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An efficient approach to calculate approximate pure-state and transition reduced density matrices in the framework of the multireference relativistic Fock-space coupled cluster (FS CC) theory is proposed. The method is based on the effective operator formalism and consists of the direct substitution of the FS CC Ansatz for a wave operator into the effective operator expression with the subsequent truncation of expansion at the terms quadratic in cluster amplitudes. The final density matrix is defined by active-space density matrices of different ranks "dressed" with contributions from cluster operators. The method gives a connected expression for pure-state density matrices, provided that the intermediate normalization condition is fulfilled. Moreover, under some additional assumptions, the connectivity can also be ensured for calculated transition property matrix elements and natural transition spinors. The developed technique allows for fast and accurate calculations of one-particle reduced density matrices for a wide range of electronic states. A pilot application of the new technique to construct averaged atomic natural orbital (ANO) basis sets for fully relativistic electronic structure calculations is presented.

The development of atomic many-body methods, capable of incorporating electron correlation effects accurately, is required for isotope shift (IS) studies. In combination with precise measurements, such calculations help to extract nuclear charge radii differences, and to probe for signatures of physics beyond the Standard Model of particle physics. We review here a few recently-developed methods in the relativistic many-body perturbation theory (RMBPT) and relativistic coupled-cluster (RCC) theory frameworks for calculations of IS factors in the highly charged ions (HCIs), and neutral or singly-charged ions, respectively. The results are presented for a wide range of atomic systems in order to demonstrate the interplay between quantum electrodynamics (QED) and electron correlation effects. In view of this, we start our discussions with the RMBPT calculations for a few HCIs by rigorously treating QED effects; then we outline methods to calculate IS factors in the one-valence atomic systems using two formulations of the RCC approach. Then we present calculations for two valence atomic systems, by employing the Fock-space RCC methods. For completeness, we briefly discuss theoretical input required for the upcoming experiments, their possibilities to probe nuclear properties and implications to fundamental physics studies., Comment: Topical Review Preprint for J. Phys. B, Comments and Suggestions are welcome!

Observations of the rotational spectrum of lead monofluoride, PbF, have been extended up to transitions in the \textit{v} = 7 level for $^{208}$PbF in the lowest $X_1\,^2\Pi_{1/2}$ state of the radical and \textit{v} = 5 for the $^{207}$Pb and $^{206}$Pb isotopologues. The data also include a few measurements for $^{204}$PbF in \textit{v} = 0. These new measurements have been combined with existing near-IR measurements of the $X_2 - X_1$ fine-structure transition and a simultaneous multi-isotope fit of the data to an effective isotope-independent ro-vibronic Hamiltonian has been carried out. The resulting parameters fully characterize the vibrational, rotational and hyperfine structure of the combined $X_1 \, / \, X_2$ state of the radical. A pair of opposite parity levels with total angular momentum quantum number, $F=1/2$, in the lowest rotational level, $J=1/2$ of \PbF \,are close in energy and their spacing decreases with vibrational excitation. The experimental results show the spacing decreases to less than 20 MHz at $v=7$ and 8. The experimental work is complemented by new \textit{ab initio} calculations which support the results and allow predictions outside the experimental data range. The calculated radiative lifetimes of the relevant vibrationally excited states are of the order of 50 ms. This work was motivated by interest in using \PbF\, as a vehicle for future probes of the standard model of physics such as placing limits on the electron's electric dipole moment (\eEDM), molecular charge-parity non-conservation and Born-Oppenheimer breakdown effects for example., Comment: 36 pages, 4 figures, 5 page Appendix. Accepted by Phys FRev A, 2024-09-04

The effects induced by the time-reversal ($\mathcal{T}$) and spatial parity ($\mathcal{P}$) violating electron-nucleus and electron-electron interactions mediated by the axion-like particles (ALPs) in the BaF molecule were estimated. Molecular parameters characterizing these interactions were calculated across a wide range of ALP masses. In case of the electron-nuclear interaction, the effect of the extended nucleus was studied and shown to be significant for heavy ALPs. Based on the calculated molecular parameters, we obtained a link between the $\mathcal{T}$,$\mathcal{P}$-violating energy shift which can be measured in the ongoing experiment designed to search for the electron electric dipole moment using the BaF molecule [A. Boeschoten et al., arXiv:2303.06402 (2023)] and the products of coupling constants of the ALP-electron or ALP-electron and ALP-nucleus interactions.

The radiative lifetime of the $A$ $^2 \Pi_{1/2}$ (v=0) state in radium monofluoride (RaF) is measured to be 35(1) ns. The lifetime of this state and the related decay rate $\Gamma = 2.86(8) \times 10^7$ $s^{-1}$ are of relevance to the laser cooling of RaF via the optically closed $A$ $^2 \Pi_{1/2} \leftarrow X$ $^2\Sigma_{1/2}$ transition, which makes the molecule a promising probe to search for new physics. RaF is found to have a comparable photon-scattering rate to homoelectronic laser-coolable molecules. Thanks to its highly diagonal Franck-Condon matrix, it is expected to scatter an order of magnitude more photons than other molecules when using just 3 cooling lasers, before it decays to a dark state. The lifetime measurement in RaF is benchmarked by measuring the lifetime of the $8P_{3/2}$ state in Fr to be 83(3) ns, in agreement with literature., Comment: Accepted as a Letter in Physical Review A; 8 pages of main text, 5 pages of supplemental material

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, $^{225}$Ra$^{19}$F. 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.

We investigate the process of photon absorption by atoms or molecules shortly interacting with a laser beam in the dipole approximation. Assuming that the interaction time $\tau$ is much smaller than the lifetime of the corresponding excited state, we examine the absorption probability as a function of $\tau$. Besides, we incorporate Doppler broadening due to nonzero temperature of the atoms (molecules). It is demonstrated that in the case of a zero detuning and without Doppler broadening, the absorption probability is quadratic in $\tau$. Once Doppler broadening is taken into account or the laser beam is off from the resonant frequency, the absorption probability becomes linear in $\tau$. Our findings are expected to be important for experimental studies in optical cells or cavities where atoms or molecules traverse continuous laser beams. The experimental prospects of searching for the electric dipole moment (EDM) of the electron are discussed in detail.

The nuclear charge radius of $^{32}$Si was determined using collinear laser spectroscopy. The experimental result was confronted with ab initio nuclear lattice effective field theory, valence-space in-medium similarity renormalization group, and mean field calculations, highlighting important achievements and challenges of modern many-body methods. The charge radius of $^{32}$Si completes the radii of the mirror pair $^{32}$Ar - $^{32}$Si, whose difference was correlated to the slope $L$ of the symmetry energy in the nuclear equation of state. Our result suggests $L \leq 60$\,MeV, which agrees with complementary observables.

The calculation of the parity-violating polarizations in the external electric field, which are associated with the electron electric dipole moment ($e$EDM) and magnetic quadrupole moment (MQM) of the $^{175}$Lu nucleus, as well as the determination of the rovibrational structure for the $^{175}$LuOH$^+$ cation, is performed. Beyond the bending of the molecule, the slight effect of the stretching of the distance between Lu and OH is taken into account. This study is required for the preparation of the experiment and for the extraction of the $e$EDM and MQM values of $^{175}$Lu from future measurements., Comment: arXiv admin note: text overlap with arXiv:2211.02112

We report the spectroscopy of the 14 lowest excited electronic states in the radioactive molecule radium monofluoride (RaF). The observed excitation energies are compared with fully relativistic state-of-the-art Fock-space coupled cluster (FS-RCC) calculations, which achieve an agreement of >=99.64% (within ~12 meV) with experiment for all states. Guided by theory, a firm assignment of the angular momentum and term symbol is made for 10 states and a tentative assignment for 4 states. The role of high-order electron correlation and quantum electrodynamics effects in the excitation energy of excited states is studied, found to be important for all states. Establishing the simultaneous accuracy and precision of calculations is an important step for research at the intersection of particle, nuclear, and chemical physics, including searches of physics beyond the Standard Model, for which RaF is a promising probe., Comment: Significant changes compared to previous version with additional experimental data. Accepted for publication under a different title

Recently, upper bounds on the static time-reversal ($\mathcal{T}$) and spatial parity ($\mathcal{P}$)-violating electron electric dipole moment ($e$EDM) and dimensionless constant characterizing the strength of the $\mathcal{T}$,$\mathcal{P}$-violating scalar-pseudoscalar nucleus-electron interaction have been updated in the JILA experiment using the HfF$^+$ cations [T.S. Roussy et al, arxiv:2212.11841 (2022)]. We considered other two sources of the $\mathcal{T}$,$\mathcal{P}$,-violation in HfF$^+$ -- axionlike-particle-mediated (ALP) scalar-pseudoscalar electron-electron and nucleus-electron interactions. To estimate the magnitude of effects, induced by such interactions in HfF$^+$ we have developed and applied a method which implies direct use of the ab-initio relativistic coupled cluster theory to calculate molecular parameters that characterize the interactions. Using these parameters, we showed that an order of magnitude updated laboratory constraints on the ALP-mediated electron-electron and nucleus-electron interactions can be derived from the experimental data on $\mathcal{T}$,$\mathcal{P}$-violating effects in HfF$^+$ for a wide range of ALP masses.

The current constraint on the electron electric dipole moment ($e$EDM), $|d_e|<4.1\times 10^{-30}$ ${e {\cdotp} {\rm cm}}$ (90\% confidence), was recently established using the trapped $^{180}$Hf$^{19}$F$^+$ molecular ions in the $J=1$ rotational level of its $ ^3\Delta_1$ electronic state [T. S. Roussy, L. Caldwell, T. Wright, et al., arxiv:2212.11841]. The extensive experimental study of the HfF$^+$ cation provides detailed spectroscopy of the $\Omega-$doublet levels in the external rotating electric and magnetic fields. We showed that previously developed theoretical approaches can fully reproduce the latest experimental data. Their justification from the first principles is very important for the examination of both modern molecular theory and possible systematic uncertainties in the interpretation of the experimental data obtained with high accuracy.

Molecules containing short-lived, radioactive nuclei are uniquely positioned to enable a wide range of scientific discoveries in the areas of fundamental symmetries, astrophysics, nuclear structure, and chemistry. Recent advances in the ability to create, cool, and control complex molecules down to the quantum level, along with recent and upcoming advances in radioactive species production at several facilities around the world, create a compelling opportunity to coordinate and combine these efforts to bring precision measurement and control to molecules containing extreme nuclei. In this manuscript, we review the scientific case for studying radioactive molecules, discuss recent atomic, molecular, nuclear, astrophysical, and chemical advances which provide the foundation for their study, describe the facilities where these species are and will be produced, and provide an outlook for the future of this nascent field.

Electronic factors for the field and mass isotope shifts in the $6p\ ^{2}P_{3/2} \to 7s\ ^{2}S_{1/2}$ (535 nm), $6p\ ^{2}P_{1/2} \to 6d\ ^{2}D_{3/2}$ (277 nm) and $6p\ ^{2}P_{1/2} \to 7s\ ^{2}S_{1/2}$ (378~nm) transitions in the neutral thallium were calculated within the high-order relativistic coupled cluster approach. These factors were used to reinterpret previous experimental isotope shift measurements in terms of charge radii of a wide range of Tl isotopes. Good agreement between theoretical and experimental King-plot parameters was found for the $6p\ ^{2}P_{3/2} \to 7s\ ^{2}S_{1/2}$ and $6p\ ^{2}P_{1/2} \to 6d\ ^{2}D_{3/2}$ transitions. It was shown that the value of the specific mass shift factor for the $6p\ ^{2}P_{3/2} \to 7s\ ^{2}S_{1/2}$ transition is not negligible compared to the value of normal mass shift in contrary to what had been suggested previously. Theoretical uncertainties in mean square charge radii were estimated. They were substantially reduced compared to the previously ascribed ones and amounted to less than 2.6%. The achieved accuracy paves the way to a more reliable comparison of the charge radii trends in the lead region.

The LuOH$^+$ cation is a promising system to search for manifestations of time reversal and spatial parity violation effects. Such effects in LuOH$^+$ induced by the electron electric dipole moment $e$EDM and the scalar-pseudoscalar interaction of the nucleus with electrons, characterized by $k_s$ constant, in LuOH$^+$ are studied. The enhancement factors, polarization in the external electric field, hyperfine interaction, rovibrational structure are calculated. The study is required for the experiment preparation and extraction of the eEDM and ks values from experimental data.

An experiment utilizing a TlF molecular beam is being developed by the CeNTREX collaboration to search for hadronic interactions that violate both time-reversal (T) and parity (P) invariance. Here we propose to use the same beam to look for a T-invariance conserving but P-nonconserving (PNC) effect induced by the anapole moment of the Tl nucleus, via a vector coupling of the two nuclear spins in TlF. To measure the nuclear anapole moment, the dc electric and magnetic fields in CeNTREX are replaced by rf fields resonant with a nuclear spin flip transition. We adapt the relativistic coupled cluster method in a combination with relativistic density functional theory for the calculation of the molecular PNC spin-spin vector coupling constant that links the experimental signal with the anapole moment. The value of the P-conserving spin-spin coupling constant calculated within the same approach is found to be in good agreement with available experimental data.

Static electric dipole moments of nondegenerate systems probe mass scales for physics beyond the Standard Model well beyond those reached directly at high energy colliders. Discrimination between different physics models, however, requires complementary searches in atomic-molecular-and-optical, nuclear and particle physics. In this report, we discuss the current status and prospects in the near future for a compelling suite of such experiments, along with developments needed in the encompassing theoretical framework., Comment: Contribution to Snowmass 2021; updated with community edits and endorsements

We estimate the effect of the tensor parity nonconserving (PNC) interaction in the $^{181}$TaO$^+$ molecular cation. It can be used to probe the unknown quadrupole distribution of the neutrons inside the Ta nucleus. To this end, we evaluate the constant which characterizes this interaction using the relativistic Fock space coupled cluster theory for electronic structure modelling. The $^3\Delta_1$ state of the TaO$^+$ cation which can be used to measure the PNC effect is found to be the ground one in agreement with the previous theoretical study.

The successive ionization potentials (IPs) and electron affinities (EAs) for superheavy elements with $111 \leq Z \leq 114$, namely, Rg, Cn, Nh, and Fl are reexamined using the relativistic Fock-space coupled-cluster method with nonperturbative single (S), double (D), and triple (T) cluster amplitudes (FS-CCSDT). For the most of considered quantities, the triple-amplitude contributions turn out to be important. The Breit and frequency-dependent Breit corrections are evaluated by means of the configuration-interaction method. The quantum-electrodynamics corrections to the IPs and EAs are taken into account within the model-QED-operator approach. The obtained results are within 0.10 eV uncertainty., Comment: 10 pages, 6 tables

The francium atom is considered as a prospective candidate system to search for the T,P-violating electron electric dipole moment [T. Aoki et al Quantum Sci. Technol. 6, 044008 (2021)]. We demonstrate that the same experiment can be used for axionlike particles (ALP) search. For this, we calculate electronic structure constants of ALP-mediated interaction for a wide range of ALP masses. Using the recently updated constraints on the ALP-electron and ALP-nucleon coupling constants, we show that the contribution of considered interactions corresponding to these constraints can give significant contribution to atomic electric dipole moment. Therefore, obtainment of stronger restrictions for ALP characteristics in the francium atom electric dipole moment experiment is possible.

Laser-induced fluorescence spectra of the $c^3\Sigma^+(v_{c},J_{c}=N_{c})\rightarrow a^3\Sigma^+(v_{a},N_{a} = J_{c} \pm 1)$ transitions excited from the ground $X^1\Sigma^+$ state of $^{39}$K$^{133}$Cs molecule were recorded with Fourier-transform spectrometer IFS125-HR (Bruker) at the highest achievable spectral resolution of 0.0063 cm${}^{-1}$. Systematic study of the hyperfine structure (HFS) of the $a^3\Sigma^+$ state for levels with $v_{a} \in [0, 27]$ and $N_{a} \in [24, 90]$ shows that the splitting monotonically increases with $v_{a}$. The spectroscopic study was supported by ab initio calculations of the magnetic hyperfine interaction in $X^1\Sigma^+$ and $a^3\Sigma^+$ states. The discovered variation of the electronic matrix elements with the internuclear distance $R$ is in a good agreement with the observed $v_{a}$-dependencies of the HFS. Overall set of available experimental data on the $a^3\Sigma^+$ state was used to improve the potential energy curve particularly near a bottom, providing the refined dissociation energy $D_e$=267.21(1) cm${}^{-1}$. The ab initio HFS matrix elements, combined with the empirical $X^1\Sigma^+$ and $a^3\Sigma^+$ PECs in the framework of the invented coupled-channel deperturbation model, reproduce the experimental term values of both ground states within 0.003 cm${}^{-1}$ accuracy up to their common dissociation limit.

Relativistic effective atomic configurations of superheavy elements Cn, Nh and Fl and their lighter homologues (Hg, Tl and Pb) in their simple compounds with fluorine and oxygen are determined using the analysis of local properties of molecular Kohn-Sham density matrices in the vicinity of heavy nuclei. The difference in populations of atomic spinors with the same orbital angular momentum and different total angular momenta is demonstrated to be essential for understanding the peculiarities of chemical bonding in superheavy element compounds. The results are fully compatible with those obtained by the relativistic iterative version of conventional projection analysis of global density matrices.

The AcOH${}^+$ molecular ion is identified as a prospective system to search for $\mathcal{CP}$-violation effects. According to our study AcOH${}^+$ belongs to the class of laser-coolable polyatomic molecular cations implying the large coherence time in the experiments to study symmetry violating effects of fundamental interactions. We perform both nuclear and high level relativistic coupled cluster electronic structure calculations to express experimentally measurable $\mathcal{T}$,$\mathcal{P}$-violating energy shift in terms of fundamental quantities such as the nuclear magnetic quadrupole moment (MQM), electron electric dipole moment ($e$EDM) and dimensionless scalar-pseudoscalar nuclear-electron interaction constant. We further express nuclear MQM in terms of the strength constants of $\mathcal{CP}$-violating nuclear forces: quantum chromodynamics vacuum angle $\bar{\theta}$ and quark chromo-EDMs. The equilibrium geometry of AcOH${}^+$ in the ground and the four lowest excited electronic states was found to be linear. The calculated Franck-Condon factors and transition dipole moments indicate that the laser cooling using optical cycle involving the first excited state is possible for the trapped AcOH${}^+$ ions with the Doppler limit estimated to be~$\sim 4$~nK. The lifetime of the (0,1$^1$,0) excited vibrational state considered as a working one for MQM and $e$EDM search experiments is estimated to be $\sim 0.4$ sec., Comment: 3 figures

Recently a breakthrough has been achieved in laser-spectroscopic studies of short-lived radioactive compounds with the first measurements of the radium monofluoride molecule (RaF) UV/vis spectra. We report results from high accuracy \emph{ab initio} calculations of the RaF electronic structure for ground and low-lying excited electronic states. Two different methods agree excellently with experimental excitation energies from the electronic ground state to the $^2\Pi_{1/2}$ and $^2\Pi_{3/2}$ states, but lead consistently and unambiguously to deviations from experimental-based adiabatic transition energy estimates for the $^2\Sigma_{1/2}$ excited electronic state and show that more measurements are needed to clarify spectroscopic assignment of the $^2\Delta$ states., Comment: 7 pages, 1 figure, 1 table

The electron affinity (EA) of superheavy element Og is calculated by the use of the relativistic Fock-space coupled cluster (FSCC) and configuration interaction methods. The FSCC cluster operator expansion included single, double, and triple excitations treated in a non-perturbative manner. The Gaunt and retardation electron-electron interactions are taken into account. Both methods yield the results that are in agreement with each other. The quantum electrodynamics correction to EA is evaluated using the model Lamb-shift operator approach. The electron affinity of Og is obtained to be 0.076(4) eV., Comment: 11 pages, 5 tables, 3 figures

The YbOH triatomic molecule can be efficiently used to measure the electron electric dipole moment, which violates time-reversal (T) and spatial parity (P) symmetries of fundamental interactions [I. Kozyryev, N.R. Hutzler, Phys. Rev. Lett. 119, 133002 (2017)]. We study another mechanism of the T,P-violation in the YbOH molecule - the electron-electron interaction mediated by the low-mass axiolike particle. For this, we calculate the molecular constant that characterizes this interaction and use it to estimate the expected magnitude of the effect to be measured. It is shown that this molecular constant has the same order of magnitude as the corresponding molecular constant corresponding to the axion-mediated electron-nucleus interaction. According to our estimation, an experiment on YbOH will allow one to set updated laboratory constraints on the CP-violating electron-axion coupling constants.

The present constraint on the space parity ($\mathcal{P}$) and time reflection invariance ($\mathcal{T}$) violating electron electric dipole moment ($e$EDM) is based on the observation of the electron spin precession in an external electric field using the ThO molecule. We propose an alternative approach: observation of the $\mathcal{P}$,~$\mathcal{T}$-odd Faraday effect in an external electric field using the cavity-enhanced polarimetric scheme in combination with a molecular beam crossing the cavity. Our theoretical simulation of the proposed experiment with the PbF and ThO molecular beams shows that the present constraint on the $e$EDM in principle can be improved by a few orders of magnitude., Comment: arXiv admin note: text overlap with arXiv:1907.11761

Three models of the nuclear magnetization distribution are applied to predict the hyperfine structure of the hydrogenlike heavy ions and neutral thallium atoms: the uniformly magnetized ball model and single-particle models for the valence nucleon with the uniform distribution and the distribution determined by the Woods-Saxon potential. Results for the hydrogenlike ions are in excellent agreement with previous studies. The application of the Woods-Saxon model is now extended to the neutral systems with the explicit treatment of the electron correlation effects within the relativistic coupled cluster theory using the Dirac-Coulomb Hamiltonian. We estimate the uncertainty for the ratio of magnetic anomalies and numerically confirm its near nuclear-model independence. The ratio is used as a theoretical input to predict the nuclear magnetic moments of short-lived thallium isotopes. We also show that the differential magnetic anomalies are strongly model dependent. The accuracy of the single-particle models significantly surpasses the accuracy of the simplest uniformly magnetized ball model for the prediction of this quantity. Skripnikov [Skripnikov, J. Chem. Phys. 153, 114114 (2020)] has shown that the Bohr-Weisskopf contribution to the magnetic dipole hyperfine structure constant for an atom or a molecule induced by a heavy nucleus can be factorized into the electronic part and the universal nuclear magnetization dependent part. We numerically confirm this factorization for the Woods-Saxon single-particle model with an uncertainty less than 1%.

Recently the YbOH molecule has been suggested as a candidate to search for the electron electric dipole moment (eEDM) which violates spatial parity (P) and time-reversal (T) symmetries [I. Kozyryev and N. R. Hutzler, Phys. Rev. Lett. 119, 133002 (2017)]. In the present paper we show that the same system can be used to measure coupling constants of the interaction of electrons and nucleus with axionlike particles. The electron-nucleus interaction produced by the axion exchange induces T,P-violating a EDM of the whole molecular system. We express the corresponding T,P-violating energy shift produced by this effect in terms of the axion mass and product of the axion-nucleus and axion-electron coupling constants.

The four-component relativistic Fock space coupled cluster method is used to describe the magnetic hyperfine interaction in low-lying electronic states of the KCs molecule. Both diagonal and off-diagonal matrix elements as functions of the internuclear separation $R$ are calculated within the finite-field scheme. The resulting matrix elements exhibit very weak dependence on $R$ for the separations exceeding 8 \AA , whereas in the vicinity of the ground-state equilibrium the deviation of molecular HFS matrix elements from the atomic values reaches 15\%. The dependence of the computed HFS couplings on the level of core correlation treatment is discussed., Comment: 3 figures

The CP-violating interaction of the nuclear magnetic quadrupole moment (MQM) of the $^{175}$Lu nucleus with electrons in the molecular cation LuOH$^+$ is studied. The resulting effect is expressed in terms of CP-odd parameters, such as quantum chromodynamics angle $\bar{\theta}$, quark electric dipole moment (EDM) and chromo-EDM. For this we have performed a calculation of the nuclear MQM as well as the molecular constant that characterises the interaction of this MQM of $^{175}$Lu with electrons. Additionally, we predict the hyperfine structure constants for the ground electronic state of LuOH$^+$. We conclude that LuOH$^+$ is a promising system to measure the nuclear MQM.

Compound-tunable embedding potential (CTEP) method developed in previous works to describe electronic structure of fragments in materials is applied to crystals containing periodically arranged lanthanide atoms, which can have open $4f$ shell. We consider YbF$_2$ and YbF$_3$ as examples such that $4f$ shell is excluded from both the crystal and cluster stages of generating the CTEP. Instead, 10 and 11 valence-electron pseudopotentials for Yb, correspondingly, are applied and the latter treats the $4f$-hole implicitly. At the next stage of the two-component embedded cluster studies of the YbF$_{2,3}$ crystals we apply the 42 valence-electron relativistic pseudopotential for Yb and, thus, $4f$ shell is treated explicitly. A remarkable agreement of the electronic density and interatomic distances within the fragment with those of the original periodic crystal calculation is attained., Comment: 5 pages, 2 figures

Relativistic study of xenotime, YPO$_4$, containing atoms thorium and uranium as point defects is performed in the framework of cluster model with using the compound-tunable embedding potential (CTEP) method proposed by us recently. The Y-(PO$_4$)$_6$-Y'$_{22}$-O'$_{104}$ cluster for xenotime is considered, in which central part, [Y-(PO$_4$)$_6$]$^{-15}$, is the main cluster, whereas outermost 22 atoms of yttrium and 104 atoms of oxygen are treated as its environment and compose electron-free CTEP with the total charge of $+15$. The P and O atoms of the orthophosphate groups nearest to the central Y atom are treated at all-electron level. The central Y, its substitutes, Th and U, together with environmental Y atoms are described within different versions of the generalized relativistic pseudopotential method. Correctness of our cluster and CTEP models, constructed in the paper, is justified by comparing the Y-O and P-O bond lengths with corresponding periodic structure values of the \ypo4 crystal, both experimental and theoretical. Using this cluster model, chemical properties of solitary point defects, X = U, Th, in xenotime are analyzed. It has been shown that the oxidation state ${+3}$ is energetically more profitable than ${+4}$ not only for thorium but for uranium as well ($\Delta E \approx 5$ eV) despite the notably higher ionic radius of U$^{+3}$ compared to Y$^{+3}$, whereas ionic radii of U$^{+4}$ and Y$^{+3}$ are close. This leads to notable local deformation of crystal geometry around the U$^{+3}$ impurity in xenotime and contradicts to widespread opinion about favorite oxidation state of uranium in such kind of minerals., Comment: 8 pages, 4 figures, 3 tables. Advanced version of this paper is available at https://doi.org/10.1039/D0CP02277B

We present state-of-the-art configuration interaction and coupled cluster calculations of the electron electric dipole moment, the nucleon-electron scalar-pseudoscalar, and the magnetic hyperfine interaction constants ($\alpha_{d_e}, \alpha_{C_S}, A_{||}$, respectively) for the thallium atomic ground state $^2P_{1/2}$. Our present best values are $\alpha_{d_e} = -559 \pm 28$, $\alpha_{C_S} = 6.77 \pm 0.34$ $[10^{-18} e$ cm$]$, and $A_{||} = 21172 \pm 1059$ [MHz]. These findings lead to a significant reduction of the theoretical uncertainties for ${\cal{P,T}}$-odd interaction constants but not to stronger constraints on the electron electric dipole moment, $d_e$, or the nucleon-electron scalar-pseudoscalar coupling constant, $C_S$., Comment: 13 pages, 2 figures

Present limit on the electron electric dipole moment ($e$EDM) is based on the electron spin precession measurement. We propose an alternative approach - observation of the $\mathcal{P}$,$\mathcal{T}$-odd Faraday effect in an external electric field on atoms and molecules using cavity-enhanced polarimetric scheme in combination with molecular (atomic) beam crossing the cavity. Our calculations of the effective electric fields and theoretical simulation of the proposed experiment on Tl and Pb atoms, PbF, YbF, ThO, and YbOH show that the present limit on the $e$EDM can be improved by 6-7 orders of magnitude.