Zhang, Liang, and Meng (Chin. Phys. Lett. 26 (2009) 092401) showed that the imaginary time step (ITS) method can solve the single-particle Dirac equation with a nonlocal, Fock-type potential directly in coordinate space, without localization, and explicitly left open its deployment in fully self-consistent relativistic Hartree-Fock or relativistic Hartree-Fock-Bogoliubov (RHF/RHFB) calculations. Later literature produced Dirac-sea-stable iterative coordinate-space Dirac solvers (inverse Hamiltonian method, preconditioned conjugate gradient, spectral and Wilson-type discretizations) and realized self-consistent three-dimensional (3D) coordinate-space covariant density functional theory, but only with local potentials; exchange correlations were brought into 3D coordinate space by localizing them (relativistic optimized effective potential, Fierz-transformed or localized Fock terms), while genuine RHF itself remains confined to basis expansions with spherical or axial symmetry. The surviving open problem is to construct and characterize a fully self-consistent coordinate-space RHF/RHFB solver that treats the Fock potential directly as a nonlocal kernel, for instance via the nonlocalized ITS algorithm: demonstrate it with density-dependent meson-nucleon couplings against basis-RHF benchmarks and establish stability with respect to the Dirac sea inside the self-consistency loop; the source work further motivates extension to deformed (including triaxial) and weakly bound nuclei where continuum coupling matters, and insight into numerical parameters and computational cost relative to basis-expansion and localization-based alternatives.

Background

Covariant density functional theory (CDFT), also called the relativistic mean-field (RMF) approach, describes atomic nuclei by nucleons moving in Lorentz scalar and vector fields generated by meson exchange or point couplings. In its usual Hartree form the exchange (Fock) contributions of the interaction are neglected and the mean fields are local functions of position; the resulting single-particle Dirac equation can be solved in coordinate space, e.g., by the shooting method. The density-dependent relativistic Hartree-Fock (DDRHF) theory retains the Fock terms of density-dependent meson-nucleon couplings, which brings in pion-exchange and rho-tensor physics and improves the description of shell evolution and spin-isospin properties; the price is that the single-particle Dirac equation acquires a nonlocal potential U(r,r′)U(\boldsymbol r,\boldsymbol r'), turning it into a set of coupled integro-differential equations that cannot be solved by the conventional shooting method without enlarging the system. The imaginary time step (ITS) method — repeatedly applying the evolution operator exp⁡(−δt h^)\exp(-\delta t\,\hat h) to trial wave functions with Gram–Schmidt reorthonormalization so that all but the lowest components are damped — is the workhorse of three-dimensional coordinate-space solvers for nonrelativistic Skyrme Hartree–Fock. For the Dirac equation the spectrum is unbounded below because of the negative-energy continuum (the Dirac sea), so a naive imaginary-time evolution collapses (the Dirac-sea 'tsunami' or variational collapse). Zhang, Liang, and Meng showed that this is avoided by evolving the Schrödinger-like equation for the upper component of the Dirac spinor (a second-order equation whose effective potentials are state- and energy-dependent), in which Fermi-sea and Dirac-sea solutions are separated; in the 2009 work cited here they extended this to nonlocal potentials, demonstrated that the ITS evolution is equivalent with or without localization (i.e., with or without converting the integro-differential system into a higher-dimensional local one), and recommended the nonlocalized algorithm for simplicity and efficiency. The demonstration used a Gaussian nonlocality of width γ\gamma multiplying a Woods–Saxon radial shape, with parameters 'chosen tentatively' to give a reasonable single-particle spectrum in 16^{16}O, inside a spherical box of radius 20 fm with radial mesh 0.1 fm; the authors concluded that the work 'demonstrates the possibility of the ITS method in the relativistic Hartree-Fock theory' and motivated deformed exotic nuclei through the relativistic continuum Hartree–Bogoliubov and DDRHFB programs, where deformed coupled-channel coordinate-space equations are notoriously difficult. After 2009 the field developed along three adjacent but distinct lines: (i) Dirac-sea-stable iterative coordinate-space Dirac solvers — the inverse Hamiltonian method of Hagino and Tanimura, which maximizes the expectation value of the inverse Dirac Hamiltonian instead of minimizing the energy, and later a preconditioned conjugate-gradient method — together with fixes for the fermion-doubling (spurious-state) problem of finite-difference Dirac discretizations, in which naive lattice differentiation of the first-order Dirac operator produces unphysical duplicate solutions; (ii) fully self-consistent 3D mesh CDFT with local point-coupling functionals (Tanimura, Hagino, and Liang 2015; Ren, Zhang, and Meng 2017 and subsequent 3D-lattice and time-dependent CDFT applications), which made arbitrarily deformed coordinate-space calculations routine but only for local potentials; and (iii) RHF itself, which remained confined to basis expansions — spherical relativistic Hartree-Fock in oscillator or Dirac Woods–Saxon bases, extended to axially deformed nuclei in the Dirac Woods–Saxon basis by Geng, Xiang, Sun, and Long (2020) and to axially deformed RHFB by Geng and Long (2022). Meanwhile the exchange physics in coordinate space was accessed by localization strategies: the localized form of Fock terms (Liang, Zhao, Ring, Roca-Maza, and Meng 2012), and the exact-exchange relativistic density functional theory (Ex-RDFT) of Zhao, Ren, Zhao, and Park (2023), in which the orbital-dependent exchange energy generates fully local Lorentz scalar and vector Kohn–Sham potentials via the relativistic optimized effective potential (ROEP) method; Zhao, Ren, Zhao, and Yoshida (2025) solved this Ex-RDFT in 3D coordinate space and applied it to triaxial neutron-rich Ru isotopes, explicitly noting that the complicated nonlocal potentials of RHF are what keep RHF limited to spherical and axial symmetry, and Dong, Ren, Zhao, and Niu (2026) extended 3D-lattice CDFT to a functional with localized exchange terms. Notably, none of these later developments solves the genuinely nonlocal RHF/RHFB equations directly on a mesh: the nonlocality is either avoided by construction (local and OEP-based functionals) or handled in a basis.

Problem Statement

Construct and characterize a fully self-consistent coordinate-space solver for the relativistic Hartree-Fock (RHF) or relativistic Hartree-Fock-Bogoliubov (RHFB) equations that treats the Fock potential directly as a nonlocal kernel U(r,r′)U(\boldsymbol r,\boldsymbol r') built self-consistently from density-dependent meson-nucleon couplings (including their rearrangement contributions, the extra mean-field terms generated by the density dependence of the couplings), rather than localizing it, expanding in a basis, or replacing it by local optimized-effective-potential or Fierz-transformed forms (a Fierz transformation here rearranges the nonlocal exchange term into equivalent local density-dependent fields). The route proposed by the source work is the nonlocalized imaginary time step (ITS) algorithm: ITS evolution of the Schrödinger-like equation for the upper component with the nonlocal terms applied as integral kernels; equivalent direct coordinate-space treatments are admissible. Within this objective, a solution must demonstrate a working self-consistent RHF or RHFB calculation with a density-dependent functional, benchmarked against existing basis-expansion RHF/RHFB results (e.g., total binding energies, charge radii, and single-particle spectra for spherical nuclei) to establish correctness, and must establish stability of the iterative evolution with respect to the Dirac sea (no variational collapse into negative-energy states) throughout the self-consistency iterations. The source work further motivates extension beyond spherical symmetry to deformed nuclei — including triaxial and weakly bound nuclei near the dripline (the limit where nuclei cease to be bound), where box-enclosed continuum coupling matters — as well as insight into the numerical parameters (mesh size, box size, time step or iteration parameters) and the computational cost relative to basis-expansion RHF and to the localization-based coordinate-space alternatives; these are directions the source discusses and useful aspects of the characterization, not acceptance conditions that must be jointly satisfied. Progress on only part of the objective (for example, a spherical-only demonstration with tentative potentials, as in the source work itself) is valuable but does not complete the problem as posed.

References

  1. Ying Zhang, Haozhao Liang, Jie Meng, "Solving the Dirac equation with nonlocal potential by Imaginary Time Step method", Chin. Phys. Lett. 26 (2009) 092401. DOI: 10.1088/0256-307X/26/9/092401; https://arxiv.org/abs/0910.4078
  2. Ying Zhang, Haozhao Liang, Jie Meng, "Avoid the Tsunami of the Dirac sea in the Imaginary Time Step method", Int. J. Mod. Phys. E 19 (2010) 55. DOI: 10.1142/S0218301310014637; https://arxiv.org/abs/0905.2505
  3. K. Hagino, Y. Tanimura, "Iterative solution of a Dirac equation with an inverse Hamiltonian method", Phys. Rev. C 82 (2010) 057301. DOI: 10.1103/PhysRevC.82.057301; https://arxiv.org/abs/1008.4995
  4. Y. Tanimura, K. Hagino, P. Ring, "Application of the inverse Hamiltonian method to Hartree-Fock-Bogoliubov calculations", Phys. Rev. C 88 (2013) 017301. DOI: 10.1103/PhysRevC.88.017301; https://arxiv.org/abs/1304.4697
  5. Y. Tanimura, K. Hagino, H. Z. Liang, "3D mesh calculations for covariant density functional theory", Prog. Theor. Exp. Phys. 2015 (2015) 073D01. DOI: 10.1093/ptep/ptv083; https://arxiv.org/abs/1411.7804
  6. Z. X. Ren, S. Q. Zhang, J. Meng, "Solving Dirac equations on a 3D lattice with inverse Hamiltonian and spectral methods", Phys. Rev. C 95 (2017) 024313. DOI: 10.1103/PhysRevC.95.024313; https://arxiv.org/abs/1612.09429
  7. B. Li, Z. X. Ren, P. W. Zhao, "Efficient solution for the Dirac equation in 3D lattice space with the conjugate gradient method", Phys. Rev. C 102 (2020) 044307. DOI: 10.1103/PhysRevC.102.044307; https://arxiv.org/abs/2007.09414
  8. Ying Zhang, Yuxuan Bao, Hong Shen, Jinniu Hu, "Resolving the spurious-state problem in the Dirac equation with the finite-difference method", Phys. Rev. C 106 (2022) L051303. DOI: 10.1103/PhysRevC.106.L051303; https://arxiv.org/abs/2211.04652
  9. Haozhao Liang, Pengwei Zhao, Peter Ring, Xavier Roca-Maza, Jie Meng, "Localized form of Fock terms in nuclear covariant density functional theory", Phys. Rev. C 86 (2012) 021302(R). DOI: 10.1103/PhysRevC.86.021302; https://arxiv.org/abs/1207.6206
  10. Jing Geng, Jian Xiang, Bao Yuan Sun, Wen Hui Long, "Relativistic Hartree-Fock model for axially deformed nuclei", Phys. Rev. C 101 (2020) 064302. DOI: 10.1103/PhysRevC.101.064302; https://doi.org/10.1103/PhysRevC.101.064302
  11. Jing Geng, Wen Hui Long, "Relativistic Hartree-Fock-Bogoliubov model for axially deformed nuclei", Phys. Rev. C 105 (2022) 034329. DOI: 10.1103/PhysRevC.105.034329; https://arxiv.org/abs/2202.00482
  12. Qiang Zhao, Zhengxue Ren, Pengwei Zhao, Tae-Sun Park, "Accurate relativistic density functional for exchange energy of atomic nuclei", Phys. Lett. B 841 (2023) 137913. DOI: 10.1016/j.physletb.2023.137913; https://arxiv.org/abs/2304.12655
  13. Qiang Zhao, Zhengxue Ren, Pengwei Zhao, Kenichi Yoshida, "Exact-exchange relativistic density functional theory in three-dimensional coordinate space", Phys. Lett. B 860 (2025) 139196. DOI: 10.1016/j.physletb.2024.139196; https://arxiv.org/abs/2412.08132
  14. Z. Y. Dong, Z. X. Ren, Q. Zhao, Z. M. Niu, "Octupole deformation in even-even Ra isotopes from covariant density functional theory with localized exchange terms in a three-dimensional lattice space", arXiv:2607.12633 (2026). DOI: 10.48550/arXiv.2607.12633; https://arxiv.org/abs/2607.12633