yy
|
T
(
C
nsr
(
225
Ra))
3
.
67 kHz
–
T
(
C
nsr
(
1
H))
15
.
3 kHz
0
.
301 kHz
α
dip
·
T
(
C
dip
(
225
Ra
−
H))
a
−
38
.
0 kHz
0
.
391 kHz
Geometry
Stark and Zeeman
r(Ra-O)
2.1949
̊
A
d
0
4.969 D
r(O-C)
1.4076
̊
A
g
N
(
225
Ra)
[68]
−
0
.
7338
μ
N
r(C-H)
1.0864
̊
A
g
N
(H)
[69]
2
.
7928
μ
N
∠
(O-C-H) 110.73
◦
g
R
(
‖
)
6
.
65
μ
N
∠
(H-C-H) 108.18
◦
g
R
(
⊥
)
0
.
619
μ
N
A
5.4010 cm
−
1
B
0.0673 cm
−
1
a
The scaling constant for the dipolar spin-spin interaction is
defined as
α
dip
=
−
√
6
μ
0
γ
H
γ
Ra
~
2
/
4
π
methanol [70]. State preparation is possible through
state-selective dissociation [34] or non-destructive quan-
tum logic [71, 72] via co-trapped Ra
+
[36].
Two schemes can be used for performing the EDM
measurement. In the spin interferometery (SI) method
[22], the molecule precesses between states of differ-
ent Schiff sensitivity. After a time
τ
, the phase
φ
=
(
ω
B
+
ω
TP
)
τ
is extracted via projective measurements,
where
ω
B
∝
g
eff
μ
N
|
B
0
|
/
~
is the Larmor precession fre-
quency and
ω
TP
∝
∆
H
sm
/
~
is the contribution from the
differential Schiff moment between the two states. The
∆
H
sm
contribution to the phase can be distinguished
from the Larmor precession by repeating the measure-
ment in different magnetic fields and with different rel-
ative orientations of the radium nuclear spin and inter-
nuclear axis, which is enabled in this case by the
K
-
doublets. A proposed alternative approach, known as the
clock-transition (CT) method [73], uses time-dependent
electromagnetic fields to drive transitions between differ-
ent hyperfine “clock” states. The
T,P
-violating inter-
action is then extracted from phase-dependent shifts to
the measured Rabi oscillations. This technique, which
is readily adapted to an ion trap, benefits from a sim-
pler state preparation scheme and better robustness to
electromagnetic noise.
In the polarized limit, the rotational manifold contains
many hyperfine states for driving the SI or CT measure-
ment scheme. Each state has different effective
g
-factors
and Schiff sensitivity (see Fig. 3). Performing an EDM
measurement with pairs of states which have unique dif-
ferential magnetic sensitivities enables one to adjust the
Larmor precession without changing the applied
B
-field.
In addition, there are multiple pairs of near-degenerate
states of opposite
m
F
with the same Schiff moment sen-
sitivity, but different magnetic moments. These add to
the set of valuable systematic checks.
BSM Sensitivity.
Calculations of the Schiff moment
of
225
Ra nuclei have been performed in the framework
of
T,P
-violating pion exchange between nucleons [15],
yielding parameterizations of
S
(
225
Ra) in terms of the
QCD
̄
θ
angle given by
|
S
(
225
Ra)
|
= 1
.
0
̄
θ
e
fm
3
[74].
The electrostatic interactions generated by the Schiff mo-
ment leads to an effective
T,P
-violating shift
H
sm
=
W
s
(
I
·
ˆn
)
|
S
|
/
|
I
|
. The species-dependent coupling con-
stant
W
s
, which is an electron-nuclear contact term, has
been calculated to be 45
,
192 atomic units in RaO [29]
and is estimated to be slightly smaller (
∼
30
,
000 a.u.)
for RaOH
+
(a.u.
≡
e/
4
π
0
a
4
0
), where the larger ligand is
assumed to reduce both electron density around Ra and
the magnitude of
W
s
[27].
To illustrate the power of a Schiff moment mea-
surement on
225
RaOCH
+
3
,
we can combine the
QCD parameterization of
S
(
225
Ra) with the estimate
W
s
(
225
RaOCH
+
3
)
≈
30
,
000 a.u. to calculate the aver-
aging time needed to reach a new model-dependent limit
on QCD
̄
θ
. We assume a single trapped
225
RaOCH
+
3
ion
5
with 5 s coherence time limited by black-body pump-
ing at 300 K, which provides a frequency sensitivity of
δω
= 7
.
5 mrad s
−
1
/
√
hour. Spin-precession measure-
ments with this setup would reach a statistical sensitiv-
ity
̄
θ <
10
−
10
with two weeks of data taking. Trapping
multiple ions and improving the coherence time through
cryogenic cooling would result in even higher sensitivity.
Conclusion and outlook.
We have considered trapped
225
RaOCH
+
3
as a sensitive platform to search for a nu-
clear Schiff moment in the octopole-deformed Ra nu-
cleus. While our theoretical calculations do not replace
the need for detailed spectroscopic studies on this partic-
ular species, they do illustrate advantageous structures
that are quite general for both symmetric and asymmet-
ric top molecules [66, 75, 76]. This approach can therefore
be used to search for a variety of fundamental symme-
try violations in many different species, including those
with exotic nuclei such as Pa [77] and U [78], and many
ligands including chiral species.
Over the years, a wide variety of metal-monohydroxide
(MOH
+
), metal-monomethoxide (MOCH
+
3
), as well as
dual-metal hypermetallic (MOM
′
+
) ions have been cre-
ated [79, 80] (and, in some cases, trapped [81, 82]), in-
cluding species with heavy nuclei such as Ba [81] and Lu
[80] in addition to Ra [35]. The rich internal complex-
ity of these molecules makes them attractive for a broad
range of studies not limited to Schiff moment measure-
ments [1]. Much of the discussion in this manuscript, for
instance, is directly applicable to searches for the electron
EDM or nuclear magnetic quadrupole moments [32, 37].
The availability of opposite parity states with diverse,
tunable splittings is particularly useful for precision mea-
surement of electroweak physics, such as nuclear spin-
dependent parity violation [83] and oscillating symmetry
violations from interactions with axion-like fields [84–86],
and the sources that generate the splittings can be sen-
sitive to variations of fundamental constants [87, 88].
Acknowledgments.
We are grateful for extensive assis-
tance from Anastasia Borschevsky and Y.A. Chamorro
Mena with the
ab initio
calculations, and to Ben Au-
genbraun, Mingyu Fan, Alex Frenett, Arian Jadbabaie,
Andrew Jayich, Ivan Kozyryev, Zack Lasner, and Tim
Steimle for helpful discussions and feedback. This re-
search was supported by a NIST Precision Measurement
Grant (60NANB18D253), the Gordon and Betty Moore
Foundation (7947), and the Alfred P. Sloan Foundation
(G-2019-12502). Computations in this manuscript were
performed on the Caltech High Performance Cluster.
∗
phelanyu@caltech.edu
†
hutzler@caltech.edu
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8
Supplemental Material
MOLECULAR STRUCTURE
As discussed in the main text, we numerically diago-
nalize the effective molecular Hamiltonian for the ground
electronic state (
̃
X
1
A
1
) of
225
RaOCH
+
3
H
total
=
H
rot
+
H
stark
+
H
zeeman
+
H
ss
+
H
nsr
+
H
sm
(S1)
where we have included the rotational (rot), Stark, Zee-
man, nuclear spin dipolar (ss), nuclear spin-rotation
terms (nsr), and Schiff moment (sm) terms. No electron
spin terms are included, as the molecule has a closed
shell. For generality, however, the matrix elements are
written in the fully decoupled basis including the elec-
tron spin
S
:
|
N,K,S,J,m
J
〉|
I
M
,m
IM
〉|
Γ
,I
H
,m
IH
〉
.
Rovibrational Structure
RaOCH
+
3
is a prolate symmetric top with point group
C
3
v
, corresponding to its three-fold cylindrical symmetry
about the principal molecular axis (
Z
). The Hamiltonian
that corresponds to the rotational energy for a prolate
top is
H
rot
=
B
N
2
+ (
A
−
B
)
N
2
Z
,
(S2)
which has eigenenergies
BN
(
N
+1)+(
A
−
B
)
K
2
.
A
and
B
are rotational constants, while
〈
N
〉
=
N
and
〈
N
Z
〉
=
K
are the canonical rotational quantum numbers.
A
corre-
sponds to the rotation of the molecule about the symme-
try axis, while
B
corresponds to end-over-end rotation.
For
|
K
|6
= 0, each
|
N,
|
K
|〉
-level in the rotational Hamil-
tonian has 2
×
(2
N
+ 1) degeneracies, which are given
by the 2
N
+ 1 lab-frame projections
m
N
and the two
projections of
|
K
|
onto the molecular axis.
The
ab initio
geometries, with corresponding rota-
tional constants, for RaOCH
+
3
(see table I in main
text) were optimized using the coupled cluster method
with single, doubles, and perturbative triples [CCSD(T)]
[90, 91] in CFOUR [47–49]. Relativistically contracted
atomic natural orbital basis sets of polarized double,
triple, and quadruple-zeta quality [ANO-RCC-V
n
ZP
(
n
=D,T,Q)] are used [41–46]. Scalar relativistic effects
are included via the one-electron variant of the spin-free
exact two component theory (SFX2C-1e) [50–52].
There are eight vibrational modes for symmetric top
molecules of the MOCH
3
form, four symmetric modes
of
a
1
character and four asymmetric modes of
e
charac-
ter. The vibrational energies and intensities of each mode
are calculated via analytic B3LYP Hessians [58, 92] with
ORCA [93]. For this calculation, we employ correlation-
consistent basis sets at the quadruple-zeta level [94], with
core-valence [95] and pseudopotential sets for the radium
Vibration
Energy [cm
−
1
]
Character
Ra-O-C bend
164
.
96
/
168
.
68
e
Ra-O stretch
390
.
78
a
1
CH
3
symmetric bend
1086
.
84
a
1
CH
3
rock
1170
.
41
/
1173
.
36
e
C-O stretch
1481
.
50
a
1
CH
3
asymmetric bend
1497
.
61
/
1498
.
56
e
CH
3
symmetric stretch
2993
.
19
a
1
CH
3
asymmetric stretch
3048
.
22
/
3059
.
15
e
Table S1. Vibrational energies, computed from B3LYP Hes-
sians [92] at DFT optimized geometry, where each mode is
classified with respect to its transformations under
C
3
v
sym-
metry. In total, there are four symmetric
a
1
states and four
doubly-degenerate
e
states. The pair of frequencies for the
degenerate vibrations is due to slight symmetry-breaking in
the computed geometry.
atom. The 78 core electrons of radium are modeled us-
ing the SK-MCDHF-RSC effective core potential (ECP)
[96]. Energies and symmetry characters are listed in Ta-
ble S1. While the proposed Schiff moment search would
take place in the ground vibrational state, excited ro-
vibrational states could be a valuable resource for state
preparation or readout [97–101]. The energy difference
between the nominally degenerate states of
e
character
indicate that the accuracy of the energies is on the few
percent level.
The vibrational states are also relevant as they will
likely present a limitation on the coherence time [102].
Black-body excitation of the Ra-O stretch mode (with
the transition dipole moment calculated to be
μ
∼
0
.
26
D) is estimated to occur with a
∼
5 second time scale in
a 300 K environment, though that can be reduced to 20
minutes in a 77 K environment. The radiative lifetime
of the
N
=
|
K
|
= 1 states are much longer than one
day due to their small energy spacing (a state with one
atomic unit of transition dipole moment at this frequency
would last around one hour) and the fact that they are
spin-forbidden to decay to the ground rotational state, so
black-body effects are likely to dominate over radiative
decay at room temperature.
Molecular Symmetries
Classification of Rovibronic and Nuclear Spin Wavefunctions
The
C
3
v
point group (which is isomorphic to
S
6
) has
three irreducible representations:
A
1
,
A
2
, and
E
.
A
1
de-
notes the fully symmetric representation, and
A
2
is the
anti-symmetric representation.
E
denotes the degener-
ate representation, which splits into positive (
E
+
) and
negative (
E
−
) components under the action of the cyclic
group
C
3
⊂
C
3
v
. The character tables for
C
3
v
and
C
3
are written in Table S2 for reference.
9
ˆ
E
2
ˆ
C
3
(
z
)
3
σ
v
A
1
+1
+1
+1
A
2
+1
+1
−
1
E
+2
−
1
0
ˆ
E
ˆ
C
3
ˆ
C
2
3
A
+1
+1
+1
E
±
+1
e
∓
2
πi/
3
e
±
2
πi/
3
Table S2. Character Tables for
C
3
v
(left) and
C
3
(right)
We start by classifying the rovibronic states. In the
electronic ground state and a vibrationally relaxed man-
ifold (
|
Λ = 0
,`
= 0
〉
), the symmetry classification of the
|
N,
|
K
|〉
rovibronic states is dependent only on
K
. The
|
N,K
= 0
〉
state transforms as
A
1
or
A
2
(depending
on
N
), while
|
N,
|
K
|
= 3
n
〉
states, where
n
is an inte-
ger and
K
6
= 0, transform as
A
1
⊕
A
2
. The remaining
|
N,
|
K
| 6
= 3
n
〉
states transform as the doubly degener-
ate
E
±
character, where the positive and negative com-
ponents correspond to the positive and negative projec-
tions of
|
K
|
. (A generalized classification for non-zero or-
bital and vibrational angular momentum (
|
Λ
6
= 0
,l
6
= 0
〉
)
can be found in [63].) The dependence of the molecular
symmetries on
K
arises from the fact that the hydrogen
atoms in the methyl group are indistinguishable and must
obey the Pauli principle, as discussed in detail later.
All states where
|
K
|
>
0 are doubly degenerate and
thus correspond to either the mixed
E
±
or
A
1
⊕
A
2
sym-
metry characters. The splitting of
K
-doublets (and thus
the breaking of
C
3
v
symmetry) is naturally associated
with the splitting of
E
±
or
A
1
⊕
A
2
into distinct repre-
sentations.
We discuss the specific mechanisms lifting the degen-
eracy below, but there is a simple and intuitive picture
how this arises due to hyperfine couplings [65]. The
K
-
doubled states
|±〉
have a different distribution of proton
spin about the azimuthal angle relative to the symmetry
axis and therefore the anisotropic nature of the dipolar
or nuclear spin-rotation interaction splits the two states.
For
|
K
|
= 1 states, the hyperfine anisotropies directly
couple states differing by ∆
K
= 2 to produce a first-order
K
-splitting, but have progressively suppressed effects on
the splitting for higher
|
K
|
>
1 [65, 103].
For states where
|
K
|
= 3
n
, the leading order source
of
K
-doubling arises from a sextic centrifugal distor-
tion term, which couples ∆
K
= 6. This term appears
as a correction to the rotational Hamiltonian
H
sextic
=
q
3
(
J
6
+
+
J
6
−
)
/
2, where
q
3
is the distortion constant [104].
In the CH
3
radical, the
q
3
distortion constant has been
measured to be 370 Hz [105]. At higher multiples of three,
the
K
-splitting generated by
q
3
is again suppressed. Us-
ing higher
K
states is therefore a general way to obtain
even smaller
K
-doublets.
The composite nuclear spin states
|
Γ
,I
H
,m
IH
〉
of the
three hydrogen atoms, where Γ denotes the symmetry,
are also classified into four states that transform as
A
1
,
and four states that transform as
E
±
:
∣
∣
∣
A,
3
2
,
±
3
2
〉
=
∣
∣
∣
±
1
2
,
±
1
2
,
±
1
2
〉
(S3)
∣
∣
∣
A,
3
2
,
±
1
2
〉
=
1
√
3
(
∣
∣
∣
±
1
2
,
±
1
2
,
∓
1
2
〉
+
∣
∣
∣
±
1
2
,
∓
1
2
,
±
1
2
〉
+
∣
∣
∣
∓
1
2
,
±
1
2
,
±
1
2
〉
)
(S4)
∣
∣
∣
E
+
,
1
2
,
±
1
2
〉
=
1
√
3
(
∣
∣
∣
∓
1
2
,
±
1
2
,
±
1
2
〉
+
e
+2
πi/
3
∣
∣
∣
±
1
2
,
∓
1
2
,
±
1
2
〉
+
e
−
2
πi/
3
∣
∣
∣
±
1
2
,
±
1
2
,
∓
1
2
〉
)
(S5)
∣
∣
∣
E
−
,
1
2
,
±
1
2
〉
=
1
√
3
(
∣
∣
∣
∓
1
2
,
±
1
2
,
±
1
2
〉
+
e
−
2
πi/
3
∣
∣
∣
±
1
2
,
∓
1
2
,
±
1
2
〉
+
e
+2
πi/
3
∣
∣
∣
±
1
2
,
±
1
2
,
∓
1
2
〉
)
(S6)
I
H
refers to the total composite spin of the three protons
(
I
H
=
I
H
1
+
I
H
2
+
I
H
3
) and
m
IH
is the corresponding
lab frame projection. Γ denotes the symmetry of the
composite spin state.
To obey Fermi-Dirac statistics, the total nuclear-
rovibronic wavefunction must transform as either
A
1
(symmetric under inversion) or
A
2
(antisymmetric un-
der inversion) [63]. Therefore, the
K
= 1 and
K
= 2 (in
the ground state) cases correspond to combined nuclear
(
n
Γ) and rovibronic (
evsr
Γ) wavefunctions that have the
symmetries
|
evsr
E
+
〉|
n
E
−
〉
or
|
evsr
E
−
〉|
n
E
+
〉
,
(S7)
whereas the fully symmetric
K
= 0 or
K
= 3 rotational
states correspond to two possibilities
|
evsr
A
1
〉|
n
A
〉
or
|
evsr
A
2
〉|
n
A
〉
.
(S8)
From eq. (S7), we can observe that any hyperfine inter-
action which couples
|
n
E
+
〉
and
|
n
E
−
〉
states also couples
the
|
evsr
E
+
〉
and
|
evsr
E
−
〉
states, and therefore splits any
residual
K
-degeneracy into the doublets,
1
√
2
(
|
evsr
E
+
〉|
n
E
−
〉±|
evsr
E
−
〉|
n
E
+
〉
)
.
(S9)
This is the case for the
K
-doubling that originates from
the nuclear dipolar spin couplings of RaOCH
+
3
, for in-
stance.
10
The symmetry assignments in eqs. (S7) and (S8) also
lead to additional selection rules for electric dipole tran-
sitions. For instance, let us consider transitions between
different
|
N,K
〉
manifolds in the electronic-vibrational
ground state. Starting from the
K
= 1 manifold, the
electric dipole operator only couples to ∆
K
= 0
,
±
1
states.
|
N,K
= 2
〉
has the same nuclear symmetry Γ
as
|
N,K
= 1
〉
and the transition is allowed.
|
N,K
= 0
〉
does not, and the transition is symmetry-forbidden (as
well as spin-forbidden), which is why the radiative life-
time of
|
N,K
= 1
〉
is so long.
Hyperfine Structure
Parameterization of the Tensor Operators
In order to evaluate hyperfine structure of the hydro-
gen spins, it is necessary to sum over the interactions in-
volving each individual individual hydrogen nucleus. A
spin-spin interaction between the hydrogen spins
I
i
and
an arbitrary spin
S
, for instance, has the form
3
∑
i
=1
S·
T
i
·
I
i
,
(S10)
where
T
i
is the interaction tensor between the
i
th hydro-
gen and
S
. In Cartesian form,
T
1
can be written as
T
1
=
T
xx
0
T
xz
0
T
yy
0
T
zx
0
T
zz
,
(S11)
while
T
2
and
T
3
can be obtained with the appropriate
rotations about the
z
axis [89].
The form of eq. (S10), however, is unwieldy for evalu-
ating matrix elements over the symmetrized nuclear spin
states derived in the previous section. A more intuitive
form can be obtained by parameterizing the
T
and
I
op-
erators, as first formulated by Hougen in cartesian form
[63] and Endo et al. in spherical tensor form [89, 106].
T
0
= (
T
1
+
T
2
+
T
3
)
/
3
,
(S12)
T
±
= (
T
1
+
e
±
2
πi/
3
T
2
+
e
∓
2
πi/
3
T
3
)
/
3
,
(S13)
I
0
=
I
1
+
I
2
+
I
3
,
(S14)
I
±
=
I
1
+
e
±
2
πi/
3
I
2
+
e
∓
2
πi/
3
I
3
.
(S15)
This allows us to rewrite eq. (S10) as
∑
α
=0
,
±
S
·
T
α
·
I
−
α
.
(S16)
Direct evaluation via the Wigner-Eckart Theorem yields
the following reduced matrix elements for the parameter-
ized nuclear spin operators over the symmetrized nuclear
spin basis:
〈
Γ
,I
0
|
T
1
(
I
0
)
|
Γ
,I
0
〉
=
√
I
0
(
I
0
+ 1)(2
I
0
+ 1)
,
(S17)
〈
E
±
,I
0
|
T
1
(
I
±
)
|
E
∓
,I
0
〉
=
−
2
√
I
0
(
I
0
+ 1)(2
I
0
+ 1)
,
(S18)
〈
E
±
,I
0
= 1
/
2
|
T
1
(
I
±
)
|
A
1
,I
0
= 3
/
2
〉
=
√
6
.
(S19)
Meanwhile, the parameterized
T
operators, in cartesian
form, are:
T
0
=
(
T
xx
+
T
yy
)
/
2
0
0
0
(
T
xx
+
T
yy
)
/
2 0
0
0
T
zz
,
(S20)
T
±
=
(
T
xx
−
T
yy
)
/
4
∓
i
(
T
xx
−
T
yy
)
/
4
T
xz
/
2
∓
i
(
T
xx
−
T
yy
)
/
4
−
(
T
xx
−
T
yy
)
/
4
±
iT
xz
/
2
T
zx
/
2
±
iT
zx
/
2
0
,
(S21)
which have the corresponding non-zero rank-2 compo-
nents in spherical tensor form (
T
α
)
2
q
[107]:
(
T
0
)
2
0
= (2
T
zz
−
T
xx
−
T
yy
)
/
√
6
,
(S22)
(
T
±
)
2
∓
=
∓
(
T
xz
+
T
zx
)
/
2
,
(S23)
(
T
±
)
2
±
2
= (
T
xx
−
T
yy
)
/
2
.
(S24)
By applying a Wigner rotation to the molecule-frame (
q
)
spherical components, we obtain
〈
N
′
K
′
|D
2
pq
(
ω
)(
T
α
)
2
q
|
NK
〉
=
∑
q
[
(
−
1)
N
′
−
K
′
×
√
(2
N
′
+ 1)(2
N
+ 1)
(
N
′
2
N
−
K
′
q K
)
(
T
α
)
2
q
]
.
(S25)
We can see that the anisotropic tensor component in
eq. (S24) will couple states with ∆
K
= 2. In the
case of
|
K
|
= 1, this term would couple
K
= +1 and
K
=
−
1 terms, providing a first-order contribution to
the
K
-doubling. Most of the sources of
K
-splitting in
the hyperfine structure of symmetric tops arise in this
manner.
Stark and Zeeman Matrix Elements
The Stark Hamiltonian is
ˆ
H
S
=
−
T
1
(
d
)
·
T
1
(
E
),
where
d
is the molecule frame dipole moment and
E
is the electric field.
Without loss of generality, we
set the electric field along the ˆ
z
direction in the lab
frame.
The matrix elements in the decoupled basis
|
N,K,S,J,m
J
〉|
I
M
,m
IM
〉|
Γ
,I
H
,m
IH
〉
are given by