# Observation of the distribution of nuclear magnetization in a molecule

S. G. Wilkins1,∗,†, S. M. Udrescu1,∗,‡, M. Athanasakis-Kaklamanakis2,3, R. F.
Garcia Ruiz1,§, M. Au4,5,
I. Belos̆ević6, R. Berger7, M. L. Bissell8, A. A. Breier9, A. J. Brinson1, K.
Chrysalidis4, T. E. Cocolios3,
R. P. de Groote3, A. Dorne3, K. T. Flanagan8,10, S. Franchoo11, K. Gaul7, S.
Geldhof3, T. F. Giesen9,
D. Hanstorp12 R. Heinke4, T. Isaev13, Á. Koszorús2, S. Kujanpää14, L.
Lalanne3, G. Neyens3, M. Nichols12,
H. A. Perrett8, J. R. Reilly8, L. V. Skripnikov13, S. Rothe4, B. van den
Borne3, Q. Wang15,
J. Wessolek8, X. F. Yang16, C. Zülch7

1Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2Experimental Physics Department, CERN, CH-1211 Geneva 23, Switzerland
3KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001 Leuven, Belgium
4Systems Department, CERN, CH-1211 Geneva 23, Switzerland
5Department Chemie, Johannes Gutenberg-Universitat Mainz, D-55099 Mainz,
Germany
6TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada
7Fachbereich Chemie, Philipps-Universität Marburg, Hans-Meerwein-Straße 4,
35032 Marburg, Germany
8School of Physics and Astronomy, The University of Manchester, Manchester M13
9PL, United Kingdom
9Laboratory for Astrophysics, Institute of Physics, University of Kassel,
34132 Kassel, Germany
10Photon Science Institute, The University of Manchester, Manchester M13 9PY,
United Kingdom
11Laboratoire Irène Joliot-Curie, F-91405 Orsay, France
12Department of Physics, University of Gothenburg, SE-412 96 Gothenburg,
Sweden
13Affiliated with an institute covered by a cooperation agreement with CERN
14Department of Physics, University of Jyväskylä, Survontie 9, Jyväskylä,
FI-40014, Finland
15School of Nuclear Science and Technology, Lanzhou University, Lanzhou
730000, People’s Republic of China
16School of Physics and State Key Laboratory of Nuclear Physics and
Technology, Peking University, Beijing 100971, China

Correspondence<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
∗These authors contributed equally to this work.
0000-0001-8897-7227 0000-0002-1989-576X 0000-0002-2926-5569
0000-0002-2062-684X

###### Abstract

Rapid progress in the experimental control and interrogation of molecules,
combined with developments in precise calculations of their structure, are
enabling new opportunities in the investigation of nuclear and particle
physics phenomena. Molecules containing heavy, octupole-deformed nuclei such
as radium are of particular interest for such studies, offering an enhanced
sensitivity to the properties of fundamental particles and interactions. Here,
we report precision laser spectroscopy measurements and theoretical
calculations of the structure of the radioactive radium monofluoride molecule,
225Ra19F. Our results allow fine details of the short-range electron-nucleus
interaction to be revealed, indicating the high sensitivity of this molecule
to the distribution of magnetization, currently a poorly constrained nuclear
property, within the radium nucleus. These results provide a direct and
stringent test of the description of the electronic wavefunction inside the
nuclear volume, highlighting the suitability of these molecules to investigate
subatomic phenomena.

††preprint: APS/123-QED

##  I Introduction

Recent developments in the synthesis and manipulation of molecular systems are
opening up a diverse range of opportunities in fundamental physics research
shuman2010laser; truppe2017molecules; safronova2018search;
andreev2018improved; roussy2023improved. Precision measurements in molecules
safronova2018search; andreev2018improved; altuntacs2018demonstration;
roussy2023improved, combined with the development of ab initio molecular
theory saue2020dirac; kallay2005approximate; kallay2020mrcc;
skripnikov2020nuclear; Skripnikov:18a, offer a compelling avenue for exploring
various aspects of nuclear and particle physics andreev2018improved;
safronova2018search. The structure of certain molecular states can be highly
sensitive to subtle details of electron-nucleon and nucleon-nucleon
interactions within the constituent nuclei of the molecule. As these effects
scale rapidly with the proton number, nuclear size, nuclear spin, and nuclear
deformation auerbach1996collective; flambaum2008electric; isaev2010laser;
kudashov2014ab; sasmal2016relativistic; safronova2018search;
flambaum2019enhanced; gaul2019systematic molecules containing heavy
radioactive nuclei, such as radium monofluoride, RaF, are of particular
interest for fundamental physics studies isaev2010laser; kudashov2014ab;
sasmal2016relativistic; garcia2020spectroscopy; udrescu2021isotope. The
radioisotope 225Ra (half-life of $14.9$ days), with 88 protons and 137
neutrons, is expected to possess a rare nuclear octupole deformation
gaffney2013studies, boosting its sensitivity to both symmetry-conserving and
symmetry-violating nuclear properties by more than three orders of magnitude
with respect to stable isotopes par15; flambaum2008electric; Chu19;
auerbach1996collective; isaev2010laser; kudashov2014ab;
sasmal2016relativistic; gaffney2013studies; garcia2020spectroscopy;
udrescu2021isotope. The former effects are critical to guide our understanding
of the nuclear force and the emergence of collective nuclear phenomena, while
the latter could provide answers to some of the most pressing questions in our
understanding of the universe safronova2018search; Chu19. Measurements of the
breaking of parity ($\mathcal{P}$)- and time-reversal ($\mathcal{T}$)
symmetries could explain the nature of dark matter, the origin of the
overwhelming imbalance between matter and anti-matter in the universe, or
settle the decades-long search for charge-conjugation and parity
($\mathcal{CP}$)-violation in the strong force andreev2018improved;
safronova2018search; roussy2023improved.

The ability to unravel nuclear and particle physics phenomena from
experimental measurements on molecules is limited by the combined precision
that can be achieved experimentally and theoretically. On the theoretical
side, a detailed understanding of the electronic wavefunction inside of the
nuclear volume is essential to reliably extract fundamental physics
information from measurements auerbach1996collective; flambaum2008electric;
isaev2010laser; kudashov2014ab; sasmal2016relativistic. Therefore, determining
observables that are sensitive to the electron-nucleon interaction within the
nucleus, such as the molecular hyperfine structure, is critical.

Here, we report precision laser spectroscopy measurements of the hyperfine
structure of the 225Ra19F molecule. With a lifetime on the order of just days,
our results represent a major milestone in precision studies of short-lived
radioactive molecules. We combine these with state-of-the-art molecular
structure calculations to reveal previously unknown details of the electron-
nucleus interaction in this molecule. This enables a clear observation of the
effect of the 225Ra nuclear magnetization distribution on the molecular energy
levels. This effect has been previously observed in atoms San23 but, to our
knowledge, has never been measured before in a molecule. These findings
exemplify the extreme sensitivity of the RaF molecule to properties of the Ra
nucleus, and provide a direct and stringent test of the description of the
electronic wavefunction within the nuclear volume.

##  II Experimental Setup

The experiment was performed using the Collinear Resonance Ionization
Spectroscopy (CRIS) setup at ISOLDE-CERN catherall2017isolde;
garcia2020spectroscopy; udrescu2021isotope. A simplified version of the
experimental setup is illustrated in Fig. 1A. The RaF molecules were created
by impinging $1.4$-GeV protons upon a uranium carbide target (UC), followed by
the injection of CF4 gas inside the target container at a temperature greater
than $2000$ K. The RaF+ isotopologues of interest were extracted, mass
selected, then trapped and bunched in a radiofrequency quadrupole (RFQ) trap
filled with He gas at room temperature for up to 20 ms. Bunches of RaF+ were
subsequently released, accelerated to $29908(1)$ eV, neutralized in a charge-
exchange cell filled with a Na vapor at a temperature of $T\approx~{}500$ K
and then entered the experimental interaction region.

There, they were collinearly overlapped with three pulsed lasers in a
resonance ionization scheme. The first laser employed was a Ti:Sapphire with a
linewidth of $20$ MHz, which was used to excite transitions between rotational
and hyperfine levels in the $X~{}^{2}\Sigma^{+}(v^{\prime\prime}=0)$
electronic ground state and the first excited
$A~{}^{2}\Pi_{1/2}(v^{\prime}=0)$ electronic state (Fig. 1B).
$v^{\prime\prime}$ and $v^{\prime}$ label the vibrational quantum numbers in
the two electronic manifolds. Then, a pulsed dye laser (PDL), with a linewidth
of $15$ GHz, was used to further excite the molecules to a higher-lying,
${}^{2}\Pi_{1/2}$, electronic state udrescu2023precision, from which the
molecules were ionized by a third, high-power $532$-nm Nd:YAG laser ($40$ mJ).
The resulting RaF+ ions were deflected from the neutral bunch and counted
using an ion detector as a function of the first laser wavenumber, leading to
the observed spectra (Fig. 1C). This laser ionization scheme allowed us to
improve our resolution by more than two orders of magnitude and increase the
signal-to-noise ratio by one order of magnitude compared to previous
experiments garcia2020spectroscopy; udrescu2021isotope, achieving a transition
linewidth of $150$ MHz. This enabled an unambiguous observation of the
hyperfine splitting in 225RaF due to the 225Ra nucleus (nuclear spin
$\mathrm{I}=1/2$), despite its short lifetime and small rates in the
interaction region (as low as 50 molecules per second in a given rotational
state). A more detailed description of the experimental setup can be found in
Ref. udrescu2023precision.

Figure 1: Experimental setup. (A) Radium fluoride molecules are produced by impinging $1.4$-GeV protons on a high-temperature (T$=2000$ K) uranium carbide target, injected with CF4 gas, then surfaced ionized and extracted using electrostatic fields (I). 225RaF is mass-selected (II) and trapped in a He-filled radiofrequency quadrupole (T = 300 K) for up to $20$ ms (III). The bunched RaF ions are guided using electrostatic deflectors (IV), neutralized in a Na-filled charge-exchange cell (V), then overlapped with 3 pulsed lasers in a collinear geometry (VI). The resulting RaF ions are deflected and detected using an ion detector (VII). (B) Example of energy levels involved in a transition between hyperfine levels in an R-branch (not to scale). N, J and F correspond to the rotational, electronic and total angular momentum quantum numbers of the molecule (N and J are not good quantum numbers when $\mathrm{I}>0$). Experimentally observed transitions are shown by upwards-pointing arrows and numbered. (C) Example of measured spectra showing the ion rate in arbitrary units (a.u.) as a function of the wavenumber of the first laser, Doppler corrected to the molecular rest frame and shifted by $T_{\Pi}$. The error bars show one standard deviation statistical uncertainty. Data points are connected by straight lines to guide the eye. The numbering on the individual peaks corresponds to the transitions shown in (B). Table 1:  Rotational and hyperfine parameters in units of cm-1 for $X~{}^{2}\Sigma^{+}(v^{\prime\prime}=0)$ and $A~{}^{2}\Pi_{1/2}(v^{\prime}=0)$ electronic manifolds of 225Ra19F. The $1\sigma$ statistical and systematic uncertainties are shown in round and square brackets, respectively. In the last two columns, values of these parameters obtained from previous experimental and theoretical studies are presented. Parameter | This work | Exp. (Lit.) | Theory (Lit.)
---|---|---|---
$B^{\prime\prime}$ | 0.192070(5)[15] | 0.19205(3)[5]111Scaled from udrescu2023precision | 0.1910222Scaled from zaitsevskii2022accurate
$A_{\parallel}$ | -0.5692(5)[20] | - | -0.5690[55]333Ref. skripnikov2020nuclear
$A_{\perp}$ | -0.5445(2)[8] | - | -0.5470[55]c
$T_{\Pi}$ | 13284.532(5)[20] | 13284.544(50)[20]444Ref. udrescu2021isotope; udrescu2023precision | -
$B^{\prime}$ | 0.191100(15)[45] | 0.19108(3)[5]a | 0.1903b
$p$ | -0.4109(15)[40] | -0.41087(9)[20]a | -
$A_{\perp}$ | -0.076(1)[2] | - | -0.074[1]c

##  III Results and Discussions

The measured transitions were fitted using a rotational and hyperfine
Hamiltonian for each of the two electronic states involved, using the software
PGOPHER western2017pgopher. The fit included $54$ transitions (see Fig. 1C and
Fig. 3) and the values of the fitted rotational and hyperfine parameters
obtained are shown in Table 1. These are in excellent agreement with previous
experiments, as well as with ab initio theoretical calculations, from which
they deviate by less than $1\%$ ($<0.5$ combined standard deviation). Examples
of the measured spectra together with a detailed description of the data
analysis and the quantum chemistry calculations can be found in the
Supplementary Materials.

The hyperfine structure parameter of the ground state, $A_{\perp}$, which
quantifies the strength of the coupling between the electron and the 225Ra
nuclear spin, can be written as the product between the magnetic dipole moment
of the 225Ra nucleus, $\mu$(225Ra), and an electronic form factor
fermi1933theorie; skripnikov2020nuclear. Using available data for the 225Ra+
cation wendt1987hyperfine, this form factor has been calculated in Ref.
skripnikov2020nuclear for 225Ra19F in two different ways: one in which
$\mu$(225Ra) is treated as a point-like dipole moment and another one, in
which the distribution of the nuclear magnetization within the 225Ra nucleus
is accounted for, in a model-independent manner. Details of the extraction of
the effect of the nuclear magnetization distribution and the relation between
Ra+ and RaF are given in the Supplementary Materials. Using these two
calculated values, together with our measured $A_{\perp}$ parameter, the value
for $\mu$(225Ra) can be precisely extracted. The obtained results are shown in
Fig. 2A, on the left, for the former case and on the right for the latter. The
black error bars correspond to the experimental uncertainty, while the blue
bands represent the combined theoretical and experimental uncertainty. The
horizontal orange band represents the literature value of $\mu$(225Ra) and
associated uncertainty, given by its thickness, obtained from an independent
experiment performed on 225Ra atoms arnold1987direct. It can be seen (Fig. 2A)
that the effect of the distribution of nuclear magnetization inside of the Ra
nucleus, $\mu_{BW}$(225Ra), amounts to almost 5$\%$ of the value of
$\mu$(225Ra). The $1\%$ uncertainty on the extracted value of $\mu$(225Ra)
therefore corresponds to $\sim 20\%$ relative uncertainty on
$\mu_{BW}$(225Ra). This level of uncertainty already has the potential to
allow discrimination between simple models of the distribution of the nuclear
magnetization inside the 225Ra nucleus (see Supplementary Materials). The
presence of this effect, known as the Bohr-Weisskopf effect (BW) in atoms
bohr1950influence, can only be clearly observed in 225Ra19F due to the
combination of high experimental resolution and precise molecular theory (Fig.
2B).

Figure 2: Nuclear effects in the RaF molecule due to the Ra nucleus. (A)
Extracted values of the magnetic moment of 225Ra,
$\mu\left({}^{225}\mathrm{Ra}\right)$, in units of nuclear magnetons,
$\mu_{N}$, assuming the Ra nucleus is a point-like magnetic dipole (left) and
accounting for the distribution of the nuclear magnetization inside of the Ra
nucleus (right). The difference between the two, $\mu_{BW}$(225Ra),
corresponds to the effect of the distribution of the nuclear magnetization and
amounts to $\sim 5\%$ of the total value of
$\mu\left({}^{225}\mathrm{Ra}\right)$. The black and blue error bars are the
experimental and total (experimental plus theoretical) uncertainties,
respectively. The center and thickness of the orange band correspond to the
previously measured value and associated uncertainty of $\mu$(225Ra) in an
atom arnold1987direct. (B) Evolution of the calculated $A_{\perp}$ for
increasing levels of theoretical sophistication (see main text and the
Supplementary Materials for more details) skripnikov2020nuclear. (C) Order-of-
magnitude estimation of nuclear effects due to Ra nucleus on the energy levels
of 223,225RaF. From left to right: changes in nuclear charge radius between Ra
isotopes udrescu2021isotope, point-like magnetic dipole moment, electric
quadrupole moment petrov2020energy, distribution of nuclear magnetization,
anapole moment, nuclear Schiff moment flambaum2019enhanced; graner2016reduced,
magnetic quadrupole moment flambaum2022enhanced. The electric and magnetic
quadrupole moments are nonzero only for Ra isotopes with nuclear spin
$\mathrm{I}>1/2$, such as 223Ra.

The remarkable agreement between experiment and theory (see Table 1), at below
the $1\%$ level, reflects the reliability of the ab initio quantum chemistry
calculations, demonstrating that state-of-the-art theoretical methods are able
to provide an accurate description of the electronic wavefunction within the
Ra nucleus. Molecular theory is an essential ingredient for extracting
fundamental physics information from precision experiments Skripnikov15b;
skripnikov2016combined; safronova2018search; skripnikov2020nuclear. Using a
computational scheme similar to that used in Ref. skripnikov2020nuclear, the
electronic parameters that provide the sensitivity of the 225Ra19F molecule to
symmetry-violating phenomena were calculated: the effective electric field
$E_{\rm eff}$ acting on the electron electric dipole moment (EDM); the
molecular parameter $W_{P,T}$ that characterizes the
$\mathcal{P},\mathcal{T}$-violating scalar-pseudoscalar nuclear-electron
interaction; the molecular constant $W_{S}$ that defines the interaction of
the $\mathcal{P},\mathcal{T}$-violating Schiff moment of 225Ra with the
electronic cloud; and the molecular parameter $W_{a}$ that captures the
interaction between electrons and the $\mathcal{P}$-violating nuclear anapole
moment (see Supplementary Materials for details). The obtained values are
shown in Table 2 (second column). They are in good agreement with previous
theoretical studies Isaev:2012; Borschevsky:13; kudashov2014ab;
sasmal2016relativistic (third column of Table 2), but are more precise, by as
much as an order of magnitude, mainly due to a more complete treatment of
correlation effects for all electrons of RaF.

The values of hyperfine structure (HFS) constants, $A_{\perp}$ and
$A_{\parallel}$, as well as the symmetry-violating electronic form factors,
$E_{\rm eff}$, $W_{P,T}$, $W_{S}$, and $W_{a}$, strongly depend on the
electronic density behaviour inside the 225Ra nucleus safronova2018search.
However, unlike the HFS constants measured in this work, the other computed
parameters, which provide the sensitivity to symmetry-violating phenomena,
cannot be measured experimentally. Hence, HFS measurements are essential for
benchmarking ab initio theoretical calculations, and critically, they allow a
reliable prediction of the molecular sensitivity to symmetry-violating
properties. As an illustrative example of the strong connection between the
HFS constants and symmetry violating electronic form factor, it can be shown
that the $A_{\perp}$ and $A_{\parallel}$ parameters can be directly related to
$E_{\rm eff}$ using a semi-empirical approximation Kozlov:97c,
$E_{\mathrm{eff}}=\alpha\sqrt{AA_{d}}$, where
$A\equiv(A_{\parallel}+2A_{\perp})/3$,
$A_{d}\equiv(A_{\parallel}-A_{\perp})/3$ and $\alpha\approx 0.0313$ GV/(cm
MHz) is an approximate proportionality constant that can be obtained by simple
numerical calculations Kozlov:97c. This approach provides a value of $E_{\rm
eff}=63.2$ GV/cm, which is within 20% of our accurate state-of-the-art ab
initio calculations, reported in Table 2.

Table 2: Calculated symmetry-violating electronic form factors in 225Ra19F.
The parameter names and their units are shown in the first column. The second
and third columns show the values of the parameters calculated in this work
and previous studies (see Supplementary Materials for details). The numbers in
square brackets correspond to one standard deviation uncertainty (where
available).

| This work | Previous work
---|---|---
$E_{\rm eff}$ (GV/cm) | -53.3[9] |  | -52.8[53]a
---
-52.5[52]b
-56.9c
-50.8d
-54.2[54]e
$W_{P,T}$ ($h$ kHz) | -144.3[14] |  | -139[14]a
---
-141.2[140]b
-152.5c
-138d
$W_{a}$ ($h$ Hz) | 1694[17] |  | 1700[170]a
---
1420[213]f
1641[246]g
$W_{S}$ $(e/(4\pi\epsilon_{0}a_{0}^{4}))$ | -20900[2100] |  | -22130[2213]a,h
---
-19148d,g

a Ref. kudashov2014ab, b Ref. sasmal2016relativistic, c Ref. Sunaga:2019, d
Ref. Gaul:2020,
e Ref. Cheng:2021, f Ref. Isaev:2012, g Ref. Borschevsky:13, h Ref.
Flambaum:2019b

Using the calculated hyperfine and $\mathcal{P},\mathcal{T}$-odd electronic
form factors, the contributions of various nuclear effects to the molecular
spectra of 223,225RaF are illustrated in Fig 2C. Electroweak nuclear
properties related to W- and Z-boson exchange within the Ra nucleus, such as
the anapole moment, are expected to be on the order of $100$ Hz. Experiments
on stable molecules are already able to achieve and exceed this level of
precision safronova2018search. Nuclear $\mathcal{CP}$-violation effects due to
beyond the SM physics are predicted to produce shifts on the order of mHz, a
level of precision that is within reach of existing atomic and molecular
techniques andreev2018improved; roussy2023improved. Molecules containing Ra
nuclei therefore represent some of the most compelling systems for discovering
$\mathcal{CP}$-violation in the strong force safronova2018search.

##  IV Conclusions and outlook

The hyperfine structure of 225Ra19F was measured, revealing the high
sensitivity of this molecule to the properties of the 225Ra nucleus. The
observation of the distribution of the nuclear magnetization effect in a
molecule was possible thanks to the combined precision of our experiment and
high accuracy of quantum chemistry calculations, which are now reaching the
sub-percent level. Improving the precision of these calculations by a factor
of $\sim 2-3$ would already enable different nuclear magnetization models to
be distinguished between, at below the $10\%$ level (see Ref.
skripnikov2020nuclear and the Supplementary Materials), facilitating stringent
tests of nuclear theory. We hope that our experimental results will motivate
the development of higher-accuracy molecular and nuclear structure
calculations.

Our findings lay the groundwork for using these molecules in future studies of
higher-order symmetry-conserving nuclear moments such as the nuclear magnetic
octupole moment de2021magnetic; de2022precision, electric hexadecapole
beloy2008hyperfine; xiao2020hyperfine or electric quadrupole shift (higher-
order correction to the electric quadrupole interaction, due to electron
penetration into the nucleus) Ste10. The former has never been measured in a
molecule, while the latter two have not been observed in any atom or molecule
so far. All of these properties are enhanced in molecular systems containing
heavy, octupole-deformed nuclei safronova2018search. Together with the
distribution of the nuclear magnetization, they can provide valuable
information about the behaviour of protons and neutrons within atomic nuclei
which is important for elucidating the microscopic origin of collective
nuclear phenomena. Observables that are sensitive to the neutron distribution
would be key to our understanding of nuclear matter and constrain properties
of neutron stars yang2022laser. Our measurements provide critical information
on the rotational and hyperfine structure of 225Ra19F, which, complemented by
the calculated electronic form factors, represent a major milestone towards
future experimental developments that aim to use these molecules for
fundamental physics studies safronova2018search.

## References

  * Shuman  _et al._ (2010) E. S. Shuman _et al._ , “Laser cooling of a diatomic molecule,” Nature 467, 820–823 (2010).
  * Truppe  _et al._ (2017) S. Truppe _et al._ , “Molecules cooled below the Doppler limit,” Nat. Phys. 13, 1173–1176 (2017).
  * Safronova  _et al._ (2018) M. S. Safronova _et al._ , “Search for new physics with atoms and molecules,” Rev. Mod. Phys. 90, 025008 (2018).
  * Collaboration  _et al._ (2018) ACME Collaboration _et al._ , “Improved limit on the electric dipole moment of the electron,” Nature 562, 355–360 (2018).
  * Roussy  _et al._ (2023) T. S. Roussy _et al._ , “An improved bound on the electron’s electric dipole moment,” Science 381, 46–50 (2023).
  * Altuntaş  _et al._ (2018) E. Altuntaş _et al._ , “Demonstration of a sensitive method to measure nuclear-spin-dependent parity violation,” Phys. Rev. Lett. 120, 142501 (2018).
  * Saue  _et al._ (2020) T. Saue _et al._ , “The DIRAC code for relativistic molecular calculations,” J. Chem. Phys. 152, 204104 (2020).
  * Kállay and Gauss (2005) M. Kállay and J. Gauss, “Approximate treatment of higher excitations in coupled-cluster theory,” J. Chem. Phys. 123, 214105 (2005).
  * Kállay  _et al._ (2020) M. Kállay _et al._ , “The MRCC program system: Accurate quantum chemistry from water to proteins,” J. Chem. Phys. 152, 074107 (2020).
  * Skripnikov (2020) L. V. Skripnikov, “Nuclear magnetization distribution effect in molecules: Ra+ and RaF hyperfine structure,” J. Chem. Phys. 153, 114114 (2020).
  * Skripnikov  _et al._ (2018) L. V. Skripnikov _et al._ , “New nuclear magnetic moment of ${}^{209}\mathrm{Bi}$: Resolving the bismuth hyperfine puzzle,” Phys. Rev. Lett 120, 093001 (2018).
  * Auerbach  _et al._ (1996) N. Auerbach _et al._ , “Collective T-and P-odd electromagnetic moments in nuclei with octupole deformations,” Phys. Rev. Lett. 76, 4316 (1996).
  * Flambaum (2008) V. V. Flambaum, “Electric dipole moments of actinide atoms and RaO molecule,” Phys. Rev. A 77, 024501 (2008).
  * Isaev  _et al._ (2010) T. A. Isaev _et al._ , “Laser-cooled RaF as a promising candidate to measure molecular parity violation,” Phys. Rev. A 82, 052521 (2010).
  * Kudashov  _et al._ (2014) A. D. Kudashov _et al._ , “Ab initio study of radium monofluoride (RaF) as a candidate to search for parity-and time-and-parity–violation effects,” Phys. Rev. A 90, 052513 (2014).
  * Sasmal  _et al._ (2016) S. Sasmal _et al._ , “Relativistic coupled-cluster study of RaF as a candidate for the parity-and time-reversal-violating interaction,” Phys. Rev. A 93, 062506 (2016).
  * Flambaum (2019) V. V. Flambaum, “Enhanced nuclear schiff moment and time-reversal violation in 229Th-containing molecules,” Phys. Rev. C 99, 035501 (2019).
  * Gaul  _et al._ (2019) K. Gaul _et al._ , “Systematic study of relativistic and chemical enhancements of P, T-odd effects in polar diatomic radicals,” Phys. Rev. A 99, 032509 (2019).
  * Garcia Ruiz  _et al._ (2020) R. F. Garcia Ruiz _et al._ , “Spectroscopy of short-lived radioactive molecules,” Nature 581, 396–400 (2020).
  * Udrescu  _et al._ (2021) S. M. Udrescu _et al._ , “Isotope shifts of radium monofluoride molecules,” Phys. Rev. Lett. 127, 033001 (2021).
  * Gaffney  _et al._ (2013) L. P. Gaffney _et al._ , “Studies of pear-shaped nuclei using accelerated radioactive beams,” Nature 497, 199–204 (2013).
  * Parker  _et al._ (2015) R. H. Parker _et al._ , “First measurement of the Atomic Electric Dipole Moment of ${}^{225}\mathrm{Ra}$,” Phys. Rev. Lett. 114, 233002 (2015).
  * Chupp  _et al._ (2019) T. E. Chupp _et al._ , “Electric dipole moments of atoms, molecules, nuclei, and particles,” Rev. Mod. Phys. 91, 015001 (2019).
  * Sanamyan  _et al._ (2023) G. Sanamyan _et al._ , “Empirical determination of the Bohr-Weisskopf effect in cesium and improved tests of precision atomic theory in searches for new physics,” Phys. Rev. Lett. 130, 053001 (2023).
  * Catherall  _et al._ (2017) R. Catherall _et al._ , “The ISOLDE facility,” J. Phys. G 44, 094002 (2017).
  * Udrescu  _et al._ (2023) S.-M. Udrescu _et al._ , “Precision spectroscopy and laser cooling scheme of a radium-containing molecule,” https://www.researchsquare.com/article/rs-2648482/v1 (2023).
  * Zaitsevskii  _et al._ (2022) A. Zaitsevskii _et al._ , “Accurate ab initio calculations of RaF electronic structure appeal to more laser-spectroscopical measurements,” J. Chem. Phys. 156, 044306 (2022).
  * Western (2017) C. M. Western, “PGOPHER: A program for simulating rotational, vibrational and electronic spectra,” J. Quant. Spectrosc. Radiat. Transf. 186, 221–242 (2017).
  * Fermi and Segrè (1933) E. Fermi and E. Segrè, “Zur theorie der hyperfeinstruktur,” Zeitschrift für Physik 82, 729–749 (1933).
  * Wendt  _et al._ (1987) K. Wendt _et al._ , “On the hyperfine structure and isotope shift of radium,” Z. Phys. D. At., Mol. Clust. 4, 227–241 (1987).
  * Arnold  _et al._ (1987) E. Arnold _et al._ , “Direct measurement of nuclear magnetic moments of radium isotopes,” Phys. Rev. Lett. 59, 771 (1987).
  * Bohr and Weisskopf (1950) A. Bohr and V. F. Weisskopf, “The influence of nuclear structure on the hyperfine structure of heavy elements,” Phys. Rev 77, 94 (1950).
  * Petrov and Skripnikov (2020) A. N. Petrov and L. V. Skripnikov, “Energy levels of radium monofluoride RaF in external electric and magnetic fields to search for P-and T, P-violation effects,” Phys. Rev. A 102, 062801 (2020).
  * Graner  _et al._ (2016) B. Graner _et al._ , “Reduced limit on the permanent electric dipole moment of 199Hg,” Phys. Rev. Lett. 116, 161601 (2016).
  * Flambaum and Mansour (2022) V. V. Flambaum and A. J. Mansour, “Enhanced magnetic quadrupole moments in nuclei with octupole deformation and their C,P-violating effects in molecules,” Phys. Rev. C. 105, 065503 (2022).
  * Skripnikov and Titov (2015a) L. V. Skripnikov and A. V. Titov, “Theoretical study of ${\mathrm{ThF}}^{+}$ in the search for $T,P$-violation effects: Effective state of a Th atom in ${\mathrm{ThF}}^{+}$ and ThO compounds,” Phys. Rev. A 91, 042504 (2015a).
  * Skripnikov (2016) L. V. Skripnikov, “Combined 4-component and relativistic pseudopotential study of ThO for the electron electric dipole moment search,” J. Chem. Phys. 145, 214301 (2016).
  * Isaev and Berger (2012) T. A. Isaev and R. Berger, “Electron correlation and nuclear charge dependence of parity-violating properties in open-shell diatomic molecules,” Phys. Rev. A 86, 062515 (2012).
  * Borschevsky  _et al._ (2013) A. Borschevsky _et al._ , “Relativistic study of nuclear-anapole-moment effects in diatomic molecules,” Phys. Rev. A 88, 022125 (2013).
  * Kozlov (1997) M. G. Kozlov, “Enhancement of the electric dipole moment of the electron in the YbF molecule,” J. Phys. B 30, L607–612 (1997).
  * Sunaga  _et al._ (2019) A. Sunaga _et al._ , “Merits of heavy-heavy diatomic molecules for electron electric-dipole-moment searches,” Phys. Rev. A 99, 062506 (2019).
  * Gaul and Berger (2020) K. Gaul and R. Berger, “Toolbox approach for quasi-relativistic calculation of molecular properties for precision tests of fundamental physics,” J. Chem Phys. 152, 044101 (2020).
  * Zhang  _et al._ (2021) C Zhang _et al._ , “Calculations of time-reversal-symmetry-violation sensitivity parameters based on analytic relativistic coupled-cluster gradient theory,” Phys. Rev. A 104, 012814 (2021).
  * Flambaum  _et al._ (2020) V. V. Flambaum _et al._ , “Time-and parity-violating effects of the nuclear Schiff moment in molecules and solids,” Phys. Rev. A 101, 042501 (2020).
  * De Groote  _et al._ (2021) R. P. De Groote _et al._ , “Magnetic octupole moment of 173Yb using collinear laser spectroscopy,” Phys. Rev. A 103, 032826 (2021).
  * de Groote  _et al._ (2022) R. P. de Groote _et al._ , “Precision measurement of the magnetic octupole moment in 45Sc as a test for state-of-the-art atomic-and nuclear-structure theory,” Phys. Lett. B 827, 136930 (2022).
  * Beloy  _et al._ (2008) K. Beloy _et al._ , “Hyperfine structure of the metastable ${}^{3}P_{2}$ state of alkaline-earth-metal atoms as an accurate probe of nuclear magnetic octupole moments,” Phys. Rev. A 77, 012512 (2008).
  * Xiao  _et al._ (2020) D. Xiao _et al._ , “Hyperfine structure of 173Yb+: Toward resolving the 173Yb nuclear-octupole-moment puzzle,” Phys. Rev. A 102, 022810 (2020).
  * Koch  _et al._ (2010) K. Koch _et al._ , “Electron penetration into the nucleus and its effect on the quadrupole interaction,” Phys. Rev. A 81, 032507 (2010).
  * Yang  _et al._ (2022) X. F. Yang _et al._ , “Laser spectroscopy for the study of exotic nuclei,” Prog. Part. Nucl. Phys. , 104005 (2022).
  * Denis  _et al._ (2022) M. Denis _et al._ , “Benchmarking of the Fock-space coupled-cluster method and uncertainty estimation: Magnetic hyperfine interaction in the excited state of BaF,” Phys. Rev. A 105, 052811 (2022).
  * Skripnikov and Prosnyak (2022) L. V. Skripnikov and S. D. Prosnyak, “Refined nuclear magnetic dipole moment of rhenium: ${}^{185}\mathrm{Re}$ and ${}^{187}\mathrm{Re}$,” Phys. Rev. C 106, 054303 (2022).
  * Neu  _et al._ (1989) W. Neu _et al._ (ISOLDE”), “Quadrupole moments of radium isotopes from the 7p ${}^{2}P_{3/2}$ hyperfine structure in Ra II,” Z. Phys. D 11, 105–111 (1989).
  * Dammalapati  _et al._ (2016) U. Dammalapati _et al._ , “Compilation of spectroscopic data of radium (Ra I and Ra II),” J. Phys. Chem. Ref. Data 45, 013101 (2016).
  * Prosnyak and Skripnikov (2021) S. D. Prosnyak and L. V. Skripnikov, “Effect of nuclear magnetization distribution within the Woods-Saxon model: Hyperfine splitting in neutral Tl,” Phys. Rev. C 103, 034314 (2021).
  * Ginges  _et al._ (2017) J. S. M. Ginges _et al._ , “Ground-state hyperfine splitting for Rb, Cs, Fr, Ba+, and Ra+,” Phys. Rev. A 96, 062502 (2017).
  * Sen’kov and Dmitriev (2002) R. A. Sen’kov and V. F. Dmitriev, “Nuclear magnetization distribution and hyperfine splitting in Bi82+ ion,” Nucl. Phys. A 706, 351 – 364 (2002).
  * Shabaev  _et al._ (1997) V. M. Shabaev _et al._ , “Ground-state hyperfine splitting of high-Z hydrogenlike ions,” Phys. Rev. A 56, 252 (1997).
  * Gomes  _et al._ (2019) A. S. P. Gomes _et al._ , “DIRAC19. DIRAC, a relativistic ab initio electronic structure program, release DIRAC19 (2019),” (2019).
  * Dyall (2016) K. G. Dyall, “Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the light elements H–Ar,” Theor. Chem. Acc. 135, 128 (2016).
  * Titov and Mosyagin (1999) A. V. Titov and N. S. Mosyagin, “Generalized relativistic effective core potential: Theoretical grounds,” Int. J. Quantum Chem. 71, 359–401 (1999).
  * Mosyagin  _et al._ (2010) N. S. Mosyagin _et al._ , “Shape-consistent relativistic effective potentials of small atomic cores,” Int. Rev. At. Mol. Phys 1, 63–72 (2010).
  * Mosyagin  _et al._ (2016) N. S. Mosyagin _et al._ , “Generalized relativistic effective core potentials for actinides,” Int. J. Quantum Chem. 116, 301–315 (2016).
  * Skripnikov  _et al._ (2013) L. V. Skripnikov _et al._ , “Relativistic coupled-cluster calculations of spectroscopic and chemical properties for element 120,” Chem. Phys. Lett. 555, 79–83 (2013).
  * Kendall  _et al._ (1992) R. A. Kendall _et al._ , “Electron affinities of the first‐row atoms revisited. Systematic basis sets and wave functions,” J. Chem. Phys. 96, 6796–6806 (1992).
  * De Jong  _et al._ (2001) W. A. De Jong _et al._ , “Parallel Douglas–Kroll energy and gradients in NWChem: Estimating scalar relativistic effects using Douglas–Kroll contracted basis sets,” J. Chem. Phys. 114, 48–53 (2001).
  * Mårtensson-Pendrill and Öster (1987) A.-M. Mårtensson-Pendrill and P. Öster, “Calculations of atomic electric dipole moments,” Phys. Scr. 36, 444 (1987).
  * Lindroth  _et al._ (1989) E. Lindroth _et al._ , “Order $\alpha^{2}$ theory of the atomic electric dipole moment due to an electric dipole moment on the electron,” J. Phys. B: At. Mol. Opt. Phys. 22, 559 (1989).
  * Khriplovich (1991) I. B. Khriplovich,  _Parity nonconservation in atomic phenomena_ (Philadelphia, PA (United States); Gordon and Breach Science Publishers, 1991).
  * Sushkov and Flambaum (1978) P. Sushkov and V. V. Flambaum, “Parity breaking effects in diatomic molecules,” Zh. Eksp. Teor. Fiz 75, 1208–1213 (1978).
  * Flambaum and Khriplovich (1985) V. V. Flambaum and I. B. Khriplovich, “On the enhancement of parity nonconserving effects in diatomic molecules,” Phys. Lett. A 110, 121–125 (1985).
  * Skripnikov and Titov (2015b) L. V. Skripnikov and A. V. Titov, “Theoretical study of thorium monoxide for the electron electric dipole moment search: Electronic properties of $H^{3}\Delta_{1}$ in ThO,” J. Chem. Phys. 142, 024301 (2015b).
  * Skripnikov  _et al._ (2017) L. V. Skripnikov _et al._ , “Scalar-pseudoscalar interaction in the francium atom,” Phys. Rev. A 95, 022507 (2017).
  * Titov  _et al._ (2006) A. V. Titov _et al._ , “P,T-Parity Violation Effects in Polar Heavy-Atom Molecules,” in  _Recent Advances in the Theory of Chemical and Physical Systems_ (Springer, 2006) pp. 253–283.
  * Skripnikov and Titov (2016) L. V. Skripnikov and A. V. Titov, “LCAO-based theoretical study of PbTiO3 crystal to search for parity and time reversal violating interaction in solids,” J. Chem. Phys. 145, 054115 (2016).
  * Visscher and Dyall (1997) L. Visscher and D. G. Dyall, “Dirac–Fock atomic electronic structure calculations using different nuclear charge distributions,” Atomic Data and Nuclear Data Tables 67, 207–224 (1997).
  * Shabaev  _et al._ (2013) V. M. Shabaev _et al._ , “Model operator approach to the lamb shift calculations in relativistic many-electron atoms,” Phys. Rev. A 88, 012513 (2013).
  * Skripnikov (2021) L. V. Skripnikov, “Approaching meV level for transition energies in the radium monofluoride molecule RaF and radium cation Ra+ by including quantum-electrodynamics effects,” J. Chem. Phys. 154, 201101 (2021).
  * Ginges and Flambaum (2004) J. S. M. Ginges and V. V. Flambaum, “Violations of fundamental symmetries in atoms and tests of unification theories of elementary particles,” Phys. Rep. 397, 63–154 (2004).
  * Flambaum and Kozlov (2012) V. V. Flambaum and A. Kozlov, “Screening and finite-size corrections to the octupole and Schiff moments,” Phys. Rev. C 85, 068502 (2012).
  * Hinds and Sandars (1980) E. A. Hinds and P. G. H. Sandars, “Electric dipole hyperfine structure of TiF,” Phys. Rev. A 21, 471 (1980).
  * Sushkov  _et al._ (1984) O. P. Sushkov _et al._ , “Possibility of investigating P-and T-odd nuclear forces in atomic and molecular experiments,” Zh. Eksp. Teor. Fiz 87, 1521 (1984).
  * Skripnikov  _et al._ (2020) L.V. Skripnikov _et al._ , “Actinide and lanthanide molecules to search for strong CP-violation,” Phys. Chem. Chem. Phys. 22, 18374–18380 (2020).
  * Becke (1993) A. D. Becke, “Density-functional thermochemistry. III. The role of exact exchange,” J. Chem. Phys. 98, 5648–5652 (1993).
  * Thierfelder  _et al._ (2010) C. Thierfelder _et al._ , “Relativistic coupled-cluster study of the parity-violation energy shift of CHFClBr,” Phys. Rev. A 81, 032513 (2010).
  * Stone (2005) N. J. Stone, “Table of nuclear magnetic dipole and electric quadrupole moments,” Atomic Data and Nuclear Data Tables 90, 75–176 (2005).
  * Pyykkö (2008) P. Pyykkö, “Year-2008 nuclear quadrupole moments,” Mol. Phys. 106, 1965–1974 (2008).

##  V Supplementary Materials

###  V.1 A. Data analysis

Each peak used in the determination of the rotational and hyperfine
Hamiltonian parameters was fit with a Voigt profile plus a constant
background, using the LMFIT Python package. The number of peaks in a given
scan was chosen based on the reduced-$\chi^{2}$ of the fit. The obtained
central value of each peak, together with its associated uncertainty, were
input into PGOPHER where they were fit with the effective Hamiltonians
described below. The main sources of systematic uncertainty in our experiment
were (expressed as an uncertainty on the wavenumber in brackets): variations
in the ion beam extraction voltage ($1.2\times 10^{-4}$ cm-1), changes in the
beam energy during the charge-exchange process ($10^{-4}$ cm-1), uncertainties
in the measurement of the Rb reference frequency by the wavemeter ($10^{-4}$
cm-1), presence of stray magnetic and electric fields ($<10^{-5}$ cm-1) and AC
Stark shifts due to the presence of the second- and third-step lasers
($5\times 10^{-4}$ cm-1). These uncertainties were added in quadrature to the
statistical uncertainty for each fitted line in the spectra, before performing
the PGOPHER fit.

For the $X~{}^{2}{}{\Sigma}^{+}$ electronic state, the employed rotational and
hyperfine Hamiltonian is given by:

$H_{X~{}^{2}{}{\Sigma}^{+}}^{\mathrm{rot}}=\left(B^{\prime\prime}-D^{\prime\prime}\bm{N}^{2}\right)\bm{N}^{2}+\gamma\bm{N}\cdot\bm{S}+b^{\prime\prime}\mathbf{I}\cdot\mathbf{S}+c^{\prime\prime}I_{z}S_{z},$
(1)

where $B^{\prime\prime}$ is the rotational constant, $D^{\prime\prime}$ is the
centrifugal distortion constant, $\gamma$ is the spin-rotation constant and
$b^{\prime\prime}$ and $c^{\prime\prime}$ are hyperfine constants due to the
225Ra nucleus. These parameters can be related to $A_{\perp}$ and
$A_{\parallel}$ of the $X~{}^{2}{}{\Sigma}^{+}$ state from
skripnikov2020nuclear using: $A_{\perp}=b^{\prime\prime}$ and
$A_{\parallel}-A_{\perp}=c^{\prime\prime}$ denis2022benchmarking.
$\bm{N}=\bm{J}-\bm{S}$, $\bm{S}$ and $\bm{I}$ are the molecular rotational
operator (excluding the electron and nuclear spin), the electron spin operator
and the nuclear spin operator, respectively, while $S_{z}$ and $I_{z}$ are the
z-component of the latter two.

The excited electronic state, $A~{}^{2}{}{\Pi}$, was described by the
effective rotational and hyperfine Hamiltonians:

$\begin{split}H_{A~{}^{2}{\Pi}}^{\mathrm{rot}}&=T_{\Pi}+A_{\Pi}L_{z}S_{z}+\left(B^{\prime}-D^{\prime}\bm{N}^{2}\right)\bm{N}^{2}-\\\
&-\frac{1}{2}\left\\{\frac{p}{2}+p_{D}\bm{N}^{2},N_{+}S_{+}e^{-2i\phi}+N_{-}S_{-}e^{2i\phi}\right\\}+\\\
&+\frac{1}{2}d\left(e^{-2i\phi}I_{+}S_{+}+e^{2i\phi}I_{-}S_{-}\right),\end{split}$
(2)

where $\left\\{O,Q\right\\}=OQ+QO$, $L_{z}$ is the z-component of the electron
orbital momentum operator in the molecular rest frame, $T_{\Pi}$ represents
the energy difference between the origins of the $\nu^{\prime}=0$ vibrational
level of the ${}^{2}\Pi$ electronic manifold and the origins of the
corresponding isovibrational level of the $X~{}^{2}{}{\Sigma}^{+}$ electronic
manifold, while $A_{\Pi}$ is the spin-orbit interaction. The ${}^{2}\Pi$
electronic manifold, gets split, in a Hund case (a) picture, into a
${}^{2}\Pi_{1/2}$ and a ${}^{2}\Pi_{3/2}$ electronic levels due to the spin-
orbit coupling, separated by $A_{\Pi}$. As rovibronic transitions to the
$A^{2}\Pi_{3/2}$ electronic state were not measured, it was not possible to
constrain both $T_{\Pi}$ and $A_{\Pi}$ simultaneously, therefore $A_{\Pi}$ was
kept fixed at its previously measured value of $2067.6$ cm-1
garcia2020spectroscopy. $p$ is the $\Lambda$-doubling parameter and $p_{D}$ is
the centrifugal distortion correction to $p$. Finally, $d$ is a hyperfine
structure constant due to the 225Ra nucleus. In a $A^{2}{}{\Pi}$ state, this
is related to the $A_{\perp}$ parameter from skripnikov2020nuclear by
$A_{\perp}=d$ denis2022benchmarking. $N_{\pm}$, $S_{\pm}$ and $I_{\pm}$ are
the raising and lowering operators for the $\bm{N}$, $\bm{S}$ and $\bm{I}$
operators defined above and $\phi$ is the polar angle around the molecular
axis, defined in the molecule’s rest frame. The molecular parameters,
extracted from fitting the above Hamiltonians to the data, correspond to the
ground vibrational level of each electronic manifold udrescu2023precision. The
hyperfine splitting due to the fluorine nucleus ($\mathrm{I}=1/2$) was
predicted to be much below $100$ MHz and hence was not observed given our
current spectroscopic resolution. Therefore, the corresponding hyperfine
Hamiltonian was not included in the analysis.

For the fitting procedure, $\gamma$, $D^{\prime}$, $D^{\prime\prime}$ and
$p_{D}$ parameters were each sampled from a Gaussian with mean and standard
deviation given by the values of the corresponding parameters of the 226Ra19F
molecule udrescu2023precision, scaled accordingly using the reduced mass of
the two isotopologues. After sampling, these parameters were kept constant
during the fitting procedure, and the values of the other rotational and
hyperfine parameters were extracted. The sampling and subsequent fitting was
repeated $1000$ times, and the obtained average value and standard deviation
of the fitted parameters are reported in Table 1.

Examples of the measured spectra of 225Ra19F together with the obtained best
fit are shown in Fig. 3, where the experimental data is shown in red, while
the best fit is shown in blue. In the center, the full simulated spectrum of
the transitions over a range of $\sim 50$ cm-1 is shown. The x-axis shows the
wavenumber of the first-step laser, Doppler corrected to the molecular rest
frame, while the y-axis shows the rate in arbitrary units (a.u.). The
splitting of each rotational line into $3$ hyperfine components can be clearly
observed in the R-branch spectra around $13300$ cm-1.

Figure 3: Example of measured spectra for the $0^{\prime}\leftarrow
0^{\prime\prime}$ transitions. In the center, in blue, the fitted combined
hyperfine and rovibronic spectrum of 225RaF obtained for $J\leq 100$, over a
range of $\sim 50$ cm-1 is presented. Figures in magnified views show measured
spectra for different regions in frequency space. The connected red dots show
the experimental data, while the continuous blue lines represents the best
fits to the data. The errorbars show one standard deviation statistical
uncertainty. The values on the x-axis correspond to the wavenumber of the
first laser used in the resonance ionization scheme, Doppler corrected to the
molecular rest frame. The rate on the y-axis is given in arbitrary units
(a.u.).

###  V.2 B. Nuclear magnetization distribution effect in Ra+ and RaF

The magnetic dipole hyperfine structure (HFS) constant can be expressed using
the following parametrization bohr1950influence:

$A=A^{(0)}-A^{\rm BW}.$ (3)

Here, $A^{(0)}$ represents the HFS constant in the point-like nuclear magnetic
dipole moment approximation, and $A^{\rm BW}$ quantifies the contribution of
the finite nuclear magnetization distribution to the HFS constant, commonly
known as the Bohr-Weisskopf (BW) effect. In Ref. skripnikov2020nuclear, it was
demonstrated that the BW correction, $A^{\rm BW}$, for heavy atoms and
molecules (including systems with complex electronic structures) can be
factorized as follows (see Eq. (29) of Ref. skripnikov2020nuclear for
details):

$\displaystyle A^{\rm BW}\approx EB_{s}.$ (4)

Here, $E$ represents a pure electronic factor, which is independent of the
nuclear magnetization distribution and is solely determined by the electronic
structure. All information regarding the nuclear magnetization distribution
can be encoded in the electronic state-independent parameter, $B_{s}$. This
parameter has a well-defined physical meaning skripnikov2020nuclear being
proportional to the BW effect contribution to the $A$ constant of the
hydrogen-like (H-like) ion. Therefore, in cases where measurements of H-like
ions are available for a given isotope, $B_{s}$ can be obtained almost
directly Skripnikov:2022 due to the very high accuracy of the theoretical
description of such ions.

In Ref. skripnikov2020nuclear, $A^{(0)}$ and $E$ were computed for the ground
electronic state $7s~{}^{2}S_{1/2}$ of the 225Ra+ cation. By combining these
theoretical values with the experimental value Neu:1988; wendt1987hyperfine;
Dammalapati:2016 of the $A$ constant, the parameter $B_{s}$ was determined.
The electronic $A^{(0)}$ and $E$ constants were also calculated for the
excited electronic state $7p~{}^{2}P_{1/2}$ of 225Ra+. Using these values,
along with the extracted value of $B_{s}$, the BW contribution to the $A$
constant for this state was computed according to Eq. (4). The resulting value
of $A$($7p~{}^{2}P_{1/2}$) was in good agreement with the available
experimental value Neu:1988; wendt1987hyperfine; Dammalapati:2016, with a
deviation of about 0.1%, though the theoretical uncertainty was estimated at
1%. The BW effect contribution to $A$($7p~{}^{2}P_{1/2})$ was $1.4\%$.
Interestingly, the Bohr-Weisskopf effects for the $7s~{}^{2}S_{1/2}$ and
$7p~{}^{2}P_{1/2}$ states are induced by different harmonics, $s_{1/2}$ and
$p_{1/2}$, respectively skripnikov2020nuclear. However, as explained in Ref.
skripnikov2020nuclear, the same constant $B_{s}$ can be utilized in both cases
due to properties of the solutions to the Dirac equation and the symmetry of
the magnetic dipole hyperfine interaction operator. This is particularly
important for systems with complex electronic structures, where both harmonics
contribute simultaneously. The validity of factorization (4) was also
numerically confirmed by considering various nuclear magnetization
distribution models skripnikov2020nuclear; Prosnyak:2021. Finally, the semi-
empirically extracted value of $B_{s}$ was employed to predict the BW effect
in the 225RaF molecule for both the ground and first excited electronic
states. The uncertainty on the deduced BW effects using this approach is
limited by the uncertainty in the electronic structure calculation of
$A^{(0)}$ and $E$ of Ra+, estimated to be $1\%$ skripnikov2020nuclear. Given
that the BW effect amounts to about $5\%$ of the $A$ value of the
$7s~{}^{2}S_{1/2}$ ground state in Ra+ skripnikov2020nuclear, the uncertainty
in the value of the BW effect obtained using this semi-empirical method is
$\sim 20$%.

It is also possible to estimate the $B_{s}$ parameter using simple nuclear
magnetization distribution models. The simplest model is a uniformly
magnetized sphere, while a more accurate model is a Woods-Saxon (WS) model.
The WS model was used in Ref. ginges2017ground to calculate the BW effect in
225Ra+. By combining the value of $B_{s}^{Ball}$ with the data from Ref.
ginges2017ground and using the factorization property (4), one can obtain
$B_{s}^{WS}$. Using Eq. (4) and the values of the $B_{s}$ parameter from
different nuclear magnetization distribution models, one can calculate the
value of the $A_{||}^{\rm BW}$ and $A_{\perp}^{\rm BW}$ constants for the
ground electronic state of the 225RaF molecule. The obtained results are given
in Table 3. The uncertainty of the BW effect depends on the nuclear model used
Senkov:02; shabaev1997ground and it is expected that future nuclear structure
calculations, employing more realistic models, will allow a prediction of this
effect with quantifiable uncertainties. An improvement in the accuracy of the
electronic structure calculations by only a factor of $\sim 2-3$, combined
with our experimental results, would allow the study of the distribution of
the nuclear magnetization with a relative precision of better than $10\%$.

Table 3: The BW contribution to the hyperfine structure constants, $A_{||}$ and $A_{\perp}$, (in MHz) for the ground electronic state of the 225RaF molecule, using different models of the nuclear magnetization. Model | $A_{||}^{\rm BW}$ | $A_{\perp}^{\rm BW}$
---|---|---
Ball | -537 | -529
WS | -830 | -818
Semi-empirical skripnikov2020nuclear | -730 | -720

###  V.3 C. Computational methods

The Hund case (c) matrix elements
$\langle^{2}\Sigma_{1/2}|J^{e}_{+}|^{2}\Sigma_{-1/2}\rangle$ and
$\langle^{2}\Pi_{1/2}|J^{e}_{+}|^{2}\Pi_{-1/2}\rangle$, where $J^{e}_{+}$ is
the $x+iy$ component of the body-fixed total electronic angular momentum, can
be related to the spin-rotational and $\Lambda$-doubling parameters in Hund
cases (b) and (a) in the above Hamiltonians as follows:
$\langle^{2}\Sigma_{1/2}|J^{e}_{+}|^{2}\Sigma_{-1/2}\rangle=1-\frac{\gamma_{e}}{2B^{\prime\prime}_{e}}$
and
$\langle^{2}\Pi_{1/2}|J^{e}_{+}|^{2}\Pi_{-1/2}\rangle=\frac{p_{e}}{2B^{\prime}_{e}}$,
where $\gamma_{e}$, $p_{e}$, $B^{\prime}_{e}$ and $B^{\prime\prime}_{e}$ refer
to vibrationally independent molecular parameters (see Ref.
udrescu2023precision for details). These matrix elements were calculated
herein and the values obtained are reported in Table 4, exhibiting an
excellent agreement with the experiment udrescu2023precision, at the $0.1\%$
level (the values of these parameters are the same for 226RaF
udrescu2023precision and 225RaF). The following scheme was used for the
calculations. First, correlation calculations were performed employing the
relativistic coupled-cluster approach with single- and double-excitation
amplitudes (CCSD) within the Dirac-Coulomb Hamiltonian gomes2019dirac;
saue2020dirac. Here, 69 electrons of RaF were included in the correlation
treatment and the extended uncontracted all-electron triple-zeta extAE3Z basis
set (based on AE3Z dyall2016relativistic by Dyall) developed in Ref.
skripnikov2020nuclear was used. It includes $[38s\,33p\,24d\,14f\,7g\,3h\,2i]$
Gaussian-type functions for Ra and corresponds to the uncontracted AE3Z
dyall2016relativistic basis set on F. To account for effects of larger basis
sets, a correction was taken as the difference between values of the matrix
elements under consideration using the extended quadruple-zeta extAE4Z
skripnikov2020nuclear basis set and the extAE3Z basis set. These calculations
were performed at the 27-electron CCSD level using the valence part of the
generalized relativistic effective-core potential approach
titov1999generalized; mosyagin2010shape; mosyagin2016generalized. Next,
higher-order correlation effects were implemented through two contributions.
The first of which was calculated as the difference between the results
obtained within the CCSD and partial iterative triple-excitation amplitudes
(CCSDT-3 kallay2005approximate) model compared to the CCSD method. In this
calculation, $35$ electrons of RaF were correlated using the special compact
basis set for Ra constructed using the approach developed in Refs.
skripnikov2013relativistic; skripnikov2016combined; skripnikov2020nuclear and
comprising $[8s\,8p\,7d\,7f\,4g\,2h]$ contracted Gaussian functions, while for
F we used the aug-cc-pVDZ-DK Kendall1992; de2001parallel basis set. Then,
correlation effects up to the CC with full iterative triple- and perturbative
quadruple-excitation amplitudes CCSDT(Q) kallay2005approximate; kallay2020mrcc
approach were implemented. For this contribution, $27$ electrons of RaF were
correlated and the reduced compact basis set reduced for Ra to
$[8s\,8p\,7d\,4f]$ contracted Gaussian functions was used. Finally, the effect
of the Gaunt interelectron interaction at the self-consistent level was
computed. In all of the calculations, the equilibrium Ra–F distance was used.

Table 4: Contribution to the theoretically calculated $\langle^{2}\Sigma_{1/2}|J^{e}_{+}|^{2}\Sigma_{-1/2}\rangle$ and $\langle^{2}\Pi_{1/2}|J^{e}_{+}|^{2}\Pi_{-1/2}\rangle$ matrix elements. The associated experimental values extracted from udrescu2023precision are shown in the last row. The numbers in round (square) brackets correspond to $1\sigma$ statistical (systematic) uncertainty. Contribution | $\langle^{2}\Sigma_{1/2}|J^{e}_{+}|^{2}\Sigma_{-1/2}\rangle$ | $\langle^{2}\Pi_{1/2}|J^{e}_{+}|^{2}\Pi_{-1/2}\rangle$
---|---|---
CCSD | 0.98355 | -1.05444
Basis correction | -0.00010 | -0.00655
CCSDT-3 $-$ CCSD | 0.00050 | -0.00728
CCSDT(Q) $-$ CCSDT-3 | 0.00001 | -0.00407
Gaunt | 0.00057 | 0.00151
Total | 0.98453[67] | -1.0708[97]
Experiment | 0.98475(13)[34] | -1.07335(21)[45]

The $\mathcal{P,T}$-breaking interaction between an $e$EDM and electrons can
be described by the following Hamiltonian MartenssonPendrill1987; Lindroth89:

$\displaystyle H_{d}^{{\rm
eff}}=d_{e}\sum_{a}2ic\gamma^{0}_{a}\gamma_{a}^{5}\bm{p}_{a}^{2},$ (5)

where index $a$ denotes each electron, $\bm{p}$ is the electron momentum
operator, $c$ is the speed of light, $d_{e}$ is the electron EDM and
$\gamma^{0}$ and $\gamma^{5}$ are Dirac matrices defined according to Ref.
Khriplovich91. These matrices are related through
$\gamma^{5}=-i\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}$. This interaction can
be characterized by the molecular constant $W_{d}$:

$W_{d}=\frac{1}{\Omega}\langle\Psi|\frac{H_{d}}{d_{e}}|\Psi\rangle.$ (6)

In these designations, the effective electric field acting on the $e$EDM is
$E_{\rm eff}=W_{d}|\Omega|$. Another possible source of the
$\mathcal{P,T}$-violation is the scalar-pseudoscalar nucleus-electron
interaction given by the following Hamiltonian (see Ref. sushkov1978parity):

$\displaystyle H_{\rm s}=i\frac{G_{F}}{\sqrt{2}}Zk_{\rm
s}\sum_{a}\gamma^{0}_{a}\gamma^{5}_{a}\rho_{N}(\textbf{r}_{a}),$ (7)

where $G_{\mathrm{F}}~{}=~{}2.22249\cdot 10^{-14}~{}a.u$ is the Fermi-coupling
constant, $Z$ is the charge of the heavy nucleus ($Z=88$ for 225Ra in our
case), $\rho_{N}(\textbf{r})$ is the nuclear density normalized to unity and
$\mathbf{r}$ is the electron radius vector with respect to the heavy atom
nucleus under consideration. This interaction is characterized by the
molecular parameter $W_{P,T}$:

$W_{P,T}=\frac{1}{\Omega}\langle\Psi|\frac{H_{\rm s}}{k_{\rm s}}|\Psi\rangle.$
(8)

The electron-nucleus $\mathcal{P}$-odd interaction Hamiltonian is defined as:

$H_{P}=\kappa\frac{G_{\mathrm{F}}}{\sqrt{2}}\bm{\alpha}\cdot\bm{I}\rho_{N}(\textbf{r}).$
(9)

The main contributions to this effect in 225RaF are the Z0-boson exchange
between the unpaired electron and the 225Ra nucleus and the interaction of the
unpaired electron with the nuclear anapole moment. These effects are
characterized by the dimensionless constant $\kappa$. By averaging this
Hamiltonian over the electronic wavefunction of the molecule, the following
rotational and hyperfine effective Hamiltonian is obtained Flambaum:85b:

$H_{\rm eff}=(W_{a}\kappa)\mathbf{n\times S}^{\prime}\cdot\mathbf{I},$ (10)

where $\mathbf{n}$ is the unit vector directed from the Ra nucleus to F and
$W_{a}$ is a molecular parameter given by:

$W_{a}=\frac{G_{\mathrm{F}}}{\sqrt{2}}\left\langle{}^{2}\Sigma_{1/2}\left|\rho_{N}(\textbf{r}){\alpha_{+}}\right|^{2}\Sigma_{-1/2}\right\rangle.$
(11)

To calculate $E_{\rm eff}$, $W_{P,T}$ and $W_{a}$, a similar scheme to that in
Ref. skripnikov2020nuclear was used to calculate hyperfine structure
constants. First, correlation calculations were performed employing the
relativistic coupled-cluster approach with single-, double- and perturbative
triple-excitation amplitudes, CCSD(T), within the Dirac-Coulomb Hamiltonian
gomes2019dirac; saue2020dirac. All 97 electrons of RaF were included in the
correlation treatment using the extAE3Z basis set. The virtual energy cutoff
was set to 10,000 $E_{h}$. The significance of the high energy cutoff for
properties that depend on the behavior of the wavefunction near the heavy-atom
nucleus has been demonstrated and analyzed in detail in Refs. Skripnikov:15a;
Skripnikov:17a. Next, higher-order correlation effects were taken as the
difference between values of the constants under consideration calculated
within the CCSDT and CCSD(T) methods correlating 27 electrons of RaF and using
the SBas basis set from Ref. skripnikov2020nuclear. Additionally, we
calculated the contribution of even higher-order correlation effects by
comparing the results obtained from CCSDT(Q) and CCSDT calculations, which
correlate 27 electrons of RaF. We used the compact basis set comprising
$[8s\,8p\,7d\,4f]$ contracted Gaussian functions for Ra, while for F, we
employed the aug-cc-pVDZ-DK basis set Kendall1992; de2001parallel and employed
the two-component two-step approach within the generalized relativistic
effective core potential (GRECP) theory Titov06amin; Skripnikov15b;
Skripnikov16a. Next, for the case of $E_{\rm eff}$, $W_{P,T}$, a basis set
correction, calculated at the 69e-CCSD(T) level within the Dirac-Coulomb
Hamiltonian was added. Here, the extended number of basis functions in the
extAE4Z basis set with respect to extAE3Z was accounted for. For the case of
$W_{a}$, an equivalent correction was calculated in a similar way, but using
the two-component 27e-CCSD(T) two-step approach Titov06amin; Skripnikov15b;
Skripnikov16a. To test the influence of further basis functions with high
angular momentum for $E_{\rm eff}$, $W_{P,T}$, additional corrections were
determined which capture the effect of $[15g\,15h\,15i\ ]$-type basis
functions within the scalar-relativistic two-step approach Titov06amin;
Skripnikov15b; Skripnikov16a and the 37e-CCSD(T) method. The Gaunt
interelectron contribution was calculated at the self-consistent level and
then rescaled by the factor 1.4 to account for correlation effects. The
calculations described above were performed at a fixed internuclear distance
of 2.24 Å, which corresponds to the equilibrium distance of the electronic
ground state kudashov2014ab; skripnikov2020nuclear. Finally, a vibrational
correction was implemented to the considered molecular constants for the
ground vibrational levels of RaF using the two-step two-component 37e-CCSD(T)
approach similar to Ref. skripnikov2020nuclear. In the calculations described
above, a Gaussian nuclear charge distribution model Visscher:1997 was used.

Table 5: Theoretical contributions to the $E_{\rm eff}$ (in units of GV/cm), $W_{P,T}$ (in units of $h$ kHz) and $W_{a}$ (in units of $h$ Hz) molecular parameters. Contribution | $E_{\rm eff}$ | $W_{P,T}$ | $W_{a}$
---|---|---|---
CCSD(T) | -53.9 | -145.2 | 1707
CCSDT(Q) $-$ CCSD(T) | -0.1 | -0.3 | 3
Basis correction | 0.0 | -0.1 | -3
Gaunt | 0.9 | 1.4 | -16
Vibr. | -0.1 | -0.4 | 4
Total | -53.3[9] | -144.3[14] | 1694[17]

The calculated values of $E_{\rm eff}$, $W_{P,T}$ and $W_{a}$ are given in
Table 5. High-order correlation effects given in the “CCSDT(Q) $-$ CCSD(T)”
lines can be seen to be quite small in addition to basis set corrections.
These two sources of the theoretical uncertainty are therefore almost
negligible for the present case. A special note should be made concerning the
Gaunt interelectron interaction contribution to $E_{\rm eff}$. The one-
electron operator (Eq. 5) is only valid within the Dirac-Coulomb Hamiltonian
Lindroth89. When the Gaunt interaction is included in the molecular
Hamiltonian, the expression (Eq. 5) should be replaced by a two-electron
operator. This is not trivial to realize computationally for molecules.
Therefore, following a previous analysis skripnikov2016combined, the whole
“Gaunt” contribution was included in the uncertainty of $E_{\rm eff}$. This
contribution represents the dominant source of uncertainty. The effect of this
approximate method for calculating the Gaunt contribution was also included in
the theoretical uncertainty of the other calculated constants. The final
uncertainty estimation of $E_{\rm eff}$, $W_{P,T}$ and $W_{a}$ values given in
the main text includes: (i) contribution of higher-order correlation effects
estimated as the value given in the “CCSDT(Q) $-$ CCSD(T)” line of Table 5;
(ii) effects of further extending the basis set which are expected to be at
the level of the values given in the “Basis correction” line of Table 5 and
(iii) effects of the Gaunt (Breit) interelectron interaction described above;
(iv) The contribution of quantum electrodynamics (QED) effects, which has been
estimated as the difference in calculated values of $E_{\rm eff}$, $W_{P,T}$,
and $W_{a}$ obtained using the Dirac-Coulomb Hamiltonian with and without the
inclusion of the model QED operator Shabaev:13 in the formulation in Ref.
Skripnikov2021. It is important to note that we did not include the obtained
QED contributions ($E_{\rm eff}$: 0.2 GV/cm, $W_{P,T}$: -0.2 kHz, $W_{a}$: 3
Hz) in the final values of the calculated constants, as the approach used is
not a rigorous QED treatment. It can however still be used to estimate the
order of magnitude of the QED effects. The final uncertainty is calculated as
the root of the sum of the squares of these uncertainties. Contributions to
these uncertainties from nuclear structure effects are not considered and
included due to the lack of corresponding nuclear structure calculations.

The effective Hamiltonian of the $\mathcal{P,T}$-odd interaction of the
nuclear Schiff moment with electrons that contains a finite nuclear size
correction is given by the following expression GFreview;
flambaum2012screening:

$H^{{\rm eff},2}=W_{S}\mathbf{S^{\prime\prime}}\cdot\mathbf{n},$ (12)

where $S^{\prime\prime}$ is the corrected nuclear Schiff moment
flambaum2012screening and $W_{S}$ can be calculated as:

$W_{S}=\langle\Psi|\sum_{a}\frac{3\mathbf{r_{a}}\cdot\mathbf{n}}{B}\rho_{N}|\Psi\rangle,$
(13)

where $B=\int\rho_{N}(r)r^{4}dr$. Direct use of Eq. (13) requires very large
Gaussian-type basis sets. Alternatively, the relation $W_{S}\approx 6X/r^{sp}$
can be used, where the coefficient $r^{sp}$ was calculated analytically in
Ref. Flambaum:2019b and the molecular parameter $X$ can be computed as follows
Hinds:80a; Sushkov:84:

$\displaystyle
X=-\frac{2\pi}{3}\langle\Psi|[\sum_{i}\mathbf{\nabla_{i}}\cdot\mathbf{n},\delta(\mathbf{R})]|\Psi\rangle.$
(14)

Following Ref. Skripnikov:2020c, the $X$ parameter was calculated at the
2-component CCSD(T) level using the two-step approach Titov06amin;
Skripnikov15b; Skripnikov16a, which allowed the use of the accurate asymptotic
behaviour of the wavefunction inside the nucleus. 37 electrons were included
in the correlation treatment using the extAE3Z basis set. According to this
calculation, correlation effects reduced the relativistic Hartree-Fock value
by a factor of 1.74. The Dirac-Hartree-Fock level of theory was also used to
directly calculate $W_{S}$ according to Eq. (13) where it was possible to use
a very large basis set. The latter was constructed by modifying the extAE3Z
basis set where all $s-$ and $p-$type functions were replaced by even-tempered
series of Gaussian functions. Here, the Gaussian exponential parameters
$\beta_{i}$ were calculated as $\beta_{i+1}=\beta_{i}\cdot 1.6$, where
$\beta_{1}=1.0\times 10^{-3}$ and the maximal $\beta_{i=64}=7.3\times 10^{9}$.
The final value of $W_{S}$ was obtained by applying the factor of 1.74 that
takes into account correlation effects at the Dirac-Hartree-Fock value. The
expected uncertainty of the final $W_{S}$ is $\sim 10\%$, similar to that
estimated in Ref. Skripnikov:2020c for $W_{S}$ constants for actinide-
containing molecules.

The calculated values of the symmetry-violating electronic form factors are
compared with literature values in the main text. Below, we provide a brief
overview of the methods used in those studies (see the corresponding
references for additional details). (i) In Ref. kudashov2014ab, the values of
all molecular parameters of the $\mathcal{P}$-$\mathcal{P}$,$\mathcal{T}$-odd
interactions were calculated for the ground electronic state of RaF using the
2-component two-step approach with the GRECP method Titov06amin;
Skripnikov15b; Skripnikov16a. Electronic correlation effects were treated
using the relativistic Fock-Space coupled cluster method with single and
double excitations, along with a correction for higher-order correlation
effects within the scalar-relativistic CCSD(T) approach. In the calculation
kudashov2014ab, 19 electrons were correlated. (ii) In Ref.
sasmal2016relativistic, the authors calculated the values of $E_{\rm eff}$ and
$W_{P,T}$ using the Dirac-Coulomb Hamiltonian and the CCSD method. They
considered correlation effects for all electrons and set the virtual energy
cutoff to 20 $E_{h}$. (iii) In Ref. Sunaga:2019, the authors calculated the
values of $E_{\rm eff}$ and $W_{P,T}$ using the Dirac-Coulomb Hamiltonian and
the CCSD method. They set the virtual energy cutoff to 80 $E_{h}$. In
comparison to the results of Ref. sasmal2016relativistic, where the so-called
$\Lambda$-equations were solved to determine the values of $E_{\rm eff}$ and
$W_{P,T}$ as analytical derivatives of the coupled cluster energy with respect
to the added perturbation (such as the interaction of the electron EDM with
the effective electric field or the scalar-pseudoscalar nucleus-electron
interaction), an expectation value approach was employed in Ref. Sunaga:2019.
There, the expectation value of a specific operator was calculated,
considering only the linear terms in the CCSD wavefunction. (iv) In Ref.
Gaul:2020, the authors calculated the values of $E_{\rm eff}$, $W_{P,T}$, and
$W_{S}$ using quasi-relativistic wavefunctions obtained within the zeroth-
order regular approximation (ZORA). They treated electronic correlation
effects using the hybrid Becke three-parameter exchange functional and the
Lee, Yang, and Parr correlation functional (B3LYP) b3lyp. A similar approach
using the local density approximation functional was used in Ref. Isaev:2012.
(v) In Ref. Cheng:2021 the value of $E_{\rm eff}$ was calculated using the
exact 2-component atomic mean-field Hamiltonian and the CCSD(T) method. The
virtual energy cutoff was set to 100 $E_{h}$ in the case of RaF. (vi) In Ref.
Borschevsky:13, the value of $W_{a}$ was calculated using the relativistic
density functional method employing the Coulomb-attenuated B3LYP functional,
the parameters of which were adjusted in Ref. Thierfelder:2010.

The values of various nuclear effects of the 223,225Ra isotopes presented in
Fig. 2C were calculated as follows. The changes in mean-square charge radii
$\delta\braket{r^{2}}$ were taken from udrescu2021isotope. The magnetic dipole
moments $\mu$ and $\mu(\mathbf{r})$ are based on the measurements presented
herein as well as electronic form factor calculations from Ref.
skripnikov2020nuclear. The effect due to the electric quadrupole moment
$Q_{z}$ was estimated using electronic form factor calculations from Ref.
petrov2020energy and the value of the 223Ra nuclear electric quadrupole moment
from Refs. stone2005table; pyykko2008year. The expected contributions from the
anapole $\mathbf{a}$ and Schiff $\mathbf{S}$ moments were estimated using the
electronic form factors calculated in this work. The value of the 225Ra
anapole moment was calculated using the nuclear shell model GFreview;
safronova2018search, while for the Schiff moment, the value from Ref.
flambaum2019enhanced was used:

$\displaystyle S(^{225}{\rm Ra})=1.0\ \bar{\theta}\mathrm{\ e\ fm^{3}},$ (15)

where $\bar{\theta}$ is the $\mathcal{CP}$-violating phase of the QCD
Hamiltonian. The upper limit on the Schiff moment in Fig. 2C, is based on the
limit on $\bar{\theta}$ from Ref. graner2016reduced. Finally, the effect of
the magnetic quadrupole moment, MQM, effect in 223RaF is based on the
calculations presented in Ref. flambaum2022enhanced.

The semi-empirical approximation

$E_{\rm eff}=\alpha\sqrt{AA_{d}},$ (16)

assumes (see Ref. Kozlov:97c for details) that there is a proportionality
relation between $E_{\rm eff}$ and a function of the HFS constants of a heavy
atom-containing diatomic molecules with a ${}^{2}\Sigma_{1/2}$ electronic
state. This expression allowed an estimation of $E_{\rm eff}$ for the YbF
molecule Kozlov:97c using the experimental values of the HFS constants and a
simple model of the electronic wavefunction, eliminating the need for large-
scale calculations. To test this approach for RaF, $E_{\rm eff}$ and the HFS
constants were calculated at the simple Dirac-Hartree-Fock level. A value of
$\alpha=0.0313$ GV/cm MHz using Eq. 16 was obtained. Next, Eq. 16 was used
again to extract the value of $E_{\rm eff}$ substituting the estimated value
of $\alpha$ and the experimental values of the HFS constants determined here.
The obtained value of $E_{\rm eff}$ in this case was found to be overestimated
by about 19% compared to the precise large-scale calculation presented here.
It is however better than the pure Dirac-Hartree-Fock value of $E_{\rm eff}$,
which is underestimated by 29$\%$.

###  V.4 Acknowledgments

###### Acknowledgements.

This work was supported by the Office of Nuclear Physics, U.S. Department of
Energy, under grants DE-SC0021176 and DE-SC0021179 (S.M.U., S.G.W., R.F.G.R.,
A.J.B.); the MISTI Global Seed Funds (S.M.U.); Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) – Projektnummer 328961117 – SFB 1319 ELCH
(A.A.B., R.B., K.G., T.G.); STFC grants ST/P004423/1 and ST/V001116/1 (M.L.B.,
K.T.F., H.A.P., J.R.R., J.W.); Belgian Excellence of Science (EOS) project No.
40007501 (G.N.); KU Leuven C1 project No. C14/22/104 (M.A.K., T.E.C., R.P.dG,
G.N); FWO project No. G081422N (M.A.K., G.N.); International Research
Infrastructures (IRI) project No. I001323N (M.A.K., T.E.C., R.P.G., A.D.,
S.G., L.L., G.N., B.vdB.); the European Unions Grant Agreement 654002
(ENSAR2); LISA: European Union’s H2020 Framework Programme under grant
agreement no. 861198 (M.A., D.H., M.N., J.W.); The Swedish Research Council
(2016-03650 and 2020-03505) (D.H., M.N.). The National Key RD Program of China
(No: 2022YFA1604800) (X.F.Y.) and the National Natural Science Foundation of
China (No:12027809). (X.F.Y.). Electronic structure calculations have been
carried out using computing resources of the federal collective usage center
Complex for Simulation and Data Processing for Mega-science Facilities at
National Research Centre “Kurchatov Institute”, http://ckp.nrcki.ru/, and
partly using the computing resources of the quantum chemistry laboratory.
Molecular electronic structure calculations performed at NRC “Kurchatov
Institute” – PNPI have been supported by Grant No. 19-72-10019. Scalar-
relativistic calculations performed at SPbU were supported by the foundation
for the advancement of theoretical physics and mathematics “BASIS” grant
according to Project No. 21-1-2-47-1.