Opportunities for fundamental physics research with radioactive molecules.

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.

Erratum: Nuclear Charge Radii of Silicon Isotopes [Phys. Rev. Lett. 132, 162502 (2024)].

This corrects the article DOI: 10.1103/PhysRevLett.132.162502.

Nuclear Charge Radii of Silicon Isotopes.

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≤60 MeV, which agrees with complementary observables.

Chiral molecule candidates for trapped ion spectroscopy by ab initio calculations: From state preparation to parity violation.

Parity non-conservation (PNC) due to the weak interaction is predicted to give rise to enantiomer dependent vibrational constants in chiral molecules, but the phenomenon has so far eluded experimental observation. The enhanced sensitivity of molecules to physics beyond the Standard Model (BSM) has led to substantial advances in molecular precision spectroscopy, and these may be applied to PNC searches as well. Specifically, trapped molecular ion experiments leverage the universality of trapping charged particles to optimize the molecular ion species studied toward BSM searches, but in searches for PNC, only a few chiral molecular ion candidates have been proposed so far. Importantly, viable candidates need to be internally cold, and their internal state populations should be detectable with high quantum efficiency. To this end, we focus on molecular ions that can be created by near threshold resonant two-photon ionization and detected via state-selective photo-dissociation. Such candidates need to be stable in both charged and neutral chiral versions to be amenable to these methods. Here, we present a collection of suitable chiral molecular ion candidates we have found, including CHDBrI+ and CHCaBrI+, that fulfill these conditions according to our ab initio calculations. We find that organo-metallic species have low ionization energy as neutrals and relatively high dissociation thresholds. Finally, we compute the magnitude of the PNC values for vibrational transitions for some of these candidates. An experimental demonstration of state preparation and readout for these candidates will be an important milestone toward measuring PNC in chiral molecules for the first time.

Relativistic study of parity-violating nuclear spin-rotation tensors.

We present a four-component relativistic approach to describe the effects of the nuclear spin-dependent parity-violating (PV) weak nuclear forces on nuclear spin-rotation (NSR) tensors. The formalism is derived within the four-component polarization propagator theory based on the Dirac-Coulomb Hamiltonian. Such calculations are important for planning and interpretation of possible future experiments aimed at stringent tests of the standard model through the observation of PV effects in NSR spectroscopy. An exploratory application of this theory to the chiral molecules H2X2 (X = 17O, 33S, 77Se, 125Te, and 209Po) illustrates the dramatic effect of relativity on these contributions. In particular, spin-free and spin-orbit effects are even of opposite signs for some dihedral angles, and the latter fully dominate for the heavier nuclei. Relativistic four-component calculations of isotropic nuclear spin-rotation constants, including parity-violating electroweak interactions, give frequency differences of up to 4.2 mHz between the H2Po2 enantiomers; on the nonrelativistic level of theory, this energy difference is 0.1 mHz only.

Systematic study and uncertainty evaluation of P, T-odd molecular enhancement factors in BaF.

A measurement of the magnitude of the electric dipole moment of the electron (eEDM) larger than that predicted by the Standard Model (SM) of particle physics is expected to have a huge impact on the search for physics beyond the SM. Polar diatomic molecules containing heavy elements experience enhanced sensitivity to parity (P) and time-reversal (T)-violating phenomena, such as the eEDM and the scalar-pseudoscalar (S-PS) interaction between the nucleons and the electrons, and are thus promising candidates for measurements. The NL-eEDM collaboration is preparing an experiment to measure the eEDM and S-PS interaction in a slow beam of cold BaF molecules [P. Aggarwal et al., Eur. Phys. J. D 72, 197 (2018)]. Accurate knowledge of the electronic structure parameters, Wd and Ws, connecting the eEDM and the S-PS interaction to the measurable energy shifts is crucial for the interpretation of these measurements. In this work, we use the finite field relativistic coupled cluster approach to calculate the Wd and Ws parameters in the ground state of the BaF molecule. Special attention was paid to providing a reliable theoretical uncertainty estimate based on investigations of the basis set, electron correlation, relativistic effects, and geometry. Our recommended values of the two parameters, including conservative uncertainty estimates, are 3.13 ±0.12×1024Hzecm for Wd and 8.29 ± 0.12 kHz for Ws.

Quantum chemical topology at the spin-orbit configuration interaction level: Application to astatine compounds.

We report a methodology that allows the investigation of the consequences of the spin-orbit coupling by means of the QTAIM and ELF topological analyses performed on top of relativistic and multiconfigurational wave functions. In practice, it relies on the "state-specific" natural orbitals (NOs; expressed in a Cartesian Gaussian-type orbital basis) and their occupation numbers (ONs) for the quantum state of interest, arising from a spin-orbit configuration interaction calculation. The ground states of astatine diatomic molecules (AtX with X = AtF) and trihalide anions (IAtI- , BrAtBr- , and IAtBr- ) are studied, at exact two-component relativistic coupled cluster geometries, revealing unusual topological properties as well as a significant role of the spin-orbit coupling on these. In essence, the presented methodology can also be applied to the ground and/or excited states of any compound, with controlled validity up to including elements with active 5d, 6p, and/or 5f shells, and potential limitations starting with active 6d, 7p, and/or 6f shells bearing strong spin-orbit couplings.

Hyperfine Structure Constants on the Relativistic Coupled Cluster Level with Associated Uncertainties.

Accurate predictions of hyperfine structure (HFS) constants are important in many areas of chemistry and physics, from the determination of nuclear electric and magnetic moments to benchmarking of new theoretical methods. We present a detailed investigation of the performance of the relativistic coupled cluster method for calculating HFS constants within the finite-field scheme. The two selected test systems are 133Cs and 137BaF. Special attention has been paid to construct a theoretical uncertainty estimate based on investigations on basis set, electron correlation and relativistic effects. The largest contribution to the uncertainty estimate comes from higher order correlation contributions. Our conservative uncertainty estimate for the calculated HFS constants is ∼5.5%, while the actual deviation of our results from experimental values is <1% in all cases.

Material Size Dependence on Fundamental Constants.

Precise experimental setups for detection of variation of fundamental constants, scalar dark matter, or gravitational waves, such as laser interferometers, optical cavities, and resonant-mass detectors, are directly linked to measuring changes in material size. Here we present calculated and experiment-derived estimates for both α and µ dependence of lattice constants and bond lengths of selected solid-state materials and diatomic molecules that are needed for interpretation of such experiments.

High accuracy theoretical investigations of CaF, SrF, and BaF and implications for laser-cooling.

The NL-eEDM collaboration is building an experimental setup to search for the permanent electric dipole moment of the electron in a slow beam of cold barium fluoride molecules [NL-eEDM Collaboration, Eur. Phys. J. D 72, 197 (2018)]. Knowledge of the molecular properties of BaF is thus needed to plan the measurements and, in particular, to determine the optimal laser-cooling scheme. Accurate and reliable theoretical predictions of these properties require the incorporation of both high-order correlation and relativistic effects in the calculations. In this work, theoretical investigations of the ground and lowest excited states of BaF and its lighter homologs, CaF and SrF, are carried out in the framework of the relativistic Fock-space coupled cluster and multireference configuration interaction methods. Using the calculated molecular properties, we determine the Franck-Condon factors (FCFs) for the A2Π1/2→X2Σ1/2 + transition, which was successfully used for cooling CaF and SrF and is now considered for BaF. For all three species, the FCFs are found to be highly diagonal. Calculations are also performed for the B2Σ1/2 +→X2Σ1/2 + transition recently exploited for laser-cooling of CaF; it is shown that this transition is not suitable for laser-cooling of BaF, due to the nondiagonal nature of the FCFs in this system. Special attention is given to the properties of the A'2Δ state, which in the case of BaF causes a leak channel, in contrast to CaF and SrF species where this state is energetically above the excited states used in laser-cooling. We also present the dipole moments of the ground and excited states of the three molecules and the transition dipole moments (TDMs) between the different states. Finally, using the calculated FCFs and TDMs, we determine that the A2Π1/2→X2Σ1/2 + transition is suitable for transverse cooling in BaF.

First Ionization Potentials of Fm, Md, No, and Lr: Verification of Filling-Up of 5f Electrons and Confirmation of the Actinide Series.

We report the first ionization potentials (IP1) of the heavy actinides, fermium (Fm, atomic number Z = 100), mendelevium (Md, Z = 101), nobelium (No, Z = 102), and lawrencium (Lr, Z = 103), determined using a method based on a surface ionization process coupled to an online mass separation technique in an atom-at-a-time regime. The measured IP1 values agree well with those predicted by state-of-the-art relativistic calculations performed alongside the present measurements. Similar to the well-established behavior for the lanthanides, the IP1 values of the heavy actinides up to No increase with filling up the 5f orbital, while that of Lr is the lowest among the actinides. These results clearly demonstrate that the 5f orbital is fully filled at No with the [Rn]5f147s2 configuration and that Lr has a weakly bound electron outside the No core. In analogy to the lanthanide series, the present results unequivocally verify that the actinide series ends with Lr.

Strong enhancement of parity violation effects in chiral uranium compounds.

The effects of parity violation (PV) on the vibrational transitions of chiral uranium compounds of the type N≡UXYZ and N≡UHXY (X, Y, Z = F, Cl, Br, I) are analysed by means of exact two-component relativistic (X2C) Hartree-Fock and density functional calculations using NUFClI and NUHFI as representative examples. The PV contributions to the vibrational transitions were found to be in the Hz range, larger than for any of the earlier proposed chiral molecules. Thus, these systems are very promising candidates for future experimental PV measurements. A detailed comparison of the N≡UHFI and the N≡WHFI homologues reveals that subtle electronic structure effects, rather than exclusively a simple Z(5) scaling law, are the cause of the strong enhancement in PV contributions of the chiral uranium molecules.

Large vibrationally induced parity violation effects in CHDBrI.

The isotopically chiral molecular ion CHDBrI+ is identified as an exceptionally promising candidate for the detection of parity violation in vibrational transitions. The largest predicted parity-violating frequency shift reaches 1.8 Hz for the hydrogen wagging mode which has a sub-Hz natural line width and its vibrational frequency auspiciously lies in the available laser range. In stark contrast to this result, the parent neutral molecule is two orders of magnitude less sensitive to parity violation. The origin of this effect is analyzed and explained. Precision vibrational spectroscopy of CHDBrI+ is feasible as it is amenable to preparation at internally low temperatures and resistant to predissociation, promoting long interrogation times (Landau et al., J. Chem. Phys., 2023, 159, 114307). The intersection of these properties in this molecular ion places the first observation of parity violation in chiral molecules within reach.

Toward Detection of the Molecular Parity Violation in Chiral Ru(acac)3 and Os(acac)3.

We present a theory-experiment investigation of the helically chiral compounds Ru(acac)3 and Os(acac)3 as candidates for next-generation experiments for detection of molecular parity violation (PV) in vibrational spectra. We used relativistic density functional theory calculations to identify optimal vibrational modes with expected PV effects exceeding by up to 2 orders of magnitude the projected instrumental sensitivity of the ultrahigh resolution experiment under construction at the Laboratoire de Physique des Lasers in Paris. Preliminary measurements of the vibrational spectrum of Ru(acac)3 carried out as the first steps toward the planned experiment are presented.

The electron affinity of astatine.

One of the most important properties influencing the chemical behavior of an element is the electron affinity (EA). Among the remaining elements with unknown EA is astatine, where one of its isotopes, 211At, is remarkably well suited for targeted radionuclide therapy of cancer. With the At- anion being involved in many aspects of current astatine labeling protocols, the knowledge of the electron affinity of this element is of prime importance. Here we report the measured value of the EA of astatine to be 2.41578(7) eV. This result is compared to state-of-the-art relativistic quantum mechanical calculations that incorporate both the Breit and the quantum electrodynamics (QED) corrections and the electron-electron correlation effects on the highest level that can be currently achieved for many-electron systems. The developed technique of laser-photodetachment spectroscopy of radioisotopes opens the path for future EA measurements of other radioelements such as polonium, and eventually super-heavy elements.