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COMMON PART

Project Number19-72-10019

Project titleTheoretical investigation of the properties and structure of the nuclei using atomic and molecular systems

Project LeadSkripnikov Leonid

AffiliationPetersburg Nuclear Physics Institute named by B.P. Konstantinov of National Research Centre "Kurchatov Institute",

Implementation period | 07.2019 - 06.2022 | extension for 07.2022 - 06.2024 |

Research area 02 - PHYSICS AND SPACE SCIENCES, 02-304 - Spectroscopy

Keywordsmagnetic and electric moments, nuclear magnetic resonance, electronic structure, hyperfine structure, relativistic effects

PROJECT CONTENT

Annotation

Due to the breakthrough in the development of experimental techniques of manipulating atoms and molecules, such systems are increasingly becoming “laboratories” for studying various physical phenomena, precision measurement of fundamental physical constants, testing theories of fundamental interactions and the nuclear structure theory. Such systems are used to build atomic clocks, search for effects of variation of the fundamental constants, and act as elements of the construction of quantum computers. Highly charged heavy ions are used to test quantum electrodynamics (QED) [Nature Communications 8, 15484 (2017); Appendix, slide 1].
However, in many cases, there are problems associated with insufficiently good knowledge of the structure and properties of the nucleus (see Appendix, slides 2,3). This causes the greatest errors in predicting the atomic and molecular properties, which depend on the properties of the nucleus. An analysis of the literature shows that these errors are greatly underestimated, and sometimes completely ignored, leading to “paradoxes” and “puzzles” (see below). At the moment, it is not possible to obtain with sufficient accuracy the properties of nuclei on the basis of direct quantum-mechanical calculations of the nuclear structure. Especially, it concerns the properties of heavy nuclei. But it is precisely the atomic and molecular systems containing such nuclei that are of most interest: many of the effects under consideration (dependence of the energies of terms, the effects of parity non-conservation, QED) are most pronounced in such compounds.
In [L.V. Skripnikov et al, "Nuclear nuclear magnetic moment of 209Bi: Resolving the bismuth hyperfine puzzle", Phys. Rev. Lett., 120, 093001 (2018)] we solved the problem (see Appendix, Slide 5) related to the discrepancy between the theoretical predictions of the specific difference of hyperfine splittings in highly charged ions [V.M. Shabaev et al. Phys. Rev. Lett. 86, 3959 (2001)] and experimental data, called "the hyperfine puzzle" [Nature Communications 8, 15484 (2017)]. The case turned out to be in the wrong "standard" value of the magnetic moment of the 209Bi nucleus, which is given in reference books and other literature with very little uncertainty. This value of the magnetic moment was obtained due to the incorrect interpretation of experimental data on nuclear magnetic resonance (Appendix, slide 2), caused by very rough estimates of the shielding constant (the calculation error was underestimated by several orders of magnitude). See also the article about our work https://physicsworld.com/a/has-the-hypefine-puzzle-been-solved/. The new value of the magnetic moment resolved the established “hyperfine puzzle”.
According to a preliminary analysis of the literature data, the reference book contains considerably underestimated errors for magnetic moment values not only for the 209Bi nucleus, but, for example, for the 203Tl, 205Tl, 207Pb, 185Re, 187Re, 199Hg nuclei, etc. At the same time for several of these nuclei the experiments were carried out to measure hyperfine splittings in hydrogen-like ions. One of the goals of the experiments was to test QED predictions in strong fields. The error in the magnetic moments of these nuclei, in fact, negates such predictions, since the contribution of QED effects may be less than this error, as was in the case of 209Bi. Knowledge the exact values of the magnetic moments is also important for other areas. In particular, in the search for effects of violating symmetries of fundamental interactions in atoms and molecules, the theoretically predicted value of the hyperfine splitting in which is used as a test of the accuracy of the calculation of the wave function. However, the error of such predictions directly depends on the error of the value of the magnetic moment used (which in most cases is completely ignored and all errors are attributed to the error of the wave function). Another example is the testing of nuclear theory predictions. For example, in [R. Sen’kov and V. Dmitriev, Nucl. Phys. A706, 351 (2002)] there was predicted the magnitude of the effect of the distribution of magnetization in the nucleus. However, even these predictions depend on the magnitude of the magnetic moment, which does not allow one to directly verify the accuracy of the predicted data.
In this project, a theory will be developed to systematically reduce the error of the extracted values of magnetic moments. It will be based on the development of a relativistic theory of coupled clusters to calculate the shielding constants, which are necessary to extract the magnetic moment values from the NMR data. In this case, the effect of magnetization distribution in the nucleus and even QED effects will be taken into account and analyzed. We are not aware of any work by other groups of theorists who would perform calculations for the screening constants in compounds of heavy elements at this level of theory (in the best works of other groups the density functional theory is used, the accuracy of which is noticeably lower). The application of the theory developed in the project will make it possible to obtain data with unique accuracy for the magnetic moments of the nuclei of heavy elements. This will allow one to resolve deviations of theoretical predictions that used inaccurate tabular data with experiment.
In the project it is supposed to consider also previously unexplored properties of nuclei. In particular, it is planned to propose an experiment to measure the quadrupole distribution of neutrons in the nucleus using molecules. As far as we know, this property of the nucleus has never been measured before. Methods of completely relativistic four-component calculations of the electronic structure of molecules that are required to predict the corresponding effect in the properties and spectra of molecules from which the desired value can be extracted will be developed. In [Phys. Rev. A 96, 012516 (2017)] this effect was considered for atoms. However, molecules can be much more promising systems for such research, because they have nearby levels that are necessary for such measurements.
In the project, methods will be developed for the first time, allowing to evaluate the effect of the finite distribution of the magnetization of the nucleus on the shielding constants and hyperfine splittings in molecules. This will bring to a fundamentally new level of prediction accuracy on the hyperfine structure, which is necessary for testing methods for predicting other molecular properties that are required for interpreting experiments to measure the fundamental properties of an electron and other particles using molecules [L.V. Skripnikov, A.V. Titov, Phys. Rev. A 91, 042504 (2015)].

Expected results

During the project, the heavy element nuclei for which incorrect or inaccurate values of magnetic moments were previously obtained will be revealed, given the fact that the magnetic dipole moment is one of the most important characteristics of the nucleus. For these nuclei, magnetic moments will be obtained at the new level of theory.
To extract the magnetic moments of the nuclei from experimental data, a completely new approach based on the relativistic coupled cluster theory will be developed. We are not aware of any work of other theoretical groups in the world that perform the calculation of the shielding constants in heavy element compounds at such a level that is required for reliable interpretation of fundamental experiments. The currently proposed approaches for calculating the shielding constants for molecules containing heavy nuclei are based on the density functional theory. Thus our approach will allow to exceed the world level in accuracy.
The magnetic moments 185Re, 203Tl, 205Tl, etc. nuclei, which are also of interest for testing various theories where magnetic moments are used as external parameters, will be refined. For nuclei 185Re, 187Re [J.R. Crespo Lo´pez-Urrutia et al Phys. Rev. A 57, 879 1998], 207Pb [Seelig et al, Phys. Rev. Lett. 81, 4824 (1998)], 165Ho [J.R. Crespo Lo´pez-Urrutia, at al, Phys. Rev. Lett. 77, 826 (1996)], 203Tl, 205Tl [Beiersdorfer et al, PRA 64 032506 (2001)] and for some others the measurements of hyperfine splitting in the hydrogen-like ions were performed. To interpret and analyze such data, magnetic moment values from standard tables are used, see, for example, [Beiersdorfer et al, PRA 64 032506 (2001)]. These tables usually contain data from NMR experiments. However, to interpret the results of such experiments carried out both on atoms and molecules, rather rough atomic estimates are used for the shielding constants, calculated with the theory developed in the second half of the 20th century by the W. R. Johnson group; this estimates no longer meet modern requirements for the accuracy of magnetic moments values. It negates the accuracy that is achieved in experimental measurements. As shown by us in [L.V. Skripnikov et al, "Nuclear nuclear magnetic moment of 209Bi: Resolving the bismuth hyperfine puzzle", Phys. Rev. Lett., 120, 093001 (2018)], the real error of the magnetic moment thus extracted can exceed that stated by several orders of magnitude. This leads to an incorrect analysis of the results of these experiments for hydrogen-like or lithium-like ions of heavy elements. Therefore, the development of a new theory for calculating the shielding constants is of fundamental interest for the physics of highly charged ions of heavy elements and the verification of fundamental theories that use the values of magnetic moments.
During the project, it is intended to clarify the magnetic moment of the stable 207Pb isotope, as well as other lead isotopes. This will allow to test the nuclear theory developed in [R. Sen’kov and V. Dmitriev, Nucl. Phys. A706, 351 (2002)]. It will also allow to resolve the contradiction in the experimental data obtained in experiments on optical pumping and nuclear magnetic resonance on an atomic beam [H.M. Gibbs et al; Phys. Rev. 188, 180 (1969); O. Lutz et al, Phys. Lett. 35A, 397 (1971)].
The nuclei with a spin > 1/2 have an electric quadrupole moment. This moment is due to the distribution of protons in the nucleus and can be measured by well-developed standard methods. The group of V. Flambaum (Australia) proposed a method for measuring quadrupole moments of neutron distribution in the nucleus using heavy atoms [Phys. Rev. A 96, 012516 (2017)]. The idea is that the weak charge of the nucleus is mainly due to neutrons: the weak charge of the neutron (−1) exceeds the weak charge of the proton (0.08) by more than an order of magnitude. Therefore, by measuring the tensor contribution to the weak interaction between the nucleus and electrons in an atom, it is possible for the first time to obtain information from the atomic data on the quadrupole distribution of neutrons in the nucleus. As far as we know, this property has never been measured before in other types of experiments. There are studies on the effect of “neutron skin” [S. Abrahamyan et al. (PREX Collaboration), Phys. Rev. Lett. 108, 112502 (2012)], however, they refer to the spherically symmetric part of the neutron distribution in the nucleus. In this project we will consider the possibility of conducting an experiment to measure the quadrupole moment of the neutron subsystem of a nucleus on molecules. For this purpose, an appropriate theory will be proposed and programs developed for performing calculations of such effects in molecules within the framework of relativistic four-component approaches. It should be noted that the theory developed by the Australian theoretical group is applicable only to the consideration of atoms. As a result, we will find the value of the effect under discussion for the TaO+ molecule (and / or other promising systems). If an experiment to measure such an effect (quadrupole distribution of neutrons in a nucleus) will have a positive result, it will be possible to bring knowledge of the properties of the nucleus to a qualitatively new level and, thus, it will have a significant impact on the development of the theory of the nucleus.
The development of methods for precision calculation of the electronic structure of atoms and molecules, assumed in this project, is of fundamental importance for the interpretation of experiments on the search for effects of spatial parity non-conservation (and / or time invariance non-conservation) of fundamental interactions. To interpret such experiments, it is necessary to develop methods by which one can calculate the electron wave function with high accuracy. We have already achieved some success in this area [L.V. Skripnikov, D.E. Maison, N.S. Mosyagin, Phys. Rev. A 95, 022507 (2017); L.V. Skripnikov, J. Chem. Phys. 145 (21) 214301 (2016)], however, upon reaching the error level of 1%, further improvement of the methods is required, which we have already encountered. Accordingly, we need methods for testing the accuracy of the wave function. A good and generally accepted test for such problems is precisely the calculation of the hyperfine structure. However, upon reaching an error level of about 1%, there is a need for reliable data for nuclear magnetic moments. The methods developed in this project can be used to solve this problem.
In the course of the project, an approach to the calculation of magnetic anomalies of the hyperfine structure in atoms and molecules will be developed taking into account the effects of electron correlation of high orders. To accomplish this, appropriate programs will be developed to calculate the contributions of the finite distribution of nuclear magnetization, and various models of the distribution of magnetization will be considered and analyzed. DIRAC [diracprogram.org] and MRCC [mrcc.hu] programs will be adapted to perform correlation calculations. The approaches and programs developed will be applied to predict nuclear magnetic moments of short-lived Pb isotopes, etc. The development of such a theory and software is of paramount importance: knowing the magnetic moments of stable isotopes of atoms and the theoretical data discussed (magnetic anomalies), one can predict the magnetic moments of other isotopes for which there is experimental data on hyperfine structure (see our work [Phys. Lett. B, 779 (10), 324-330 (2018)]). The magnetic moments of stable isotopes can be extracted from NMR experimental data at a completely new level of accuracy, as was noted above.
The scale of this task seems to be much broader. For a number of molecules, there are also data on hyperfine splitting for various isotopes. As far as we know, magnetic anomalies in molecular systems have never been calculated before. In this project it is planned to consider a combination of such effects. This is the peculiarity of our approaches compared with the approaches used by the best world groups in this field - their approaches are developed only for atoms. Our approaches will allow to consider molecules.
The aim of the project is basic research, which includes the development of theory and software, as well as calculations of specific compounds of heavy elements. This study will allow to obtain various properties of nuclei at a new level of accuracy. This will allow to test various nuclear theories, to solve problems that have arisen while testing quantum electrodynamics, etc. However, the developed theory and programs aimed at calculating compounds of heavy elements at a completely new level of accuracy can be used to understand the many physical and chemical processes occurring in the environment with molecules containing atoms of heavy elements at the molecular level. In the future, this may be important for solving a variety of scientific and technological problems in physics, chemistry, medicine and materials science, that is, our developments will be interdisciplinary.

REPORTS

Annotation of the results obtained in 2021

We have refined the value of the magnetic dipole moments of rhenium nuclei. To do this, we have performed a new interpretation of the nuclear magnetic resonance experiment [F. Alder and F. C. Yu, Phys. Rev. 82, 105 (1951)] on the solution of the NaReO4 molecule. To solve the molecular problem of calculating the shielding constant, we have proposed to use the relativistic coupled clusters theory. For the ReO4^- anion, we were able to take into account the contribution of correlation effects to this constant up to the iterative consideration of triple cluster amplitudes. We were able to take into account an important effect - the contribution to the shielding constant from the finite distribution of magnetization over the nucleus. This contribution was calculated in the single-particle nuclear model using the Woods-Saxon potential. Note that for many-electron molecules we have not found even attempts to take this effect into account, at least within the framework of the model of a uniformly magnetized ball. The contribution from the finite distribution of magnetization to the shielding constant for the ReO4- anion that we took into account turned out to be several times bigger than the effect of the solvent. The latter effect, for example, was studied in detail in [Phys. Chem. Chem. Phys., 22, 7065-7076 (2020)], but the effect of the magnetization distribution (which turns out to be several times larger) was not mentioned there. Using the data for the NMR experiment and the shielding constant calculated by us, the following results were obtained for the magnetic moments mu in units of nuclear magnetons muN:
mu(185Re) = 3.1564(3)(12)muN , mu(187Re) = 3.1887(3)(12)muN.
Here the first error is the experimental one, and the second is the theoretical one. The new values of magnetic moments have a significant disagreement with tabulated ones:
mu(185Re) = 3.1871(3)muN , mu(187Re) = 3.2197(3)muN.
Note that the tabulated magnetic moments are given with a very small error, but this error corresponds exclusively to the experiment. For example, the fact that the shielding constant for the atomic cation Re^7+ differs from the shielding constant for the molecular anion ReO4^- (on which the experiment was performed) is completely ignored. According to our calculations, the shielding constant of ReO4^- is approximately three times less than the shielding constant of the Re^7+ ion. This has led to serious consequences: the actual error in the values of the magnetic moments of the 185Re and 187Re nuclei, which is given in the reference book, turned out to be underestimated by a factor of 100. Earlier that caused a contradiction between the theoretical results on hyperfine splitting in hydrogen-like rhenium ions and the experiment. For example, the prediction for hyperfine splitting in hydrogen-like 185Re using the tabulated value of the magnetic moment is 2.7456(102) eV. Experiment [J. R. Crespo Lo´pez-Urrutia et al., Phys. Rev. A 57, 879 (1998)] gives 2.7189(18) eV. If we use the new value of the magnetic moment, then the theoretical prediction for the splitting is 2.7192(102), which is in excellent agreement with the experiment (the prediction error is related to the error in considering the finite magnetization distribution effect in the hyperfine splitting, the contribution from the magnetic moment error is much smaller than this error).
We also performed new calculations of shielding constants for Hg and Au atoms, for which experimental data are available. In this case, our results agreed with previous theoretical works. Although the considered electronic problem in our case was vastly more sophisticated, the effects of electron correlation happened to be insignificant.
The method for calculating the shielding constants that we have developed for molecules has now been generalized to the calculation of another important property: the constants of the indirect spin-spin interaction of nuclei in a molecule containing heavy atoms (the indirect interaction of nuclear magnetic moments). The proposed approach is based on the relativistic coupled cluster theory. Our calculation of the TlF molecule showed that this approach is much more accurate than the approach within the framework of the relativistic density functional theory, which is used by other groups. The result obtained by us coincides with the experimental one within its uncertainty As far as we know, the approach within the framework of the relativistic (with the Dirac-Coulomb Hamiltonian) coupled cluster theory has never been used before.
In the paper[L.V. Skripnikov, A.V. Oleynichenko, A.V. Zaitsevskii, D.E. Maison, A.E. Barzakh, Phys. Rev. C 104, 034316 (2021)], we have solved the eight-year-old puzzle of the contradiction between the values of the electric quadrupole moments of the 209Bi nucleus, extracted from molecular and atomic data. It was shown that the contradiction was due to the low accuracy of the value of the electric field gradient on the nucleus of the Bi atom in the ground electronic state, which was previously used to interpret atomic experimental data. To solve this problem, we were the first to apply the relativistic coupled cluster theory in the Fock space for three particles (the author of this program is A.V. Oleynichenko, one of the main executors of the project). The corrected value of the quadrupole moment was -418(6) mb (which differs significantly from the value of -516(15) mb given in the reference book). The value we obtained was found not only from the data for the ground state of bismuth, which was considered earlier but also independently from the data for the excited state. Our interpretation of three molecular experiments led to the same value of the quadrupole moment (within the calculation error). The results obtained for the electric field gradient allowed us in cooperation with experimenters to refine the electric quadrupole short-lived moments of neutron-deficient isotopes 187Bi, 188Bi^g, 188Bi^m, 189Bi, 191Bi [A. Barzakh,… Skripnikov, Oleynichenko, Maison et al, Phys. Rev. Lett. 127, 192501 (2021)].
Earlier in the project, we performed very accurate predictions of the hyperfine structure in the RaF molecule, taking into account the effect of the finite nuclear magnetization distribution. To support experiments on this molecule, we performed precise prediction of its spectra with a record-breaking error of only 40 cm-1. Our results indicate that in [Nature 581, 396 (2020)], the assignment of spectral lines, provided by experimentalists, needs to be reconsidered.
For the TaO^+ cation, we predicted the energies and spectroscopic constants of low-lying states. In particular, we have shown that the state 3De_1, which is suitable to measure the quadrupole distribution of neutrons (we estimated this effect in the current project), is the ground one [G. Penyazkov, L.V. Skripnikov, A.V. Oleynichenko, A.V. Zaitsevskii, Chem. Phys. Lett., 793, 139448 (2022)].
In the papers [D.E. Maison, L.V. Skripnikov, A.V. Oleynichenko, A.V. Zaitsevskii, J. Chem. Phys., 154(22), 224303 (2021); D.E. Maison, L.V. Skripnikov Phys. Rev. A, 105(3), 032813 (2022)], we studied the effect of the exchange of an axion (a hypothetical particle introduced in QCD extensions) between the nucleus and electrons (as well as between two electrons) in the Fr atom and the ytterbium monohydroxide molecule. According to our estimates, the expected sensitivity of experiments on these systems may be sufficient to establish new constraints on the product of the interaction constants of an axion with a nucleus and an electron. This particular study is of interest for solving topical issues of quantum chromodynamics.

Publications

**1.** *L.V. Skripnikov, A.V. Oleynichenko, A.V. Zaitsevskii, D.E. Maison, A.E. Barzakh* **Relativistic Fock space coupled-cluster study of bismuth electronic structure to extract the Bi nuclear quadrupole moment** PHYSICAL REVIEW C, Phys. Rev. C 104, 034316 (2021) (year - 2021).

**2.** *A. Barzakh, ..., D.E. Maison, A.V. Oleynichenko, L.V. Skripnikov...* **Large Shape Staggering in Neutron-Deficient Bi Isotopes** PHYSICAL REVIEW LETTERS, Phys. Rev. Lett. 127, 192501 (2021) (year - 2021).

**3.** *D.E. Maison, L.V. Skripnikov, A.V. Oleynichenko, A.V. Zaitsevskii* **Axion-mediated electron–electron interaction in ytterbium monohydroxide molecule** The Journal of Chemical Physics, J. Chem. Phys. 154, 224303 (2021) (year - 2021).

**4.** *G. Penyazkov, L.V. Skripnikov, A.V. Oleynichenko, A.V. Zaitsevskii* **Effect of the neutron quadrupole distribution in the TaO+ cation** Chemical Physics Letters, Chem. Phys. Lett. 793, 139448 (2022) (year - 2022).

**5.** *V. Fella, L.V. Skripnikov, W. Nortershauser, M.R. Buchner, H.L. Deubner, F. Kraus, A.F. Privalov, V.M. Shabaev, M. Vogel* **Magnetic moment of 207Pb and the hyperfine splitting of 207Pb81+** Physical Review Research, Phys. Rev. Research 2, 013368 (2020) (year - 2020).

**6.** *A. Zaitsevskii, L.V. Skripnikov, N.S. Mosyagin, T. Isaev, R. Berger, A.A. Breier, T.F. Giesen* **Accurate ab initio calculations of RaF electronic structure appeal to more laser-spectroscopical measurements** The Journal of Chemical Physics, J. Chem. Phys. 156, 044306 (2022) (year - 2022).

**7.** *D.E. Maison, L.V. Skripnikov* **Static electric dipole moment of the francium atom induced by axionlike particle exchange** Phys. Rev. A, Phys. Rev. A 105, 032813 (2022) (year - 2022).

**8.** *L.V. Skripnikov, S.D. Prosnyak* **Refined nuclear magnetic dipole moment of rhenium: 185Re and 187Re** arxiv, arXiv:2204.13015 [physics.atom-ph] (year - 2022).

**9.** *-* **Петербургский институт ядерной физики обнародовал подходы к исследованиям сверхтонкого расщепления в спектрах ионов, атомов и молекул** Научный портал «Атомная энергия 2.0», https://www.atomic-energy.ru/news/2022/02/22/122208 (year - ).

**10.** *-* **Новые подходы к исследованиям сверхтонкого расщепления в спектрах ионов, атомов и молекул** сайт НИЦ «Курчатовский Институт» - ПИЯФ, - (year - ).

**11.** *L. V. Skripnikov and S. D. Prosnyak* **Refined nuclear magnetic dipole moment of rhenium: 185Re and 187Re** Physical Review C, Phys. Rev. C 106, 054303 (2022) (year - 2022).

Annotation of the results obtained in 2019

The main results obtained at the first year are summarized below.
In [V. Fella, L.V. Skripnikov, et al "Magnetic moment of ^207Pb and the hyperfine splitting of ^207Pb^81+", Phys. Rev. Res. 2, 013368 (2020)] the nuclear magnetic moment of ^207Pb has been clarified (http://www.pnpi.spb.ru/press-center/novosti/1814-issledovanie-svojstv-yadra-s-ispolzovaniem-atomnykh-i-molekulyarnykh-sistem). Theoretical part of this work was aimed at the extraction of the ^207Pb nuclear magnetic moment from the NMR experimental data for the [PbF6]^2- anion. Only the value of the ^207Pb nuclear magnetic moment shielded by the molecular environment can be extracted directly from the NMR experiment. To extract the magnetic moment itself, it is necessary to calculate the shielding constant of the magnetic moment in this anion. The shielding tensor corresponding to a given nucleus in a molecule can be defined as the mixed derivative of the energy with respect to the nuclear magnetic moment and external magnetic field. To calculate such a derivative, the contributions from the hyperfine interaction of electrons with the nucleus and from their interaction with an external magnetic field are to be added to the electronic Hamiltonian. In the relativistic one-electron case, the shielding tensor can be calculated by the sum-over-states method in the perturbation theory framework; the summation should include both positive and negative energy states of the Dirac one-particle spectrum. The part associated with the former states is called the paramagnetic contribution, and the part associated with negative energy states is called the diamagnetic contribution. To solve the many-electron problem, the approaches to calculate these contributions based on the Dirac-Fock and relativistic density functional methods were developed earlier. According to our calculations, the diamagnetic contribution can be fairly accurately calculated within the framework of these approaches; however, they are not accurate enough to calculate paramagnetic contribution. Therefore, we have adapted the four-component relativistic coupled cluster method to solve this problem. The possibility of systematic evaluation of accuracy of the calculations performed is one of the main advantages of the coupled cluster approach. The paramagnetic contribution to the shielding constant of the ^207Pb magnetic moment in the [PbF6]^2- anion was calculated in the four-component coupled cluster framework, accounting for single, double and perturbative triple excitations. The contribution from the latter was found to be only -29 ppm. Using this value, we can estimate the error of the obtained value of the shielding constant with respect to the level of the models used to account for effects of electron correlation. This error (29 ppm) is more than an order of magnitude smaller than the error typical for the relativistic density functional approach; this clearly shows the developed approach is promising.
The final value of the shielding constant obtained is equal to 13393 ppm. The value of the nuclear magnetic moment extracted from the NMR experiment using this constant is equal to \mu(^207Pb) = 0.59102(18) \mu_N. This value differs markedly from that given in the tables [N. Stone, Table of nuclear magnetic dipole and electric quadrupole moments, INDC(NDS)–0658, International Atomic Energy Agency (IAEA) (2014).]: 0.592583(9) \mu_N and does not coincide with it within the specified error of the latter. This is due to the fact that the error of the shielding constant value used was not included in the error given in the reference book; the theoretical value of the shielding constant was only roughly estimated theoretically.
Using the approach developed in the project, the magnetic moments of the stable nuclei ^203Tl and ^205Tl were also refined.
In [S. D. Prosnyak, D. E. Maison, and L. V. Skripnikov., J. Chem. Phys. 152, 044301 (2020)] the dependence of the hyperfine structure constants in the ground and excited states of the thallium atom on the spatial distribution of the magnetization in the nucleus was studied. We have shown that the correction for accounting for this effect can reach 16% (for the 6P_3/2 state) and cannot be discarded in high-precision studies of hyperfine interactions in systems containing heavy nuclei. It has been demonstrated that for the 6P_3/2 electronic state, the high level of accounting for electronic correlation is of crucial importance. At the Dirac-Fock level the calculated value differs from the experimental one by 5 times, and in order to achieve an error of 10%, one has to use the CCSDT(Q) coupled cluster method.
The study of the so-called hyperfine magnetic anomalies is also of a great interest. For a point nucleus, the ratio of the hyperfine splitting of two different isotopes 1 and 2 is proportional to the ratio of the nuclear g-factors of these isotopes. However, this ratio does not hold for real nuclei due to their finite size. The corresponding correction is called the magnetic anomaly, 1_Delta_2 = (A_1 g_2) / (A_2 g_1) - 1, where A_1 and A_2 are the hyperfine constants for each electronic state, and g_1 and g_2 are the g-factors of the isotopes under consideration. Using the Bohr-Weisskopf corrections calculated within the uniformly magnetized ball model together with available experimental data, we predicted the value of the hyperfine magnetic anomaly for the pair of nuclei ^203Tl and ^205Tl in the 7S_1/2 state for the neutral Tl atom and in the 1S_1/2 state for the hydrogen-like atom. In addition, the ratio of hyperfine magnetic anomalies in the 7S_1/2 and 6P_1/2 states was calculated. Combining this quantity with the known experimental data, we predicted the magnetic moments of the metastable thallium isotopes ^191Tl^m and ^193Tl^m. The results are consistent with an independent study [A. E. Barzakh, et. al., Phys. Rev. C 86, 014311 (2012)]. However, in our work, a more accurate calculation of the ratio of magnetic anomalies was performed.
Within the project, we are developing a technique for taking into account the contribution of the nuclear structure to various properties of atoms and molecules. In particular, for these purposes we are developing the program for calculating the Bohr-Weisskopf correction within the single-particle nuclear model; this approach can be applied to the study of neutral atoms. Within this model, the distribution of magnetization is determined by a valence nucleon, whose wave function is found from the solution of the radial Schrödinger equation with the Woods-Saxon potential. The solution procedure is performed numerically on a one-dimensional grid. We used the computer program developed to perform the first calculation of the hyperfine splitting in the thallium atom employing the nuclear model different from the uniformly magnetized ball model; note that both electronic correlation and finite magnetization distribution effects were treated simultaneously. It was shown that the model employed allows to predict the magnitude of the Bohr-Weisskopf correction quite accurately.

Publications

**1.** *Fleig T., Skripnikov L.V.* **P,T -Violating and Magnetic Hyperfine Interactions in Atomic Thallium** Symmetry, 12(4), 498 (2020) (year - 2020).

**2.** *-* **Исследование свойств ядра с использованием атомных и молекулярных систем** НИЦ КИ - ПИЯФ, Сайт НИЦ КИ - ПИЯФ, новость от 27.04.2020 (year - ).

**3.** *-* **Новые атомно-молекулярные системы для изучения свойств ядра** сайт Национального исследовательского центра "Курчатовский институт", - (year - ).

**4.** *-* **Созданы новые атомно-молекулярные системы для изучения свойств ядра** сайт РНФ, - (year - ).

**5.** *-* **В ПИЯФ исследуют свойства атомных ядер с использованием атомных и молекулярных систем** сайт www.atomic-energy.ru, Этот проект был недавно поддержан в рамках Президентской программы исследовательских проектов РНФ # 19-72-10019. (year - ).

**6.** *-* **Новые атомно-молекулярные подходы для изучения свойств ядра** НАУЧНАЯ РОССИЯ, - (year - ).

**7.** *Prosnyak S.D., Maison D.E., Skripnikov L.V.* **Hyperfine structure in thallium atom: Study of nuclear magnetization distribution effects** Journal of Chemical Physics, J. Chem. Phys. 152, 044301 (2020) (year - 2020).

Annotation of the results obtained in 2020

The main results obtained in the first year are summarized below (4 papers in the Q1 peer-reviewed journals have been published).
One of the objectives of the project was the calculation of the hyperfine structure constant in a diatomic molecule taking into account the finite nature of the distribution of the nuclear magnetization. The 225RaF molecule was chosen as the main object of research since the prediction of the hyperfine structure in this molecule is important for preparing the experiment at CERN aiming at direct spectroscopic measurements of this property [https://cds.cern.ch/record/2717941/files/INTC-P-555.pdf; Nature 581, 396 (2020)]. Previously other theoretical groups performed calculations of the hyperfine structure in diatomic molecules containing heavy atoms including RaF. However, the effect of the finite distribution of the nuclear magnetization has not been systematically considered yet as well as rather large uncertainties of theoretical predictions were obtained in these studies. Therefore, in [L.V. Skripnikov, J. Chem. Phys. 153, 114114 (2020)] we examined this effect in the 225RaF molecule. It is shown that for an accurate prediction of the hyperfine structure in the RaF molecule it is necessary to take into account the finite distribution of magnetization inside the radium nucleus. For atoms, this effect is known as the Bohr-Weisskopf (BW) effect. Its value in the calculation depends on the chosen nuclear magnetization distribution model, which is usually known rather poorly. We have shown theoretically that it is possible to express this contribution from the nuclear magnetization distribution to the hyperfine structure constant in terms of a single universal parameter (matrix element) depending on the magnetization distribution. This is the matrix element of some (possibly unknown) "BW operator” for the 1s state of the corresponding hydrogen-like ion. This element can be extracted from highly precise experimental data on hyperfine structure and theoretical calculations of the electronic structure of an atom, ion, or molecule. It is important that the approach described does not require any assumptions about the distribution of magnetization. The general form of the operator valid for a wide class of models of the distribution of magnetization is only required. After the designated universal parameter of the magnetization distribution is found, it can be then used to predict the contribution of the Bohr-Weisskopf effect in other systems (atoms, ions, molecules) containing the nucleus under consideration. Thus it has been shown that the contribution of the effect of the finite distribution of the nuclear magnetization on the hyperfine structure in heavy-element compounds with good accuracy can be factorized into the purely electronic part and the part depending on the nuclear magnetization distribution. An important feature of the formulated approach consists in the fact that the matrix element over the 1s function for the corresponding hydrogen-like ion can be easily evaluated simply by the solution of the Dirac equation without accounting for the electron correlation. Moreover, one can use well-developed techniques for calculating the quantum electrodynamics (QED) contributions to the quantities under consideration.
The practical implementation of this method is described in [L.V. Skripnikov J. Chem. Phys. 153, 114114 (2020)]. It is required to find projectors onto special functions coinciding with the hydrogen-like 1s1/2 and 2p1/2 functions inside the nucleus and equal to zero outside the nucleus. After calculating the expectation values of these projectors over the atomic and molecular wave functions, their linear combination is constructed with fixed coefficients that do not depend on a particular molecule or ion (electronic factor). Finding the full contribution of the effect of the finite nuclear magnetization distribution from experimental and accurate theoretical data for the case of a point magnetic dipole, together with the previously calculated electronic factor, we find a universal parameter of the nuclear magnetization distribution. This parameter is then used to predict the hyperfine structure of other heavy-element compounds under consideration. This scheme was first applied to the atomic 225Ra^+ cation. The nuclear magnetization distribution parameter was found from the data on the hyperfine structure for the ground state of this ion and then used to calculate the hyperfine splitting in its excited electronic state. The theoretical results taking into account the magnetization distribution are in excellent agreement with the experiment with an error of 0.15%. The contribution of the finite nuclear magnetization effect was found to be equal to 1.3% for the excited state and 4% for the ground state, so the effect is rather significant. Calculating finally the electronic factors for the RaF molecule in the ground and excited electronic states and then using the established finite nuclear magnetization parameter, we have obtained the values of the hyperfine interaction constants in the ground and first excited electronic states of the 225RaF molecule. The contribution from the finite magnetization distribution was found to be equal to 4% for the ground state and 1% for the excited state. Thus, the calculated contribution for the ground state is rather large and significantly exceeds the theoretical error for the calculated value of the hyperfine interaction constant. Therefore, it should be taken into account. Such a systematic study of the effect under consideration in molecular systems was performed for the first time. The theory developed was implemented in the computer program.
The theory developed was also tested for a number of specific models of the magnetization distribution. In [S. D. Prosnyak, L. V. Skripnikov, Phys. Rev. C 103, 034314 (2021)] the approach based on the consideration of a one-particle model of the magnetization distribution, in which the valence nucleon state is obtained by solving the Schrödinger equation with the Woods-Saxon potential, was implemented and studied. The feature of this approach consists in its applicability for many-electron systems (e.g. neutral thallium atom). It can also be combined with the explicit consideration of electron correlation effects. The approach was implemented in the computer program (the registration certificate was obtained). For this model, it was shown that the typical error of the factorization technique developed is less than 1%, which is very good. Using the values of the magnetic anomalies ratios calculated within the described approach of the one-particle model of the nucleus, the values for the short-lived Tl isotopes (187Tlm, 189Tlm, 191Tlm, 193Tlm) were refined. The shielding constant was also calculated for the thallium atom, taking into account the various effects studied in the current project.
The possibility of using approximate Hamiltonians to calculate the correlation contributions to the paramagnetic components of the shielding constants was investigated. It is shown that such approaches can be successfully used to take into account corrections for high-order electron correlation effects. Thallium atom was used as the object of research.
Using experimental data on the nuclear magnetic resonance and calculated shielding constants for two lead compounds, the values of the magnetic dipole moment of the 207Pb nucleus were obtained. They are in good agreement with the data obtained by interpreting the NMR spectra of the [PbF6]- anion.
We have investigated the problem of the discrepancy between the values of the quadrupole moment of the 209Bi nucleus obtained in different theoretical works using the available experimental data. In the modern reference tables [Mol. Phys. 116, 1328 (2018)] the value of -516 (15) mbarn obtained in [Phys. Rev. Lett. 87, 133003 (2001)] from the interpretation of the experiment on the bismuth atom is given. However, in the later paper [Phys. Rev. A 88, 052504 (2013)] the value of -420 (8) mbarn was reported. In the latter paper, the data obtained for the BiN and BiP molecules were used and very close values of the ^209Bi nuclear quadrupole moment were obtained for both systems. To interpret the experimental data in terms of the electric quadrupole moment of the nucleus, it is necessary to calculate the gradient of the electric field on the nucleus. It should be noted that there are certain questions to the previous atomic and molecular calculations of this quantity. For instance, the contribution of core electrons was not considered in the papers cited. According to our calculations, a more accurate value is closer to that reported in [Phys. Rev. A 88, 052504 (2013)]. Our calculation accounted for the correlations of all electrons at the more advanced level of theory.
In the paper [Skripnikov et al Phys.Chem.Chem.Phys., 22, 18374 (2020)] we performed theoretical estimates of the effect induced in molecules due to the presence of the nuclear Schiff moment. This moment corresponds to the presence inside the nucleus of an approximately constant electric field directed along the spin of the nucleus. Molecules are very sensitive to this effect (more sensitive than atoms). In the case of heavy nuclei of finite size, the electrons of the molecule can penetrate into the nucleus and interact with this field, leading to a level shift in molecular spectra. To the moment the CENTREX collaboration is preparing such an experiment on the 205TlF molecule. However, it was previously shown that, due to the collective nuclear effects, the Schiff moment of many deformed actinide nuclei is enhanced by one or two orders of magnitude [Phys. Rev. A, 101, 042504 (2020)] compared to the case of spherical nuclei (e.g. Tl). However, until now, no direct calculation of this effect has been performed for molecules containing actinide or lanthanide atoms. Using the theoretical approaches developed in this project, we solved this problem for a number of molecules interesting for the experiment: 227AcF, 227AcN, 227AcO+, 229ThO, 153EuO+, and 153EuN. The estimates obtained indicate that experiments with these systems are highly promising. In the considered systems, the expected energy shift, which can be recorded experimentally, is more than two orders of magnitude higher than the analogous effect in the TlF molecule.
In [D.E. Maison, V.V. Flambaum, N.R. Hutzler, L.V. Skripnikov, Phys. Rev A, 103 (2), 022813 (2021)] the study of the exchange of pseudoscalar particles between the nucleus and electrons in molecular systems was carried out. This study is of interest for solving problems of quantum chromodynamics.

Publications

**1.** *Leonid V. Skripnikov, Nikolai S. Mosyagin, Anatoly V. Titov, Victor V. Flambaum* **Actinide and lanthanide molecules to search for strong CP-violation** Physical Chemistry Chemical Physics, Phys.Chem.Chem.Phys., 22, 18374 (2020) (year - 2020).

**2.** *Leonid V. Skripnikov* **Nuclear magnetization distribution effect in molecules: Ra+ and RaF hyperfine structure** Journal of Chemical Physics, J. Chem. Phys. 153, 114114 (2020) (year - 2020).

**3.** *Maison D.E., Flambaum V.V., Hutzler N.R., Skripnikov L.V.* **Electronic structure of the ytterbium monohydroxide molecule to search for axionlike particles** Physical Review A, Phys. Rev. A 103, 022813 (2021) (year - 2021).

**4.** *-* **Программа для расчёта матричных элементов поправки на распределение намагниченности к сверхтонкой структуре тяжёлых атомов в одночастичной модели ядра с потенциалом Вудса-Саксона** -, 2020666971 (year - ).

**5.** *S. D. Prosnyak, L. V. Skripnikov* **Effect of nuclear magnetization distribution within the Woods-Saxon model: Hyperfine splitting in neutral Tl** Physical Review C, Phys. Rev. C 103, 034314 (2021) (year - 2021).