Supplemental Material for 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: November 24, 2020)
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)] [1, 2]
in CFOUR [3–5]. Relativistically contracted atomic nat-
ural orbital basis sets of polarized double, triple, and
quadruple-zeta quality [ANO-RCC-V
n
ZP (
n
=D,T,Q)]
are used [6–11]. Scalar relativistic effects are included
via the one-electron variant of the spin-free exact two
component theory (SFX2C-1e) [12–14].
There are eight vibrational modes for symmetric top
molecules of the MOCH
3
form, four symmetric modes
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 [15] 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.
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 [15, 16] with
ORCA [17]. For this calculation, we employ correlation-
consistent basis sets at the quadruple-zeta level [18], with
core-valence [19] and pseudopotential sets for the radium
atom. The 78 core electrons of radium are modeled us-
ing the SK-MCDHF-RSC effective core potential (ECP)
[20]. 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 [21–25]. 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 [26].
Black-body excitation [27] 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 life-
time 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
2
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.
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 [28].) 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 [29]. 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 [29, 30].
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 [31].
In the CH
3
radical, the
q
3
distortion constant has been
measured to be 370 Hz [32]. 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) [28]. 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
3
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.
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 [33].
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
[28] and Endo et al. in spherical tensor form [33, 34].
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
[35]:
(
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.
4
0
5
10
15
-
30
-
20
-
10
0
10
20
30
Electric Field
(
mV
/
cm
)
Stark Shift
(
kHz
)
0
2
4
6
8
10
-
50
0
50
Magnetic Field
(
G
)
Zeeman Shift
(
kHz
)
Figure S1. (left) Stark plot of hyperfine states in the
|
N
= 1
,
|
K
|
= 1
〉
manifold, up to 15 mV cm
−
1
electric field. (right)
Zeeman plot of hyperfine states in the
|
N
= 1
,
|
K
|
= 1
〉
manifold, up to 10 G magnetic field.
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
〈
N
′
K
′
S
′
J
′
m
′
J
;
I
M
m
IM
; Γ
I
H
m
IH
|
ˆ
H
S
|
NKSJm
J
;
I
M
m
IM
; Γ
I
H
m
IH
〉
=
−
E
〈
N
′
K
′
S
′
J
′
m
′
J
|
T
1
0
(
d
)
|
NKSJm
J
〉
=
−
Ed
0
(
−
1)
J
′
−
m
′
J
(
J
′
1
J
−
m
′
J
0
m
J
)
(
−
1)
J
+
N
′
+1+
S
′
√
(2
J
+ 1)(2
J
′
+ 1)
{
N J S
′
J
′
N
′
1
}
×
δ
S,S
′
(
−
1)
N
′
−
K
′
√
(2
N
′
+ 1)(2
N
+ 1)
(
N
′
1
N
−
K
′
0
K
)
.
(S26)
Note that the dipole moment is along the molecular axis
such that
T
1
0
(
d
) =
d
0
and that the matrix element is
diagonal with respect to the nuclear and electron spins.
The Zeeman Hamiltonian is written as
ˆ
H
z
=
ˆ
H
z,I
+
ˆ
H
z,R
, with two terms corresponding to the coupling of
the nuclear spin and molecular rotation to the external
magnetic field.
The nuclear spin term
ˆ
H
z,I
is given by
ˆ
H
z,I
=
−
g
N
μ
N
∑
i
T
1
(
I
i
)
·
T
1
(
B
)
=
−
g
N
μ
N
∑
i,p
(
−
1)
p
T
1
p
(
I
i
)
T
1
−
p
(
B
)
(S27)
where we note that we need to sum over all the hydrogen
and radium nuclear spins. Evaluating the matrix ele-
ment in the decoupled basis for the radium spin, which
is diagonal in
|
N,K,S,J,m
J
〉|
Γ
,I
H
,m
IH
〉
:
〈
I
′
M
m
′
IM
|
ˆ
H
z,I
M
|
I
M
m
IM
〉
=
−
g
N
μ
N
∑
p
(
−
1)
p
〈
I
′
M
,m
′
IM
|
T
1
p
(
I
M
)
|
I
M
,m
IM
〉〈
T
1
−
p
(
B
)
〉
,
(S28)
where the nuclear spin angular momentum matrix ele-
ment for a single nucleus is, generically,
〈
I
′
,m
′
I
|
T
1
p
(
I
)
|
I,m
I
〉
= (
−
1)
I
′
−
m
′
I
(
I
′
1
I
−
m
′
I
p m
I
)
×
δ
I
′
,I
√
I
(
I
+ 1)(2
I
+ 1)
.
(S29)
For the coupling to hydrogen spins, we evaluate the pa-
rameterized
T
1
(
I
0
) matrix elements derived in the previ-
ous section, which results in a near-identical expression
5
to eq. (S29),
〈
I
M
m
IM
|
ˆ
H
z,I
M
|
I
M
m
IM
〉
=
−
g
N
μ
N
∑
p
[
(
−
1)
p
〈
Γ
′
,I
′
H
,m
′
IH
|
T
1
p
(
I
0
)
|
Γ
,I
H
,m
IH
〉
×〈
T
1
−
p
(
B
)
〉
]
.
(S30)
The rotational Zeeman term [36] is:
ˆ
H
z,R
=
g
R
μ
N
N
·
B
(S31)
=
μ
N
B
0
∑
k
(
−
1)
k
√
2
k
+ 1
3
T
1
0
(
g
k
r
,
N
)
=
μ
N
B
0
∑
k
(
−
1)
2
k
+1
√
2
k
+ 1
×
∑
p
1
2
[
T
k
p
(
g
r
)
T
1
−
p
(
N
) + (
−
1)
k
T
1
p
(
N
)
T
k
−
p
(
g
r
)
]
×
(
k
1 1
p
−
p
0
)
.
(S32)
Relativistic
ab initio
values for the rotational g-tensor
g
k
r
have been computed using the four-component linear re-
sponse within elimination of small component (LRESC)
approach of Aucar et al. [37] as implemented in the
DIRAC19 code [38]. Electron correlation is treated at
the level of DFT with a B3LYP functional [16] and a
dyall.v4z basis is used [39]. See table I in main text for
specific values.
Nuclear Spin-Rotation Coupling
The nuclear spin-rotation coupling is expressed simi-
larly to the electron spin-rotation elements:
H
nsr
=
1
2
∑
α,β
C
α,β
(
N
α
I
β
+
I
β
N
α
)
(S33)
=
1
2
2
∑
k
=0
[
T
k
(
C
)
·
T
k
(
N
,
I
) +
T
k
(
N
,
I
)
·
T
k
(
C
)
]
(S34)
=
1
2
2
∑
k
=0
k
∑
p
[
T
k
p
(
C
)
T
k
−
p
(
N
,
I
) +
T
k
p
(
N
,
I
)
T
k
−
p
(
C
)
]
,
(S35)
where the product is decomposed as
T
k
p
(
N
,
I
) = (
−
1)
p
√
2
k
+ 1
∑
p
1
,p
2
T
1
p
1
(
N
)
T
1
p
2
(
I
)
(
1 1
k
p
1
p
2
−
p
)
.
(S36)
In
225
RaOCH
+
3
, the nuclear spin rotation interaction is
divided into two components: the interaction between
rotation and the spin-1
/
2
225
Ra nucleus, and the inter-
action between rotation and the hydrogen spins in the
methyl group.
Let us consider the spin-rotation coupling for the single
radium spin
I
M
. We write out eq. (S35), in the decoupled
basis
|
N,K,S,J,m
J
〉|
I
M
,m
IM
〉|
Γ
,I
H
,m
IH
〉
, taking care
to sum over a complete set of states between
T
(
C
) and
T
(
N
),
〈
N
′
K
′
S
′
J
′
m
′
J
;
I
′
M
m
′
IM
; Γ
I
H
m
IH
|
H
nsr-M
|
NKSJm
J
;
I
M
m
IM
; Γ
I
H
m
IH
〉
=
1
2
∑
k,p
(
−
1)
p
√
2
k
+ 1
∑
p
1
,p
2
[
∑
η
′′
δ
J
′
,J
(
−
1)
J
′
−
J
′′
2
J
′
+ 1
〈
N
′
K
′
S
′
J
′
m
′
J
|
T
k
p
(
C
)
|
N
′′
K
′′
S
′′
J
′′
m
′′
J
〉
×〈
N
′′
K
′′
S
′′
J
′′
m
′′
J
|
T
1
p
1
(
N
)
|
NKSJm
J
〉〈
I
′
M
m
′
IM
|
T
1
p
2
(
I
M
)
|
I
M
m
IM
〉
(
1 1
k
p
1
p
2
p
)
+
∑
η
′′
δ
J
′
,J
(
−
1)
J
′
−
J
′′
2
J
′
+ 1
〈
N
′
K
′
S
′
J
′
m
′
J
|
T
1
p
1
(
N
)
|
N
′′
K
′′
S
′′
J
′′
m
′′
J
〉
×〈
Γ
′
I
′
M
m
′
IM
|
T
1
p
2
(
I
M
)
|
Γ
I
M
m
IM
〉〈
N
′′
K
′′
S
′′
J
′′
m
′′
J
|
T
k
−
p
(
C
)
|
NKSJm
J
〉
(
1 1
k
p
1
p
2
−
p
)]
.
(S37)
Note that eq. (S37) is diagonal with respect to the
|
Γ
,I
H
,m
IH
〉
spin states.
Evaluating the nuclear spin-rotation and rotational an-
gular momentum matrix elements yields
〈
N
′
K
′
S
′
J
′
m
′
J
|
T
1
p
(
N
)
|
NKSJm
J
〉
=
×
(
−
1)
J
′
−
m
′
J
(
J
′
1
J
−
m
′
J
p m
J
)
×
δ
S
′
,S
(
−
1)
J
+
N
′
+1+
S
′
√
(2
J
′
+ 1)(2
J
+ 1)
×
{
N J S
J
′
N
′
1
}
δ
N
′
,N
√
(
N
(
N
+ 1)(2
N
+ 1)
,
(S38)
6
〈
N
′
K
′
S
′
J
′
m
′
J
|
T
k
p
(
C
)
|
NKSJm
J
〉
=
(
−
1)
J
′
−
m
′
J
(
J
′
k J
−
m
′
J
p m
J
)
×
δ
S
′
,S
(
−
1)
J
+
N
′
+
k
+
S
′
√
(2
J
′
+ 1)(2
J
+ 1)
{
N J S
J
′
N
′
k
}
×
∑
q
[
(
−
1)
N
′
−
K
′
√
(2
N
′
+ 1)(2
N
+ 1)
×
(
N
′
k N
−
K
′
q K
)
T
k
q
(
C
)
]
,
(S39)
where, for an on-axis spin, the non-zero spherical tensor
elements for the nuclear spin-rotation coupling are:
T
0
0
(
C
) =
−
(
C
xx
+
C
yy
+
C
zz
)
/
√
3
,
(S40)
T
2
0
(
C
) = (2
C
zz
−
C
xx
−
C
yy
)
/
√
6
,
(S41)
T
2
±
2
(
C
) = (
C
xx
−
C
yy
)
/
2
.
(S42)
The lab-frame matrix element
T
1
p
(
I
M
) can be evalu-
ated as eq. (S29) for a single spin.
Now, we consider the coupling to rotation for the com-
posite hydrogen spins (
I
H
). We can utilize the param-
eterized nuclear spin operators and coupling tensors de-
rived earlier. This adds an additional summation over
the parameter
α
,
H
nsr-H
=
1
2
∑
α
2
∑
k
=0
k
∑
p
[
T
k
p
(
C
α
)
T
k
−
p
(
N
,
I
−
α
)
+
T
k
p
(
N
,
I
−
α
)
T
k
−
p
(
C
α
)
]
.
(S43)
We can thus generalize eq. (S37) to the case of multiple
spins,
〈
N
′
K
′
S
′
J
′
m
′
J
; Γ
′
I
′
m
′
I
|
H
nsr-H
|
NKSJm
J
; Γ
Im
I
〉
=
∑
α
1
2
∑
k,p
(
−
1)
p
√
2
k
+ 1
×
∑
p
1
,p
2
[
∑
η
′′
δ
J
′
,J
(
−
1)
J
′
−
J
′′
2
J
′
+ 1
〈
N
′
K
′
S
′
J
′
m
′
J
|
T
k
p
(
C
α
)
|
N
′′
K
′′
S
′′
J
′′
m
′′
J
〉
×〈
N
′′
K
′′
S
′′
J
′′
m
′′
J
|
T
1
p
1
(
N
)
|
NKSJm
J
〉〈
Γ
′
I
′
m
′
I
|
T
1
p
2
(
I
−
α
)
|
Γ
,I,m
I
〉
(
1 1
k
p
1
p
2
p
)
+
∑
η
′′
δ
J
′
,J
(
−
1)
J
′
−
J
′′
2
J
′
+ 1
〈
N
′
K
′
S
′
J
′
m
′
J
|
T
1
p
1
(
N
)
|
N
′′
K
′′
S
′′
J
′′
m
′′
J
〉
×〈
Γ
′
I
′
m
′
I
|
T
1
p
2
(
I
−
α
)
|
Γ
Im
I
〉〈
N
′′
K
′′
S
′′
J
′′
m
′′
J
|
T
k
−
p
(
C
α
)
|
NKSJm
J
〉
(
1 1
k
p
1
p
2
−
p
)]
.
(S44)
where
〈
Γ
′
I
′
m
′
I
|
T
1
p
(
I
α
)
|
Γ
Im
I
〉
is evaluated using Wigner-
Eckart and the reduced matrix elements listed in eqs.
(S17) – (S19). The
T
k
p
(
C
α
) tensors can be obtained with
the appropriate parameterization listed in eqs. (S17) –
(S18).
Relativistic
ab initio
values for the
T
(
C
) nuclear spin-
rotation tensors are also obtained with a four-component
LRESC approach [40, 41] in the DIRAC19 code [38] using
a dyall.v4z basis set [39]. Correlation effects are treated
at the level of DFT with the local density approximation
and popular hybrid functions (B3LYP [16], PBE0 [42]).
The full results are listed in Table S3. While validating
these methods is challenging, due to the absence of exper-
imental data (particularly for atoms heavier than group
4), benchmarks with diatomic species in refs. [37, 41]
suggests that these predictions are accurate within 10
percent.
The contributions from nuclear spin rotation couplings
to the
K
-doubling can be extracted from the anisotropy
in the
T
(
C
) tensors. As the radium spin-rotation cou-
pling is diagonal in
|
Γ
,I
H
,m
IH
〉
, the
H
nsr-M
Hamiltonian
does not couple off-diagonally between
|
E
+
〉
and
|
E
−
〉
hy-
drogen spin states, and therefore does not contribute to
the breaking of
K
-degeneracy. By contrast, the hydrogen
spin-rotation coupling does couple
|
E
+
〉
and
|
E
−
〉
hydro-
gen spin states. The computed anisotropic contribution
to
K
-doubling is
|
T
xx
−
T
yy
|∼
0
.
3 kHz, as noted in Table
I of the main text.
Nuclear Spin Dipolar Coupling
For coupling two generic spins
I
1
and
I
2
, the matrix
element is
7
H
II
=
μ
0
γ
1
γ
2
~
2
4
π
[
I
1
·
I
2
r
3
−
3(
I
1
·
r
)(
I
2
·
r
)
r
5
]
(S45)
=
−
√
6
μ
0
γ
1
γ
2
~
2
4
π
T
2
(
C
)
·
T
2
(
I
1
,
I
2
)
(S46)
=
−
√
6
μ
0
γ
1
γ
2
~
2
4
π
∑
p
T
2
p
(
C
)
T
2
−
p
(
I
1
,
I
2
)
,
(S47)
where
μ
0
is the vacuum permeability and
γ
1
,
2
are the
nuclear gyromagnetic ratios.
T
2
q
(
C
) is the spherical
harmonic tensor
〈
C
2
q
(
θ,φ
)
r
−
3
〉
, where the polar angles
(
θ,φ
) parameterize the vector from
I
1
to
I
2
. Meanwhile,
T
2
(
I
1
,
I
2
) can be decomposed as
T
2
p
(
I
1
,
I
2
) = (
−
1)
p
√
5
∑
p
1
,p
2
T
1
p
1
(
I
1
)
T
1
p
2
(
I
2
)
(
2
1 1
−
p p
1
p
2
)
.
(S48)
Let us consider the spin-spin interaction between
the
225
Ra nucleus and the hydrogen nuclei.
We
write the matrix elements in the decoupled basis
|
N,K,S,J,m
J
〉|
I
M
,m
IM
〉|
Γ
,I
H
,m
IH
〉
and then evalu-
ate the spherical harmonic and nuclear spin-spin tensors
〈
N
′
K
′
S
′
J
′
m
′
J
;
I
′
M
m
′
IM
; Γ
′
,I
′
H
m
′
IH
|
H
ss
|
NKSJm
J
;
I
M
m
IM
; Γ
I
H
m
IH
〉
=
−
√
6
μ
0
γ
1
γ
2
~
2
4
π
∑
p
〈
N
′
K
′
S
′
J
′
m
′
J
|
T
2
p
(
C
)
|
NKSJm
J
〉
×〈
I
′
M
m
′
IM
; Γ
′
I
′
H
m
′
IH
|
T
2
−
p
(
I
M
,
I
H
)
|
I
M
m
IM
; Γ
I
M
m
IM
〉
=
−
√
6
μ
0
γ
1
γ
2
~
2
4
π
∑
α
∑
p
〈
N
′
K
′
S
′
J
′
m
′
J
|
T
2
p
(
C
α
)
|
NKSJm
J
〉
×〈
I
′
M
m
′
IM
; Γ
′
I
′
H
m
′
IH
|
T
2
−
p
(
I
M
,
I
−
α
)
|
I
M
m
IM
; Γ
I
M
m
IM
〉
,
(S49)
〈
N
′
K
′
S
′
J
′
m
′
J
|
T
2
p
(
C
α
)
|
NKSJm
J
〉
=(
−
1)
J
′
−
m
′
J
(
J
′
2
J
−
m
′
J
p m
J
)
δ
S
′
,S
(
−
1)
J
+
N
′
+
S
′
+2
√
(2
J
′
+ 1)(2
J
+ 1)
{
N J S
J
′
N
′
2
}
×〈
N
′
K
′
|
∑
q
D
2
pq
(
ω
)
·
T
2
q
(
C
α
)
|
NK
〉
(S50)
=(
−
1)
J
′
−
m
′
J
(
J
′
2
J
−
m
′
J
p m
J
)
δ
S
′
,S
(
−
1)
J
+
N
′
+
S
′
+2
√
(2
J
′
+ 1)(2
J
+ 1)
{
N J S
J
′
N
′
2
}
×
∑
q
(
−
1)
N
′
−
K
′
√
(2
N
′
+ 1)(2
N
+ 1)
(
N
′
2
N
−
K
′
q K
)
T
2
q
(
C
α
)
(S51)
〈
I
′
M
m
′
IM
; Γ
′
I
′
H
m
′
IH
|
T
2
p
(
I
M
,
I
α
)
|
I
M
m
IM
; Γ
I
H
m
IH
〉
=(
−
1)
p
√
5
∑
p
1
,p
2
〈
I
′
M
m
′
IM
|
T
1
p
1
(
I
M
)
|
I
M
m
IM
〉〈
Γ
′
I
′
H
m
′
IH
|
T
1
p
2
(
I
α
)
|
Γ
I
H
m
IH
〉
(
2
1 1
−
p p
1
p
2
)
.
(S52)
The molecule-frame
T
2
q
(
C
α
) reduced matrix element
can be evaluated using the appropriate sums of spherical
harmonics that parameterize the vector between the
i
th
hydrogen and the radium atom,
T
2
q
(
C
0
) =
3
∑
i
=1
√
4
π
5
Y
2
q
(
θ
i
,φ
i
)
r
−
3
,
(S53)
T
2
q
(
C
±
) =
√
4
π
5
[
Y
2
q
(
θ
1
,φ
1
) +
e
±
2
πi/
3
Y
2
q
(
θ
2
,φ
2
)
+
e
∓
2
πi/
3
Y
2
q
(
θ
3
,φ
3
)
]
r
−
3
.
(S54)
By convention [33], the Cartesian form of the single-
hydrogen dipolar tensor
T
1
(of the form eq. (S11)) is
taken to be traceless tr[
T
] = 0 and symmetric, which
8
yields the following simplifications from eqs.
(S22)–
(S23), which are invoked in table I in the main text:
α
dip
·
T
2
0
(
C
0
) = 3
T
zz
/
√
6
,
(S55)
α
dip
·
T
2
±
(
C
∓
) =
∓
T
xz
,
(S56)
α
dip
·
T
2
±
2
(
C
±
) = (
T
xx
−
T
yy
)
/
2
.
(S57)
where we define a scaling parameter
α
dip
=
−
√
6
μ
0
γ
H
γ
Ra
~
2
/
4
π
.
The radium-hydrogen spin-spin interaction couples
the opposite symmetry states
|
E
+
〉
and
|
E
−
〉
.
This
leads to
|
T
xx
−
T
yy
| ∼
0
.
4 kHz
K
-doubling terms
which couple the
|
N,K,S,J,m
J
〉|
I
M
,m
IM
〉|
I
H
,m
IH
〉
and
|
N,
−
K,S,J,m
J
〉|
I
M
,m
IM
〉|
I
H
,m
IH
〉
states. The
spin-spin interaction between the hydrogen spins them-
selves are only non-zero between the “ortho” stretched
spin states of
A
character. This term can therefore be
ignored in the
|
N,
|
K
|
= 1
〉
manifold.
SCHIFF MOMENT
Effective molecular sensitivity
The effective Schiff moment sensitivity is proportional
to
〈
I
M
·
ˆn
〉
, where
I
M
is the metal spin and ˆ
n
is the inter-
nuclear axis [43]. In the decoupled basis, this is written
as:
〈
N
′
,K
′
,S
′
,J
′
,m
′
J
;
I
′
M
,m
′
IM
;
I
′
H
,m
′
IH
|
I
M
·
ˆn
|
N,K,S,J,m
J
;
I
M
,m
IM
;
I
H
,m
IH
〉
=
δ
I
′
H
,I
H
δ
m
′
IH
,m
IH
〈
I
′
M
,m
′
IM
|
I
M
|
I
M
,m
IM
〉〈
N
′
,K
′
,S
′
,J
′
,m
′
J
|
ˆ
n
|
N,K,S,J,m
J
〉
=
δ
I
′
H
,I
H
δ
m
′
IH
,m
IH
〈
I
′
M
,m
′
IM
|
I
M
|
I
M
,m
IM
〉
(
−
1)
J
′
−
m
′
J
(
J
′
1
J
−
m
J
0
m
J
)
×
δ
S
′
,S
(
−
1)
J
+
N
′
+
S
′
+1
√
(2
J
′
+ 1)(2
J
+ 1)
{
N J S
′
J
′
N
′
1
}
〈
N
′
|
ˆn
|
N
〉
=
δ
I
′
H
,I
H
δ
m
′
IH
,m
IH
(
−
1)
I
′
M
−
m
′
IM
(
I
′
M
1
I
M
−
m
′
IM
0
m
IM
)
δ
I
′
M
,I
M
√
I
M
(
I
M
+ 1)(2
I
M
+ 1)
×
(
−
1)
J
′
−
m
′
J
(
J
′
1
J
−
m
J
0
m
J
)
δ
S
′
,S
(
−
1)
J
+
N
′
+
S
′
+1
√
(2
J
′
+ 1)(2
J
+ 1)
{
N J S
′
J
′
N
′
1
}
×
(
−
1)
N
′
−
K
′
√
(2
N
′
+ 1)(2
N
+ 1)
(
N
′
1
N
−
K
′
0
K
)
(S58)
In zero field, there is no defined orientation of the
molecule, and the Schiff moment sensitivity is zero. In
the decoupled limit (
&
1 V/cm), the value of
〈
I
M
·
ˆn
〉
is
K
·
m
N
·
m
IM
·
1
/
2 for the
N
= 1 and
|
K
|
= 1 manifold.
The 1
/
2 factor arises from the
K/N
(
N
+ 1) pre-factor to
the Stark energy. The states with “stretched” sensitivity
therefore take on the values
〈
I
M
·
ˆn
〉
=
±
1
/
4.
BSM Sensitivity
The frequency sensitivity of a spin-precession measure-
ment on a molecule or atom with coherence time
τ
and
repetition
n
is expressed as
δω
= [
τ
√
n
]
−
1
. In the main
text, we assume a single trapped
225
RaOCH
+
3
ion with 5
sec coherence time, which provides a frequency sensitiv-
ity of
δω
= 7
.
5 mrad s
−
1
/
√
hour.
Dobaczewski and Engel calculated
|
S
(
225
Ra)
|
in the
framework of
T,P
-violating interactions between nu-
cleons mediated by a pion using a variety of Skyrme
energy functionals [44]. The average of their results,
which was expressed in terms of
πNN
vertices, was re-
parameterized by ref. [43] in terms of QCD
̄
θ
and the
quark-chromo EDMs
̃
d
u
and
̃
d
d
:
|
S
(
225
Ra)
|
= 1
.
0
̄
θ
e
fm
3
(S59)
= 10
4
(0
.
50
̃
d
u
−
0
.
54
̃
d
d
)
e
fm
2
(S60)
The energy shift resulting from a Schiff moment
S
and
a coupling constant
W
s
is
H
sm
=
W
s
(
I
M
·
ˆn
)
|
S
|
/
|
I
|
. For
a differential measurement performed between states of
opposite effective sensitivity (
〈
I
M
·
ˆn
〉
=
±
1
/
4), we arrive
at
δ
̄
θ
=
2
~
δω
W
S
(
S/
̄
θ
)
(S61)
where
S/
̄
θ
= 1
.
0
e
fm
3
and the factor of two arises from
∆[
〈
I
M
·
ˆn
〉
] = 1
/
2. Given the estimate of
W
s
(RaOCH
+
3
)
≈
30
,
000 a.u. [43] (where a.u.
≡
e/
4
π
0
a
4
0
), we have
δ
̄
θ/δω
= 24
×
10
−
8
s. Two weeks of data taking would
thus result in the model-dependent sensitivity of
δ
̄
θ <
×
10
−
10
.