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Fine-structure splittings in low-lying triplet states of the carbon atom

Pavel Rzhevskii1,*, Dmitry Tumakov2,1,†, Toreniyaz Shomenov3,1,‡, and Sergiy Bubin1,§

  • *Contact author: pavel.rzhevskii@nu.edu.kz
  • †Contact author: dm.tumakov@gmail.com
  • ‡Contact author: toreniyaz.shomenov@nu.edu.kz
  • §Contact author: sergiy.bubin@nu.edu.kz
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Phys. Rev. Research 8, 013126 – Published 4 February, 2026

DOI: https://doi.org/10.1103/v4jk-j7c9

Abstract

We present benchmark theoretical values of the fine-structure splittings in the ground 3Pe and low-lying excited 3Pe and 3De states of the C12, C13, and C14 isotopes of the neutral carbon atom. We handle the correlations in the dynamics of all particles (six electrons and a nucleus) in all considered states in the practically exact manner by employing very large expansions of the wave functions in terms of all-particle explicitly correlated Gaussian basis sets, whose nonlinear parameters were extensively optimized. The resulting wave functions were used to compute the fine-structure splittings for the three isotopes by including the leading-order spin-orbit and electron spin-spin corrections (∝α2), contribution from the electron anomalous magnetic moment (∝α3), and by taking into account the most significant couplings between the targeted and neighboring electronic states (∝α4). Our results show a notable improvement in accuracy compared to any previous theoretical calculations, are in good agreement with the best available experimental data, and demonstrate that the calculations of six-electron systems are now sufficiently advanced for precision measurement applications.

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Phys. Rev. Research 8, 013126– Published 4 February, 2026
  • Received 29 October 2024
  • Accepted 22 October 2025
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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.

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