Diatomic molecules with an energetically low-lying 3A1 state are attractive platforms to detect new physics beyond the Standard Model, such as parity-and time-reversal-violating phenomena. One of the advantages of using a 3A1 state is its tiny A splitting due to the coupling between the electronic and rotational angular momenta, which facilitates polarizing the molecules in small external electric fields. Theoretical estimation of the magnitude of the A splitting is helpful for planning new experiments. In this study, we present a theoretical model to calculate the A splitting. Our model integrates the relativistic four-component wave function and the traditional rotational Hamiltonian based on Hund's case (a). The multireference character of the wave function is taken into account. Our calculations for PtH and ThF+ molecules qualitatively agree with experiment. The A splitting of TaO+ for the rotational ground state is predicted to be around 9 kHz. This tiny splitting can reduce the systematic uncertainty, but in a practical experiment, it may cause depolarization during rotation ramp-up.

We present a systematic study of the nucleon-electron tensor-pseudotensor (Ne-TPT) interaction in the francium-silver (FrAg), cesium-silver (CsAg), and francium-lithium (FrLi) molecules which are candidates for next-generation experimental searches for new sources of charge-parity violation. The considered molecules are all amenable to assembly from laser-cooled atoms, with the FrAg molecule previously shown to be the most sensitive to the Schiff moment interaction in this set. Interelectron correlation effects are treated through relativistic general-excitation-rank configuration interaction theory in the framework of the Dirac–Coulomb Hamiltonian. We find in FrAg the Ne-TPT interaction constant to be W_T(Fr) = 2.57 ± 0.21 [<Σ>_A kHz] , considering the Francium atom as target of the measurement. Taking into account nuclear structure in a multi-source interpretation of a measured electric dipole moment, FrAg is found to be an excellent probe of physics beyond the standard model as this system will in addition to its sizeable Ne-TPT interaction constant greatly constrain fundamental parameters such as the semileptonic four-fermion interaction C_lequ from which nuclear and atomic 𝒞𝒫 -violating properties arise. Nucleon–electron tensorpseudotensor interaction on the Francium target nucleus in the Francium-Silver molecule, where the corresponding Hamiltonian operator is parity- and time-reversal violating. Details of the interaction are explained in the paper body. These interactions give rise to molecular charge-parity violating effects that can be detected by experiments.

An electronically variational approach to the calculation of atomic hyperfine structure transition energies under the influence of static external electric fields is presented. The method avoids the calculation of intermediate atomic states entirely and requires only the wave functions of the electronic states involved in the respective hyperfine levels. These wave functions are obtained through relativistic general-excitation-rank configuration-interaction theory. A variant of the method also enables calculations on atoms with the most complicated of shell structures. Applications to 87Rb, 133Cs, and a specific clock transition in 169Tm are presented. The final results kRb = -1.234 +/- 0.023 [10-10 Hz/(V/m)2] and kCs = -2.347 +/- 0.084 [10-10 Hz/(V/m)2] obtained under inclusion of up to quintuple excitations in the atomic wave-function expansion are compatible with previous calculations and, in the case of Cs, confirm that one of the earlier experimental measurements is not reliable. For 169Tm that is used in the development of atomic clocks the differential static scalar electric dipole polarizability between ground levels J = 72 and 52 is calculated to be A alpha s0 = -0.134 +/- 0.122 a.u. This result from a pure ab initio calculation supports the result of A alpha s0 = -0.063+0.01-0.005 a.u. obtained recently [A. Golovizin, E. Fedorova, D. Tregubov, D. Sukachev, K. Khabarova, V. Sorokin, and N. Kolachevsky, Nat. Commun. 10, 1724 (2019)] where a combination of measurement and theoretical modeling has been used.

Leptoquark models may explain deviations from the Standard Model observed in decay processes involving heavy quarks at high-energy colliders. Such models give rise to low-energy parity- and time-reversal-violating phenomena in atoms and molecules. One of the leading effects among these phenomena is the nucleon-electron tensor-pseudotensor interaction when the low-energy experimental probe uses a quantum state of an atom or molecule predominantly characterized by closed electron shells. In the present paper the molecular interaction constant for the nucleon-electron tensor-pseudotensor interaction in the thallium-fluoride molecule -- used as such a sensitive probe by the CeNTREX collaboration [Quantum Sci. Technol., 6:044007, 2021] -- is calculated employing highly-correlated relativistic many-body theory. Accounting for up to quintuple excitations in the wavefunction expansion the final result is $W_T({\text{Tl)}} = -6.25 \pm 0.31\, $[$10^{-13} {\langle\Sigma\rangle}_A$ a.u.] Interelectron correlation effects on the tensor-pseudotensor interaction are studied for the first time in a molecule, and a common framework for the calculation of such effects in atoms and molecules is presented.

A method for highly accurate calculations of atomic electric quadrupole moments (EQM) is presented, using relativistic general-excitation-rank configuration interaction wave functions based on Dirac spinors. Application to the clock transition states of the thulium atom employing up to full Quadruple excitations for the atomic wave function yields a final value of Q(zz) (F-2(7/2)) = 0.07(-0.00)(+0.07) a.u., establishing that the thulium electronic ground state has an exceptionally small EQM. A detailed analysis of this result is presented which has implications for EQMs of other atoms with unpaired f electrons.

We systematically study a set of strongly polar heteronuclear diatomic molecules composed of laser-coolable atoms for their suitability as sensitive probes of new charge-parity violation in the hadron sector of matter. Using relativistic general-excitation-rank configuration interaction theory we single out the molecule francium-silver (FrAg) as the most promising system in this set and calculate its nuclear Schiff-moment interaction constant to $W^\mathrm{FrAg}_{SM}(\mathrm{Fr}) = 30168 \pm 2504\mathrm{a.u.}$ for the target nucleus Fr. Our work includes the development of system-tailored atomic Gaussian basis sets for the target atom in each respective molecule.

A method for highly accurate calculations of atomic electric quadrupole moments (EQM) is presented, using relativistic general-excitation-rank configuration interaction wavefunctions based on Dirac spinors. Application to the clock transition states of the thulium atom employing up to full Quadruple excitations for the atomic wavefunction yields a final value of $Q_{zz}({^2F}_{7/2}) = 0.07 \pm 0.07$ a.u., establishing that the thulium electronic ground state has an exceptionally small EQM. A detailed analysis of this result is presented which has implications for EQMs of other atoms with unpaired $f$ electrons.

The TaO cation is an attractive molecular species to search for parityand time-reversal-violating interactions, in particular of hadronic origin. For the spectroscopic detection and preparation of TaO cation in a desired state detailed knowledge of spectroscopic and electric properties in excited states is essential information. In this work we present spectroscopy constants for TaO in the electronic ground and 29 excited states calculated with relativistic configuration interaction theory. The equilibrium bond lengths (Re), harmonic vibrational frequencies (ωe), transition dipole moments (TDM), vertical excitation energies and static molecular dipole moments (PDM) are summarized. We include a detailed characterization of all electronic states in terms of their spinor occupations. This work supports the realization of experiments using TaO ions to search for new physics beyond the standard model of elementary particles.

We present relativistic many-body calculations of atomic and molecular Schiff-moment interaction constants including interelectron correlation effects using atomic Gaussian basis sets specifically optimized for the Schiff interaction. Our present best results employing a Gaussian nuclear density function are ${\ensuremath{\alpha}}_{\text{SM}}=(0.362\ifmmode\pm\else\textpm\fi{}0.025)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}17}\frac{e\phantom{\rule{0.16em}{0ex}}\text{cm}}{e\phantom{\rule{0.16em}{0ex}}{\text{fm}}^{3}}$ for atomic $^{129}\mathrm{Xe}, {\ensuremath{\alpha}}_{\text{SM}}=(\ensuremath{-}2.26\ifmmode\pm\else\textpm\fi{}0.23)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}17}\frac{e\phantom{\rule{0.16em}{0ex}}\text{cm}}{e\phantom{\rule{0.16em}{0ex}}{\text{fm}}^{3}}$ for atomic $^{199}\mathrm{Hg}$, and ${W}_{\text{SM}}=(39\phantom{\rule{0.16em}{0ex}}967\ifmmode\pm\else\textpm\fi{}3600)$ a.u. for the thallium nucleus in the molecule $^{205}\mathrm{TlF}$. We discuss agreements and discrepancies between our present results and those from earlier calculations on the atoms $^{129}\mathrm{Xe}$ and $^{199}\mathrm{Hg}$. Using the most recent measurements of $\mathcal{P},\mathcal{T}$-odd electric dipole moments and the present interaction constants, reliable upper bounds on the Schiff moments of the $^{199}\mathrm{Hg}$ and $^{205}\mathrm{Tl}$ nuclei are determined in the context of a single-source assumption.

We present relativistic many-body calculations of atomic and molecular Schiff-moment interaction constants including interelectron correlation effects using atomic Gaussian basis sets specifically optimized for the Schiff interaction. Our present best results employing a Gaussian nuclear density function are alpha SM=(0.362 +/- 0.025)x10(-17)ecm/efm(3) for atomic Xe-129, alpha SM=(-2.26 +/- 0.23)x10(-17)ecm/efm(3) for atomic Hg-199, and WSM=(39967 +/- 3600) a.u. for the thallium nucleus in the molecule (TlF)-Tl-205. We discuss agreements and discrepancies between our present results and those from earlier calculations on the atoms Xe-129 and Hg-199. Using the most recent measurements of P,T-odd electric dipole moments and the present interaction constants, reliable upper bounds on the Schiff moments of the Hg-199 and Tl-205 nuclei are determined in the context of a single-source assumption.

We explore the possibilities for a next-generation electron-electric-dipole-moment experiment using ultracold heteronuclear diatomic molecules assembled from a combination of radium and another laser-coolable atom. In particular, we calculate their ground state structure and their sensitivity to parity- and time-reversal (P, T) violating physics arising from flavor-diagonal charge-parity (CP) violation. Among these species, the largest P, T-violating molecular interaction constants-associated for example with the electron electric dipole moment-are obtained for the combination of radium (Ra) and silver (Ag) atoms. A mechanism for explaining this finding is proposed. We go on to discuss the prospects for an electron EDM search using ultracold, assembled, optically trapped RaAg molecules, and argue that this system is particularly promising for rapid future progress in the search for new sources of CP violation.

We calculate interaction constants for the contributions from $\mathcal{P},\mathcal{T}$-odd scalar-pseudoscalar and tensor-pseudotensor operators to the electric dipole moment of $^{129}\mathrm{Xe}$ using relativistic many-body theory including the effects of dynamical electron correlations. These interaction constants are necessary ingredients to relating the corresponding measurements to fundamental parameters in models of physics beyond the standard model. We obtain ${\ensuremath{\alpha}}_{{C}_{S}}=\left(0.71\ifmmode\pm\else\textpm\fi{}0.18\right)({10}^{\ensuremath{-}23}\phantom{\rule{0.16em}{0ex}}e\phantom{\rule{3.33333pt}{0ex}}\text{cm})$ and ${\ensuremath{\alpha}}_{{C}_{T}}=\left(0.520\ifmmode\pm\else\textpm\fi{}0.049\right)({10}^{\ensuremath{-}20}\phantom{\rule{0.16em}{0ex}}{\ensuremath{\langle}\mathrm{\ensuremath{\Sigma}}\ensuremath{\rangle}}_{\text{Xe}}\phantom{\rule{0.16em}{0ex}}e\phantom{\rule{3.33333pt}{0ex}}\text{cm})$, respectively. We apply our results to test a phenomenological relation between the two quantities, commonly used in the literature, and discuss their present and future phenomenological impact.

DIRAC is a freely distributed general-purpose program system for one-, two-, and four-component relativistic molecular calculations at the level of Hartree-Fock, Kohn-Sham (including range-separated theory), multiconfigurational self-consistent-field, multireference configuration interaction, electron propagator, and various flavors of coupled cluster theory. At the self-consistent-field level, a highly original scheme, based on quaternion algebra, is implemented for the treatment of both spatial and time reversal symmetry. DIRAC features a very general module for the calculation of molecular properties that to a large extent may be defined by the user and further analyzed through a powerful visualization module. It allows for the inclusion of environmental effects through three different classes of increasingly sophisticated embedding approaches: the implicit solvation polarizable continuum model, the explicit polarizable embedding model, and the frozen density embedding model.

We present state-of-the-art string-based relativistic general-excitation-rank configuration interaction and coupled cluster calculations of the electron electric dipole moment, the nucleon–electron scalar-pseudoscalar, and the magnetic hyperfine interaction constants ( α d e , α C S , A | | , respectively) for the thallium atomic ground state 2 P 1 / 2 . Our present best values are α d e = − 558 ± 28 , α C S = 6.77 ± 0.34 [ 10 − 18 e cm], and A | | = 21172 ± 1059 [MHz]. The central value of the latter constant agrees with the experimental result to within 0.7% and serves as a measurable probe of the P , T -violating interaction constants. Our findings lead to a significant reduction of the theoretical uncertainties for P , T -odd interaction constants for atomic thallium but not to stronger constraints on the electron electric dipole moment, d e , or the nucleon–electron scalar-pseudoscalar coupling constant, C S .

Highly correlated pure ab initio relativistic configuration-interaction theory is in the present paper applied to the calculation of the tensor-pseudotensor P, T-violating nucleon-electron interaction constant in the electronic ground states of atomic mercury and radium. The final best obtained results are R-T(Hg) = -4.43 [10(-20) e cm] and R-T(Ra) = -15.0 [10(-20) e cm]. The accuracy of the employed electronic-structure models are confirmed by determining the static electric-dipole polarizability alpha(d)(Hg) = 35.7 a.u. which is in accord with the experimental value to about 5%. R-T(Ra) will be useful for constraining (or obtaining) the CP-violating parameter C-T when combined with future measurements of the electric-dipole moment of the radium atom.

We perform model-independent analyses extracting limits for the electric dipole moment of the electron and the P,T-odd scalar-pseudoscalar (S-PS) nucleon-electron coupling from the most recent measurements with atoms and molecules. The analysis using paramagnetic systems, only, is improved substantially by the inclusion of the recent measurement on HfF+ ions, but complicated by the fact that the corresponding constraints are largely aligned, owing to a general relation between the coefficients for the two contributions. Since this same relation does not hold in diamagnetic systems, it is possible to find atoms that provide essentially orthogonal constraints to those from paramagnetic ones. However, the coefficients are suppressed in closed-shell systems and enhancements of P,T-odd effects are only prevalent in the presence of hyperfine interactions. We formulate the hyperfine-induced time-reversal-symmetry breaking S-PS nucleon-electron interaction in general atoms in a mixed perturbative and variational approach, based on electronic Dirac-wavefunctions including the effects of electron correlations. The method is applied to the Hg atom, yielding the first direct calculation of the coefficient of the S-PS nucleon-electron coupling in a diamagnetic system. This results in additionally improved model-independent limits for both the electron EDM and the nucleon-electron coupling from the global fit. Finally we employ this fit to provide indirect limits for several paramagnetic systems under investigation.

Parity-and time-reversal-symmetry violating interaction constants required for the interpretation of a recent measurement [W.B. Cairncross et al., Phys. Rev. Lett. 119, 153001 (2017)] of corresponding symmetry violations in the Omega = 1 ((3)Delta(1)) science state of the HfF+ molecular ion are reported. Using a relativistic four-component all-electron multireference configuration interaction model the nucleon-electron scalar-pseudoscalar interaction constant is determined as W-S = 20.0 [kHz]. An updated result for the electron electric-dipole-moment effective electric field of vertical bar E-eff vertical bar = 22.7[GV/cm] is obtained. Further results of relevance in the context of the search for leptonic charge-parity violation such as the Hf-177 magnetic hyperfine interaction constant and electronic G tensor for HfF+ are presented.

The TaO$^+$ molecular ion is proposed as a candidate system for detecting signatures of charge parity (${\cal{CP}}$) violating physics beyond the standard model of elementary particles. The electron electric dipole moment (EDM) effective electric field $E_{\text{eff}} = 20.2 \left[\frac{\rm GV}{\rm cm}\right]$, the nucleon-electron scalar-pseudoscalar (ne-SPS) interaction constant $W_{S} = 17.7$ [kHz] and the nuclear magnetic quadrupole interaction constant $W_M = 0.45$ [$\frac{10^{33} {\text{Hz}}}{e\, {\text{cm}}^2}$] are found to be sizeable ${\cal{P,T}}$-odd enhancements. The ratio of the leptonic and semi-leptonic enhancements differs strongly from the one for the ThO system which may provide improved limits on the electron EDM, $d_e$, and the SPS coupling constant, $C_S$. TaO$^+$ is found to have a ${^3\Delta_1}$ electronic ground state like the previously proposed ThF$^+$ molecular ion, but an order of magnitude smaller parallel G-tensor component which makes it less vulnerable to systematic errors in experiment.

We present an updated EDM effective electric field of E_eff = 75.2[ GV/ cm] and the electron-nucleon scalar-pseudoscalar interaction constant W_S=107.8 [kHz] for the ^3Δ_1 science state of ThO. The criticisms made in reference [J. Chem. Phys. 142, 024301 (2015)] are addressed and largely found to be unsubstantiated within the framework of our approach.