- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

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

Ultracold High-Spin Σ-State Polar Molecules for New Physics Searches

Alessio Ciamei1,2,3,*,†, Adam Koza4,*, Marcin Gronowski4, and Michał Tomza4,‡

  • *These authors contributed equally to this work.
  • †Contact author: ciamei@lens.unifi.it
  • ‡Contact author: michal.tomza@fuw.edu.pl
PDF

Phys. Rev. X 16, 011052 – Published 9 March, 2026

DOI: https://doi.org/10.1103/wlq7-9p41

Abstract

We propose high-spin Σ-state polar molecules assembled from ultracold atoms to probe charge-parity-violating physics beyond the standard model. We identify YbCr as a prime candidate to search for the electric dipole moment of the electron. We show that the combination of relativistic ytterbium and high-spin chromium, amenable to magnetoassociation, leads to molecules with easy-to-polarize parity doublets and large intramolecular electric fields. Based on ab initio results for molecular constants, we predict a sensitivity of δde=(6×10−31/nday)  e cm via standard spin-precession measurements, assess the experimental feasibility, and discuss potential extensions to more advanced quantum control as well as searches of the nuclear magnetic quadrupole moment. This work paves the way to next-generation searches for new physics with ultracold molecules in both the leptonic and hadronic sectors.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

A major mystery in modern physics is why the Universe contains far more matter than antimatter, which motivates searches for tiny violations of fundamental symmetries such as the electric dipole moment of the electron. We identify YbCr as a candidate that combines relativistic ytterbium with high-spin chromium to feature large internal electric fields and a rotational structure controllable with modest laboratory fields. We show that assembling these molecules from laser-cooled atoms allows for long interrogation times and large sample sizes that are essential for high-precision measurements. Our results indicate that experiments based on ultracold YbCr could reach sensitivities beyond current leading approaches using standard spin-precession measurements. This proposal, extendable to the hadronic sector and advanced quantum control, supports next-generation searches for physics beyond the standard model.

Article Text

Supplemental Material

References (114)

  1. 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).
  2. T. E. Chupp, P. Fierlinger, M. J. Ramsey-Musolf, and J. T. Singh, Electric dipole moments of atoms, molecules, nuclei, and particles, Rev. Mod. Phys. 91, 015001 (2019).
  3. M. Dine and A. Kusenko, Origin of the matter-antimatter asymmetry, Rev. Mod. Phys. 76, 1 (2004).
  4. L. Canetti, M. Drewes, and M. Shaposhnikov, Matter and antimatter in the universe, New J. Phys. 14, 095012 (2012).
  5. A. D. Sakharov, Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe, Pis’ma Zh. Eksp. Teor. Fiz. 32, 392 (1967).
  6. C. Patrignani, Review of Particle Physics, Chin. Phys. C 40, 100001 (2016).
  7. T. S. Roussy, L. Caldwell, T. Wright, W. B. Cairncross, Y. Shagam, K. B. Ng, N. Schlossberger, S. Y. Park, A. Wang, J. Ye, and E. A. Cornell, An improved bound on the electron’s electric dipole moment, Science 381, 46 (2023).
  8. V. Andreev, D. G. Ang, D. DeMille, J. M. Doyle, G. Gabrielse, J. Haefner, N. R. Hutzler, Z. Lasner, C. Meisenhelder, B. R. O’Leary, C. D. Panda, A. D. West, E. P. West, and X. Wu (ACME Collaboration), Improved limit on the electric dipole moment of the electron, Nature (London) 562, 355 (2018).
  9. R. Alarcon, J. Alexander, V. Anastassopoulos et al., Electric dipole moments and the search for new physics, arXiv:2203.08103.
  10. 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).
  11. J. Ye and P. Zoller, Essay: Quantum sensing with atomic, molecular, and optical platforms for fundamental physics, Phys. Rev. Lett. 132, 190001 (2024).
  12. M. Saffman, Quantum computing with atomic qubits and Rydberg interactions: Progress and challenges, J. Phys. B 49, 202001 (2016).
  13. L. Henriet, L. Beguin, A. Signoles, T. Lahaye, A. Browaeys, G.-O. Reymond, and C. Jurczak, Quantum computing with neutral atoms, Quantum 4, 327 (2020).
  14. A. Browaeys and T. Lahaye, Many-body physics with individually controlled Rydberg atoms, Nat. Phys. 16, 132 (2020).
  15. A. M. Kaufman and K.-K. Ni, Quantum science with optical tweezer arrays of ultracold atoms and molecules, Nat. Phys. 17, 1324 (2021).
  16. A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Practical quantum advantage in quantum simulation, Nature (London) 607, 667 (2022).
  17. D. González-Cuadra, D. Bluvstein, M. Kalinowski, R. Kaubruegger, N. Maskara, P. Naldesi, T. V. Zache, A. M. Kaufman, M. D. Lukin, H. Pichler, B. Vermersch, J. Ye, and P. Zoller, Fermionic quantum processing with programmable neutral atom arrays, Proc. Natl. Acad. Sci. U.S.A. 120, e2304294120 (2023).
  18. T. Kusano, Y. Nakamura, R. Yokoyama, N. Ozawa, K. Shibata, T. Takano, Y. Takasu, and Y. Takahashi, Plane-selective manipulations of nuclear spin qubits in a three-dimensional optical tweezer array, Phys. Rev. Res. 7, L022045 (2025).
  19. A. Ludlow, “Tweezer clock” offers new possibilities in timekeeping, Physics 12 (2019).
  20. L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys. 90, 035005 (2018).
  21. B. K. Malia, Y. Wu, J. Martínez-Rincón, and M. A. Kasevich, Distributed quantum sensing with mode-entangled spin-squeezed atomic states, Nature (London) 612, 661 (2022).
  22. G. Bornet, G. Emperauger, C. Chen, B. Ye, M. Block, M. Bintz, J. A. Boyd, D. Barredo, T. Comparin, F. Mezzacapo, T. Roscilde, T. Lahaye, N. Y. Yao, and A. Browaeys, Scalable spin squeezing in a dipolar Rydberg atom array, Nature (London) 621, 728 (2023).
  23. W. J. Eckner, N. Darkwah Oppong, A. Cao, A. W. Young, W. R. Milner, J. M. Robinson, J. Ye, and A. M. Kaufman, Realizing spin squeezing with Rydberg interactions in an optical clock, Nature (London) 621, 734 (2023).
  24. J. M. Robinson, M. Miklos, Y. M. Tso, C. J. Kennedy, T. Bothwell, D. Kedar, J. K. Thompson, and J. Ye, Direct comparison of two spin-squeezed optical clock ensembles at the 10−17 level, Nat. Phys. 20, 208 (2024).
  25. C. Zhang, P. Yu, A. Jadbabaie, and N. R. Hutzler, Quantum-enhanced metrology for molecular symmetry violation using decoherence-free subspaces, Phys. Rev. Lett. 131, 193602 (2023).
  26. 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).
  27. N. J. Fitch, J. Lim, E. A. Hinds, B. E. Sauer, and M. R. Tarbutt, Methods for measuring the electron’s electric dipole moment using ultracold YbF molecules, Quantum Sci. Technol. 6, 014006 (2020).
  28. 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).
  29. L. Anderegg, N. B. Vilas, C. Hallas, P. Robichaud, A. Jadbabaie, J. M. Doyle, and N. R. Hutzler, Quantum control of trapped polyatomic molecules for eEDM searches, Science 382, 665 (2023).
  30. Y. Chamorro, A. Borschevsky, E. Eliav, N. R. Hutzler, S. Hoekstra, and L. c. v. 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).
  31. T. Köhler, K. Góral, and P. S. Julienne, Production of cold molecules via magnetically tunable Feshbach resonances, Rev. Mod. Phys. 78, 1311 (2006).
  32. C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82, 1225 (2010).
  33. N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. Bergmann, Stimulated raman adiabatic passage in physics, chemistry, and beyond, Rev. Mod. Phys. 89, 015006 (2017).
  34. G. Valtolina, K. Matsuda, W. G. Tobias, J.-R. Li, L. De Marco, and J. Ye, Dipolar evaporation of reactive molecules to below the Fermi temperature, Nature (London) 588, 239 (2020).
  35. M. Duda, X.-Y. Chen, A. Schindewolf, R. Bause, J. von Milczewski, R. Schmidt, I. Bloch, and X.-Y. Luo, Transition from a polaronic condensate to a degenerate Fermi gas of heteronuclear molecules, Nat. Phys. 19, 720 (2023).
  36. N. Bigagli, W. Yuan, S. Zhang, B. Bulatovic, T. Karman, I. Stevenson, and S. Will, Observation of Bose–Einstein condensation of dipolar molecules, Nature (London) 631, 289 (2024).
  37. T. Karman, M. Tomza, and J. Pérez-Ríos, Ultracold chemistry as a testbed for few-body physics, Nat. Phys. 20, 722 (2024).
  38. S. L. Cornish, M. R. Tarbutt, and K. R. A. Hazzard, Quantum computation and quantum simulation with ultracold molecules, Nat. Phys. 20, 730 (2024).
  39. C. M. Holland, L. Yukai, and L. W. Cheuk, On-demand entanglement of molecules in a reconfigurable optical tweezer array, Science 382, 1143 (2023).
  40. E. R. Meyer and J. L. Bohn, Electron electric-dipole-moment searches based on alkali-metal- or alkaline-earth-metal-bearing molecules, Phys. Rev. A 80, 042508 (2009).
  41. V. Barbé, A. Ciamei, B. Pasquiou, L. Reichsöllner, F. Schreck, P. S. Żuchowski, and J. M. Hutson, Observation of Feshbach resonances between alkali and closed-shell atoms, Nat. Phys. 14, 881 (2018).
  42. M. Verma, A. M. Jayich, and A. C. Vutha, Electron electric dipole moment searches using clock transitions in ultracold molecules, Phys. Rev. Lett. 125, 153201 (2020).
  43. J. D. Polet, Y. Chamorro, L. F. Pašteka, S. Hoekstra, M. Tomza, A. Borschevsky, and I. A. Aucar, P, T-odd effects in YbCu, YbAg, and YbAu, J. Chem. Phys. 161, 234302 (2024).
  44. T. Fleig and D. DeMille, Theoretical aspects of radium-containing molecules amenable to assembly from laser-cooled atoms for new physics searches, New J. Phys. 23, 113039 (2021).
  45. I. B. Khriplovich and S. K. Lamoreaux, General features of EDM experiments, in CP Violation Without Strangeness: Electric Dipole Moments of Particles, Atoms, and Molecules (Springer, Berlin, Heidelberg, 1997) pp. 19–51.
  46. A. Vutha, A search for the electric dipole moment of the electron using thorium monoxide, Ph.D. thesis, Yale University, 2011.
  47. J. Baron, W. C. Campbell, D. DeMille, J. M. Doyle, G. Gabrielse, Y. V. Gurevich, P. W. Hess, N. R. Hutzler, E. Kirilov, I. Kozyryev, B. R. O’Leary, C. D. Panda, M. F. Parsons, B. Spaun, A. C. Vutha, A. D. West, and E. P. West (ACME Collaboration), Methods, analysis, and the treatment of systematic errors for the electron electric dipole moment search in thorium monoxide, New J. Phys. 19, 073029 (2017).
  48. J. M. Brown and A. Carrington, Rotational Spectroscopy of Diatomic Molecules, Cambridge Molecular Science (Cambridge University Press, Cambridge, England, 2003).
  49. See Supplemental Material at http://link.aps.org/supplemental/10.1103/wlq7-9p41 for detailed information and complementary results, which includes Refs. [32,40–44,48,50–71].
  50. F. Schreck and K. v. Druten, Laser cooling for quantum gases, Nat. Phys. 17, 1296 (2021).
  51. J. Lee, J. H. Lee, J. Noh, and J. Mun, Core-shell magneto-optical trap for alkaline-earth-metal-like atoms, Phys. Rev. A 91, 053405 (2015).
  52. A. Ciamei, S. Finelli, A. Cosco, M. Inguscio, A. Trenkwalder, and M. Zaccanti, Double-degenerate Fermi mixtures of Li6 and Cr53 atoms, Phys. Rev. A 106, 053318 (2022).
  53. M. D. Frye, P. S. Żuchowski, and M. Tomza, Spin–spin interaction and magnetic Feshbach resonances in collisions of high-spin atoms with closed-shell atoms, Phys. Rev. Res. 6, 023254 (2024).
  54. M. Tomza, Prospects for ultracold polar and magnetic chromium–closed-shell-atom molecules, Phys. Rev. A 88, 012519 (2013).
  55. H. Ladjimi and M. Tomza, Diatomic molecules of alkali-metal and alkaline-earth-metal atoms: Interaction potentials, dipole moments, and polarizabilities, Phys. Rev. A 109, 052814 (2024).
  56. S. Finelli, A. Ciamei, B. Restivo, M. Schemmer, A. Cosco, M. Inguscio, A. Trenkwalder, K. Zaremba-Kopczyk, M. Gronowski, M. Tomza, and M. Zaccanti, Ultracold LiCr: A new pathway to quantum gases of paramagnetic polar molecules, PRX Quantum 5, 020358 (2024).
  57. K. G. Dyall, Core correlating basis functions for elements 31–118, Theor. Chem. Acc. 131, 1 (2012).
  58. Y. J. Franzke, L. Spiske, P. Pollak, and F. Weigend, Segmented contracted error-consistent basis sets of quadruple-ζ valence quality for one-and two-component relativistic all-electron calculations, J. Chem. Theory Comput. 16, 5658 (2020).
  59. A. Sunaga, V. S. Prasannaa, M. Abe, M. Hada, and B. P. Das, Ultracold mercury–alkali-metal molecules for electron-electric-dipole-moment searches, Phys. Rev. A 99, 040501(R) (2019).
  60. C. Baroni, G. Lamporesi, and M. Zaccanti, Quantum mixtures of ultracold gases of neutral atoms, Nat. Rev. Phys. 6, 736 (2024).
  61. H. R. Larsson, H. Zhai, C. J. Umrigar, and G. K.-L. Chan, The chromium dimer: Closing a chapter of quantum chemistry, J. Am. Chem. Soc. 144, 15932 (2022).
  62. E. Tiesinga, J. Kłos, M. Li, A. Petrov, and S. Kotochigova, Relativistic aspects of orbital and magnetic anisotropies in the chemical bonding and structure of lanthanide molecules, New J. Phys. 23, 085007 (2021).
  63. M. L. González-Martínez and P. S. Żuchowski, Magnetically tunable Feshbach resonances in Li+Er, Phys. Rev. A 92, 022708 (2015).
  64. M. B. Kosicki, M. Borkowski, and P. S. Żuchowski, Quantum chaos in Feshbach resonances of the ErYb system, New J. Phys. 22, 023024 (2020).
  65. P. Hamilton, Preliminary results in the search for the electron electric dipole moment in PbO*, Ph.D. thesis, Yale University, 2010.
  66. S. Bickman, P. Hamilton, Y. Jiang, and D. DeMille, Preparation and detection of states with simultaneous spin alignment and selectable molecular orientation in PbO, Phys. Rev. A 80, 023418 (2009).
  67. R. F. Curl, The relationship between electron spin rotation coupling constants and g-tensor components, Mol. Phys. 9, 585 (1965).
  68. F. Neese, Prediction of electron paramagnetic resonance g values using coupled perturbed Hartree–Fock and Kohn–Sham theory, J. Chem. Phys. 115, 11080 (2001).
  69. M. Denis, Y. Hao, E. Eliav, N. R. Hutzler, M. K. Nayak, R. G. E. Timmermans, and A. Borschesvky, Enhanced P,T-violating nuclear magnetic quadrupole moment effects in laser-coolable molecules, J. Chem. Phys. 152, 084303 (2020).
  70. T. Fleig and M. K. Nayak, Electron electric dipole moment and hyperfine interaction constants for ThO, J. Mol. Spectrosc. 300, 16 (2014).
  71. D. Wu, C. Zhou, J. J. Bao, L. Gagliardi, and D. G. Truhlar, Zero-field splitting calculations by multiconfiguration pairdensity functional theory, J. Chem. Theory Comput. 18, 2199 (2022).
  72. I. Kopp and J. T. Hougen, Rotational energy levels of states and intensities in transitions: Applications to some heavier hydrides, Can. J. Phys. 45, 2581 (1967).
  73. D. T. Halfen and L. M. Ziurys, The submillimeter spectrum of MnH and MnD (X7Σ+), Astrophys. J. 672, L77 (2007).
  74. D. T. Halfen and L. M. Ziurys, Molecules in high spin states III: The millimeter/submillimeter-wave spectrum of the MnCl radical (X7Σ+), J. Chem. Phys. 122, 054309 (2005).
  75. C. He, E. Hajiyev, Z. Ren, B. Song, and G.-B. Jo, Recent progresses of ultracold two-electron atoms, J. Phys. B 52, 102001 (2019).
  76. L. Chomaz, I. Ferrier-Barbut, F. Ferlaino, B. Laburthe-Tolra, B. L. Lev, and T. Pfau, Dipolar physics: A review of experiments with magnetic quantum gases, Rep. Prog. Phys. 86, 026401 (2022).
  77. L. Gabardos, S. Lepoutre, O. Gorceix, L. Vernac, and B. Laburthe-Tolra, Cooling all external degrees of freedom of optically trapped chromium atoms using gray molasses, Phys. Rev. A 99, 023607 (2019).
  78. M. Kitagawa, K. Enomoto, K. Kasa, Y. Takahashi, R. Ciuryło, P. Naidon, and P. S. Julienne, Two-color photoassociation spectroscopy of ytterbium atoms and the precise determinations of s-wave scattering lengths, Phys. Rev. A 77, 012719 (2008).
  79. J. Werner, A. Griesmaier, S. Hensler, J. Stuhler, T. Pfau, A. Simoni, and E. Tiesinga, Observation of Feshbach resonances in an ultracold gas of Cr52, Phys. Rev. Lett. 94, 183201 (2005).
  80. B. Pasquiou, G. Bismut, Q. Beaufils, A. Crubellier, E. Maréchal, P. Pedri, L. Vernac, O. Gorceix, and B. Laburthe-Tolra, Control of dipolar relaxation in external fields, Phys. Rev. A 81, 042716 (2010).
  81. R. Roy, A. Green, R. Bowler, and S. Gupta, Rapid cooling to quantum degeneracy in dynamically shaped atom traps, Phys. Rev. A 93, 043403 (2016).
  82. A. Farolfi, D. Trypogeorgos, G. Colzi, E. Fava, G. Lamporesi, and G. Ferrari, Design and characterization of a compact magnetic shield for ultracold atomic gas experiments, Rev. Sci. Instrum. 90, 115114 (2019).
  83. G. Colzi, E. Fava, M. Barbiero, C. Mordini, G. Lamporesi, and G. Ferrari, Production of large Bose-Einstein condensates in a magnetic-shield-compatible hybrid trap, Phys. Rev. A 97, 053625 (2018).
  84. A. Farolfi, Spin dynamics in two-component Bose-Einstein condensates, Ph.D. thesis, University of Trento, 2021.
  85. E. Fava, Static and dynamics properties of a miscible two-component Bose–Einstein Condensate, Ph.D. thesis, University of Trento, 2018.
  86. S. Eckel, P. Hamilton, E. Kirilov, H. W. Smith, and D. DeMille, Search for the electron electric dipole moment using Ω-doublet levels in PbO, Phys. Rev. A 87, 052130 (2013).
  87. R. Bause, N. Balasubramanian, T. Fikkers, E. H. Prinsen, K. Steinebach, A. Jadbabaie, N. R. Hutzler, I. A. Aucar, L. c. v. F. Pašteka, A. Borschevsky, and S. Hoekstra, Prospects for measuring the electron’s electric dipole moment with polyatomic molecules in an optical lattice, Phys. Rev. A 111, 062815 (2025).
  88. K. Zhu, N. Solmeyer, C. Tang, and D. S. Weiss, Absolute polarization measurement using a vector light shift, Phys. Rev. Lett. 111, 243006 (2013).
  89. N. Vanhaecke and O. Dulieu, Precision measurements with polar molecules: The role of the black body radiation, Mol. Phys. 105, 1723 (2007).
  90. Y. Takahashi, C. Zhang, A. Jadbabaie, and N. R. Hutzler, Engineering field-insensitive molecular clock transitions for symmetry violation searches, Phys. Rev. Lett. 131, 183003 (2023).
  91. N. Hutzler (private communication).
  92. M. Tomza, Ab initio properties of the ground-state polar and paramagnetic europium–alkali-metal-atom and europium–alkaline-earth-metal-atom molecules, Phys. Rev. A 90, 022514 (2014).
  93. A. Ciamei, A. Koza, M. Gronowski, and M. Tomza, Supporting data for “Ultracold high-spin Σ-state polar molecules for new physics searches”, 10.5281/zenodo.18467046.
  94. S. Boys and F. Bernardi, The calculation of small molecular interactions by the differences of separate total energies. some procedures with reduced errors, Mol. Phys. 19, 553 (1970).
  95. Y. Wang and M. Dolg, Pseudopotential study of the ground and excited states of Yb2, Theor. Chem. Acc. 100, 124 (1998).
  96. M. Dolg, U. Wedig, H. Stoll, and H. Preuss, Energy-adjusted ab initio pseudopotentials for the first row transition elements, J. Chem. Phys. 86, 866 (1987).
  97. J. G. Hill, M. Gronowski, and M. Tomza (to be published).
  98. P. J. Knowles, C. Hampel, and H.-J. Werner, Coupled cluster theory for high spin, open shell reference wave functions, J. Chem. Phys. 99, 5219 (1993).
  99. H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, and M. Schütz, Molpro: A general-purpose quantum chemistry program package, WIREs Comput. Mol. Sci. 2, 242 (2012).
  100. H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. Schütz et al., Molpro, version 2022, a package of ab initio programs, see https://www. molpro.net.
  101. M. Kállay, P. R. Nagy, D. Mester, Z. Rolik, G. Samu, J. Csontos, J. Csóka, P. B. Szabó, L. Gyevi-Nagy, B. Hégely, I. Ladjánszki, L. Szegedy, B. Ladóczki, K. Petrov, M. Farkas, P. D. Mezei, and A. Ganyecz, Mrcc, a quantum chemical program, see https://www.mrcc.hu.
  102. J. F. Stanton and R. J. Bartlett, The equation of motion coupled‐cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited state properties, J. Chem. Phys. 98, 7029 (1993).
  103. H. Werner and P. J. Knowles, An efficient internally contracted multiconfiguration–reference configuration interaction method, J. Chem. Phys. 89, 5803 (1988).
  104. D. A. Matthews, L. Cheng, M. E. Harding, F. Lipparini, S. Stopkowicz, T.-C. Jagau, P. G. Szalay, J. Gauss, and J. F. Stanton, Coupled-cluster techniques for computational chemistry: The CFOUR program package, J. Chem. Phys. 152, 214108 (2020).
  105. J. F. Stanton, J. Gauss, L. Cheng, M. E. Harding, D. A. Matthews, and P. G. Szalay, CFOUR, Coupled-Cluster techniques for Computational Chemistry, a quantum-chemical program package, With contributions from A. Asthana, A. A. Auer, R. J. Bartlett, U. Benedikt, C. Berger, D. E. Bernholdt, S. Blaschke, Y. J. Bomble, S. Burger, O. Christiansen et al., and the integral packages MOLECULE (J. Almlöf and P. R. Taylor), PROPS (P. R. Taylor), ABACUS (T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, and J. Olsen), and ECP routines by A. V. Mitin and C. van Wüllen. For the current version, see http://www.cfour.de.
  106. T. Saue et al., The Dirac code for relativistic molecular calculations, J. Chem. Phys. 152, 204104 (2020).
  107. S. Knecht, H. J. A. Jensen, and T. Fleig, Large-scale parallel configuration interaction. II. Two- and four-component double-group general active space implementation with application to BiH, J. Chem. Phys. 132, 014108 (2010).
  108. E. Lindroth, B. Lynn, and P. Sandars, Order α2 theory of the atomic electric dipole moment due to an electric dipole moment on the electron, J. Phys. B 22, 559 (1989).
  109. M. Kozlov, Semiempirical calculations of P- and P, T-odd effects in diatomic molecules-radicals, Zh. Eksp. Teor. Fiz. 89, 1933 (1985).
  110. V. Flambaum, Spin hedgehog and collective magnetic quadrupole moments induced by parity and time invariance violating interaction, Phys. Lett. B 320, 211 (1994).
  111. L. V. Skripnikov, A. V. Titov, and V. V. Flambaum, Enhanced effect of CP-violating nuclear magnetic quadrupole moment in a HfF+ molecule, Phys. Rev. A 95, 022512 (2017).
  112. D. Peng, N. Middendorf, F. Weigend, and M. Reiher, An efficient implementation of two-component relativistic exact-decoupling methods for large molecules, J. Chem. Phys. 138, 184105 (2013).
  113. F. Neese, The ORCA program system, WIREs Comput. Mol. Sci. 2, 73 (2012).
  114. F. Neese, Software update: The ORCA program system—version 6.0, Comput. Mol. Sci. 15, e70019 (2025).

Outline

Information

Phys. Rev. X 16, 011052– Published 9 March, 2026
  • Received 23 July 2025
  • Accepted 27 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation