Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Magnetic hyperfine structure constants of BaF137 in the Π1/22 and Π3/22 excited states

Yuly Chamorro1,*, Felix Kogel2, Tim Langen2,3, and Anastasia Borschevsky1

  • 1Van Swinderen Institute for Particle Physics and Gravity, University of Groningen, 9747 AG Groningen, The Netherlands
  • 25. Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
  • 3Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, A-1020 Vienna, Austria

  • *Contact author: y.a.chamorro.mena@rug.nl

Phys. Rev. A 112, 042808 – Published 9 October, 2025

DOI: https://doi.org/10.1103/sksm-t85m

Abstract

High-precision molecular experiments testing the Standard Model of particle physics require an accurate understanding of the molecular structure at the hyperfine level, both for the control of the molecules and for the interpretation of the results. In this work, we calculate the hyperfine structure constants for the excited states Π1/22 and Π3/22 of BaF137 due to the Ba137 nucleus. We use the four-component relativistic Fock-space coupled-cluster method, extrapolating our results to the complete basis set limit. We investigate the effect of the basis sets and electron correlation, and estimate the uncertainty in our final results. Our results are used in the interpretation of the experimental spectroscopy of the hyperfine and rovibrational spectra of BaF, and the planning of laser-cooling schemes for future parity-violating anapole moment measurements [Kogel et al., Phys. Rev. A 112, 042807 (2025)].

View figure in article

Physics Subject Headings (PhySH)

See Also

High-resolution spectroscopy of barium monofluoride: Odd isotopologues, hyperfine structure, and isotope shifts

Felix Kogel, Yuly Chamorro, Mangesh Bhattarai, Marian Rockenhäuser, Tatsam Garg, David DeMille, Anastasia Borschevsky, and Tim Langen
Phys. Rev. A 112, 042807 (2025)

Article Text

References (47)

  1. B. M. Roberts and J. S. M. Ginges, Hyperfine anomaly in heavy atoms and its role in precision atomic searches for new physics, Phys. Rev. A 104, 022823 (2021).
  2. J. S. M. Ginges and A. V. Volotka, Testing atomic wave functions in the nuclear vicinity: The hyperfine structure with empirically deduced nuclear and quantum electrodynamic effects, Phys. Rev. A 98, 032504 (2018).
  3. U. D. Jentschura and V. A. Yerokhin, Quantum electrodynamic corrections to the hyperfine structure of excited s states, Phys. Rev. A 73, 062503 (2006).
  4. S. G. Karshenboim, Precision physics of simple atoms: Qed tests, nuclear structure and fundamental constants, Phys. Rep. 422, 1 (2005).
  5. V. V. Flambaum and V. A. Dzuba, Search for variation of the fundamental constants in atomic, molecular, and nuclear spectra, Can. J. Phys. 87, 25 (2009).
  6. M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, Search for new physics with atoms and molecules, Rev. Mod. Phys. 90, 025008 (2018).
  7. D. DeMille, N. R. Hutzler, A. M. Rey, and T. Zelevinsky, Quantum sensing and metrology for fundamental physics with molecules, Nat. Phys. 20, 741 (2024).
  8. A. Boeschoten, V. R. Marshall, T. B. Meijknecht, A. Touwen, H. L. Bethlem, A. Borschevsky, S. Hoekstra, J. W. F. van Hofslot, K. Jungmann, M. C. Mooij, R. G. E. Timmermans, W. Ubachs, and L. Willmann, Spin-precession method for sensitive electric dipole moment searches, Phys. Rev. A 110, L010801 (2024).
  9. J. Lim, J. R. Almond, M. A. Trigatzis, J. A. Devlin, N. J. Fitch, B. E. Sauer, M. R. Tarbutt, and E. A. Hinds, Laser cooled ybf molecules for measuring the electron's electric dipole moment, Phys. Rev. Lett. 120, 123201 (2018).
  10. I. Kozyryev and N. R. Hutzler, Precision measurement of time-reversal symmetry violation with laser-cooled polyatomic molecules, Phys. Rev. Lett. 119, 133002 (2017).
  11. Y. Hao, L. F. Pašteka, L. Visscher, P. Aggarwal, H. L. Bethlem, A. Boeschoten, A. Borschevsky, M. Denis, K. Esajas, S. Hoekstra, K. Jungmann, V. R. Marshall, T. B. Meijknecht, M. C. Mooij, R. G. E. Timmermans, A. Touwen, W. Ubachs, L. Willmann, Y. Yin, and A. Zapara (NL-eEDM Collaboration), High accuracy theoretical investigations of CaF, SrF, and BaF and implications for laser-cooling, J. Chem. Phys. 151, 034302 (2019).
  12. F. Kogel, T. Garg, M. Rockenhäuser, S. A. Morales-Ramírez, and T. Langen, Molecular laser cooling using serrodynes: Implementation, characterization and prospects, New J. Phys. 27, 055001 (2025).
  13. E. Altuntaş, J. Ammon, S. B. Cahn, and D. DeMille, Demonstration of a sensitive method to measure nuclear-spin-dependent parity violation, Phys. Rev. Lett. 120, 142501 (2018).
  14. F. Kogel, Y. Chamorro, M. Bhattarai, M. Rockenhäuser, T. Garg, D. DeMille, A. Borschevsky, and T. Langen, preceding paper, High-resolution spectroscopy of barium monofluoride: Odd isotopologues, hyperfine structure, and isotope shifts, Phys. Rev. A 112, 042807 (2025).
  15. F. Kogel, T. Garg, M. Rockenhäuser, and T. Langen, Laser-cooled BaF137 molecules for measuring nuclear-spin-dependent parity violation, Phys. Rev. Res. 7, L022041 (2025).
  16. W. E. Ernst, J. Kändler, and T. Törring, Hyperfine structure and electric dipole moment of baf BaFXΣ+2, J. Chem. Phys. 84, 4769 (1986).
  17. M. Denis, P. A. B. Haase, M. C. Mooij, Y. Chamorro, P. Aggarwal, H. L. Bethlem, A. Boeschoten, A. Borschevsky, K. Esajas, Y. Hao et al., Benchmarking of the fock-space coupled-cluster method and uncertainty estimation: Magnetic hyperfine interaction in the excited state of BaF, Phys. Rev. A 105, 052811 (2022).
  18. C. Ryzlewicz, H.-U. Schütze-Pahlmann, J. Hoeft, and T. Törring, Rotational spectrum and hyperfine structure of the 2σ radicals BaF and BaCl, Chem. Phys. 71, 389 (1982).
  19. P. A. Haase, E. Eliav, M. Ilias, and A. Borschevsky, Hyperfine structure constants on the relativistic coupled cluster level with associated uncertainties, J. Phys. Chem. A 124, 3157 (2020).
  20. T. Langen, G. Valtolina, D. Wang, and J. Ye, Quantum state manipulation and cooling of ultracold molecules, Nat. Phys. 20, 702 (2024).
  21. F. Kogel, M. Rockenhäuser, R. Albrecht, and T. Langen, A laser cooling scheme for precision measurements using fermionic barium monofluoride (Ba137F19) molecules, New J. Phys. 23, 095003 (2021).
  22. P. Pyykko, Relativistic effects in structural chemistry, Chem. Rev. 88, 563 (1988).
  23. A.-M. Mårtensson-Pendrill and S. Salomonson, Hyperfine structure of the 4s, 4p, and 3d states in Ca+ evaluated by many-body perturbation theory, Phys. Rev. A 30, 712 (1984).
  24. S. Raeder, D. Ackermann, H. Backe, R. Beerwerth, J. C. Berengut, M. Block, A. Borschevsky, B. Cheal, P. Chhetri, C. E. Düllmann, V. A. Dzuba, E. Eliav, J. Even, R. Ferrer, V. V. Flambaum, S. Fritzsche, F. Giacoppo, S. Götz, F. P. Heßberger, M. Huyse et al., Probing sizes and shapes of nobelium isotopes by laser spectroscopy, Phys. Rev. Lett. 120, 232503 (2018).
  25. S. G. Porsev, C. Cheung, and M. S. Safronova, Calculation of energies and hyperfine-structure constants of U+233 and U233, Phys. Rev. A 106, 042810 (2022).
  26. F. P. Gustafsson, C. M. Ricketts, M. L. Reitsma, R. F. Garcia Ruiz, S. W. Bai, J. C. Berengut, J. Billowes, C. L. Binnersley, A. Borschevsky, T. E. Cocolios, B. S. Cooper, R. P. de Groote, K. T. Flanagan, A. Koszorús, G. Neyens, H. A. Perrett, A. R. Vernon, Q. Wang, S. G. Wilkins, and X. F. Yang, Tin resonance-ionization schemes for atomic- and nuclear-structure studies, Phys. Rev. A 102, 052812 (2020).
  27. T. Fleig and M. K. Nayak, Electron electric dipole moment and hyperfine interaction constants for tho, J. Mol. Spectrosc. 300, 16 (2014).
  28. A. V. Oleynichenko, L. V. Skripnikov, A. Zaitsevskii, E. Eliav, and V. M. Shabaev, Diagonal and off-diagonal hyperfine structure matrix elements in kcs within the relativistic fock space coupled cluster theory, Chem. Phys. Lett. 756, 137825 (2020).
  29. K. G. Dyall and K. Fægri, Jr., Introduction to Relativistic Quantum Chemistry (Oxford University Press, New York, 2007).
  30. DIRAC, a relativistic ab initio electronic structure program, Release DIRAC19 (2019), written by A. S. P. Gomes, T. Saue, L. Visscher, H. J. Aa. Jensen, and R. Bast, with contributions from I. A. Aucar, V. Bakken, K. G. Dyall, S. Dubillard, U. Ekström, E. Eliav, T. Enevoldsen, E. Faßhauer, T. Fleig, O. Fossgaard, L. Halbert, E. D. Hedegård, T. Helgaker, B. Helmich-Paris, J. Henriksson, M. Iliaš, Ch. R. Jacob, S. Knecht, S. Komorovský, O. Kullie et al., available at http://dx.doi.org/10.5281/zenodo.3572669; see also http://www.diracprogram.org.
  31. T. Saue, R. Bast, A. S. P. Gomes, H. J. A. Jensen, L. Visscher, I. A. Aucar, R. Di Remigio, K. G. Dyall, E. Eliav, E. Fasshauer et al., The DIRAC code for relativistic molecular calculations, J. Chem. Phys. 152, 204104 (2020).
  32. K. Huber, Molecular Spectra and Molecular Structure: IV. Constants of Diatomic Molecules (Springer Science & Business Media, New York, 2013).
  33. N. J. Stone, Table of recommended nuclear magnetic dipole moments: Part I–Long-lived states, Report No. INDC (NDS)-0794 (IAEA, Vienna, Austria, 2019), https://inis.iaea.org/records/whf2k-6h240.
  34. A. Antušek, P. Rodziewicz, D. Ke¸dziera, A. Kaczmarek-Ke¸dziera, and M. Jaszuński, Ab initio study of nmr shielding of alkali earth metal ions in water complexes and magnetic moments of alkali earth metal nuclei, Chem. Phys. Lett. 588, 57 (2013).
  35. L. Visscher, E. Eliav, and U. Kaldor, Formulation and implementation of the relativistic fock-space coupled cluster method for molecules, J. Chem. Phys. 115, 9720 (2001).
  36. K. G. Dyall, Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the 4s, 5s, 6s, and 7s elements, J. Phys. Chem. A 113, 12638 (2009).
  37. K. G. Dyall, Core correlating basis functions for elements 31–118, Theor. Chem. Acc. 131, 1217 (2012).
  38. K. G. Dyall, Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the light elements H–Ar, Theor. Chem. Acc. 135, 128 (2016).
  39. A. V. Oleynichenko, A. Zaitsevskii, and E. Eliav, Towards high performance relativistic electronic structure modelling: The exp-t program package, in Supercomputing, edited by V. Voevodin and S. Sobolev (Springer International, Cham, Switzerland, 2020), pp. 375–386.
  40. T. Helgaker, W. Klopper, H. Koch, and J. Noga, Basis-set convergence of correlated calculations on water, J. Chem. Phys. 106, 9639 (1997).
  41. J. M. Martin, Ab initio total atomization energies of small molecules—towards the basis set limit, Chem. Phys. Lett. 259, 669 (1996).
  42. M. Lesiuk and B. Jeziorski, Complete basis set extrapolation of electronic correlation energies using the Riemann zeta function, J. Chem. Theory Comput. 15, 5398 (2019).
  43. Y. Chamorro, A. Borschevsky, E. Eliav, N. R. Hutzler, S. Hoekstra, and L. F. Pašteka, Molecular enhancement factors for the P, T-violating electric dipole moment of the electron in BaCH3 and YbCH3 symmetric top molecules, Phys. Rev. A 106, 052811 (2022).
  44. Y. Chamorro, V. V. Flambaum, R. F. Garcia Ruiz, A. Borschevsky, and L. F. Pašteka, Enhanced parity and time-reversal-symmetry violation in diatomic molecules: LaO, LaS, and LuO, Phys. Rev. A 110, 042806 (2024).
  45. P. A. Haase, D. J. Doeglas, A. Boeschoten, E. Eliav, M. Iliaš, P. Aggarwal, H. L. Bethlem, A. Borschevsky, K. Esajas, Y. Hao et al., Systematic study and uncertainty evaluation of p, t-odd molecular enhancement factors in BaF, J. Chem. Phys. 155, 034309 (2021).
  46. L. Skripnikov and A. Titov, Theoretical study of thorium monoxide for the electron electric dipole moment search: Electronic properties of HΔ13 in ThO, J. Chem. Phys. 142, 024301 (2015).
  47. J. Lang, M. Przybytek, and M. Lesiuk, Estimating the complete basis set extrapolation error through random walks, J. Phys. Chem. Lett. 16, 4952 (2025).

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation