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Exact solutions of the nuclear shell-model secular problem: Discrete Non-Orthogonal Shell Model within a Variation After Projection approach

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Pith's one-line read This paper claims that a small set of non-orthogonal Slater determinants, optimized by variation after angular-momentum projection, reproduces exact shell-model eigensolutions; in $^{78}$Ni it converges to $-372.73275$ MeV, below the…

desk verdict Real variational progress on the shell-model secular problem, but the 'exact recovery' claim overstates what the reported numbers show. read the letter →

arxiv 2507.09073 v2 pith:VT43EC3U submitted 2025-07-11 nucl-th

Duy Duc Dao , Frédéric Nowacki This is my paper · ORCID
classification nucl-th
keywords NuclearStructureNon-orthogonalShellModelVariationafterProjectionSymmetrybreakingandrestorationSlaterdeterminantsPairingcorrelations78NiBackbending
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that a small set of non-orthogonal Slater determinants, chosen by minimizing the angular-momentum-projected energy, reproduces the exact solutions of the nuclear shell-model secular problem. In sd-shell nuclei and in the yrast band of $^{48}$Cr, the variation-after-projection energies match full shell-model diagonalization to within a few keV, including the backbending caused by pairing. In $^{78}$Ni, whose full $pf$--$sdg$ valence space contains about $2\times 10^{11}$ Slater determinants, the method converges to $-372.73275$ MeV, lower than the $10p10h$ Lanczos value ($-372.71668$ MeV) and lower than the exponentially extrapolated estimate ($-372.72850$ MeV). If correct, this means exact-quality nuclear-structure calculations no longer require diagonalizing the full shell-model space: a few dozen optimized determinants can carry the same physics, and a previously formal theorem on discrete non-orthogonal spanning sets becomes a practical numerical tool.

What carries the argument

The machinery is the Discrete Non-Orthogonal Shell Model in its variation-after-projection variant (DNO-SM(VAP)). Trial states are linear combinations $|\psi^{J\pi}_n\rangle = \sum_{q,K} C^{J\pi}_{n,qK}\hat P^J_{MK}\hat P^\pi|\phi_q\rangle$ of non-orthogonal Slater determinants projected onto good angular momentum $J$ and parity $\pi$. Each determinant is written in Thouless form $|\phi_q\rangle = \mathcal N_0 e^{\sum_{ij} Z^{(q)}_{ij} a_i^\dagger a_j}|\phi^{(q)}_0\rangle$, with $Z^{(q)}$ a skew-symmetric complex matrix, and the projected energy $E = \langle\psi|H|\psi\rangle/\langle\psi|\psi\rangle$ is minimized with respect to both the mixing coefficients $C$ and the matrices $Z$. The stationarity condition is a generalized Brillouin condition, meaning the energy is stable against one-particle--one-hole excitations of all determinants in the set, and a quasi-Newton algorithm carries out the minimization. A greedy basis-selection procedure grows the non-orthogonal set from the Hamiltonian itself, so parity and rotational symmetry are built in from the start rather than imposed afterward; this is the mechanism that turns the formal spanning theorem into a converging numerical solver.

What would settle it

An independent calculation of the $^{78}$Ni ground state with the same PFSDG-U interaction that reaches an energy below $-372.73275$ MeV, for example a Lanczos diagonalization carried beyond the $10p10h$ truncation or a different exact method in the full $pf$--$sdg$ space, would refute the claim that the VAP result is the exact eigensolution. In the small spaces where exact diagonalization is available, restarting the VAP optimization from many random initial Slater determinants and finding multiple distinct converged energies would reveal local-minimum trapping and undermine the global-convergence assumption.

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Extended reading notes

Core claim

The central discovery is that symmetry-restored non-orthogonal Slater determinants exactly span the full shell-model valence space. Applying the Ritz variational principle to angular-momentum- and parity-projected linear combinations of Slater determinants, and optimizing the determinants themselves at the same time, yields energies indistinguishable from exact shell-model diagonalization wherever that comparison can be made. In $^{48}$Cr the entire yrast band is reproduced with 12--57 determinants, capturing proton--neutron pairing without breaking particle number. In $^{78}$Ni the converged ground-state energy $-372.73275$ MeV lies below the value obtained by the largest Lanczos diagonalization and below the extrapolated Lanczos limit; the paper reads this as the first variational recovery of the exact diagonalization at this scale. This is presented as numerical proof that a completeness theorem for discrete non-orthogonal bases is realized in realistic shell-model calculations, and that variation after projection is numerically equivalent to including very high particle--hole excitations on top of a deformed or spherical reference state.

Load-bearing premise

The load-bearing premise is that the numerical minimization of the projected energy reaches the true ground state rather than stopping in a local minimum; for $^{78}$Ni, where no independent full diagonalization exists, this is inferred from the plateau of the convergence curve rather than proved.

Editorial extensions

If this is right

  • Exact shell-model ground and yrast states can be represented by a dozen to a few dozen non-orthogonal Slater determinants rather than by the $\sim 2\times 10^{11}$ orthonormal basis states of a conventional M-scheme diagonalization.
  • Variation after projection with non-orthogonal Slater determinants is numerically equivalent to including very high particle--hole excitations on top of deformed or spherical reference states, which explains why it recovers the correlation energy missing in simpler projected mean-field treatments.
  • Strong proton--neutron pairing, including the $^{48}$Cr backbending, is fully captured without breaking particle number, so the method provides a particle-number-conserving alternative to symmetry-restored quasiparticle approaches.
  • Because well-deformed heavy nuclei show nearly perfect $J(J+1)$ rotational spectra, where pairing is weaker than deformation, the present variational strategy is expected to be even more efficient there, opening exact-quality shell-model calculations in the heavy-mass region.
  • In the $^{78}$Ni region, where shape coexistence is pervasive, the demonstrated convergence allows low-lying states and effective interactions to be studied without waiting for larger Lanczos diagonalizations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The authors do not state it, but if the $^{78}$Ni VAP energy is indeed the exact eigensolution, the extrapolated Lanczos value ($-372.72850$ MeV) underestimates the binding energy by about 4 keV, implying the truncation error of the $10p10h$ diagonalization is slightly larger than the exponential extrapolation assumed.
  • A natural and testable extension is to compute excited states and electromagnetic transitions in $^{78}$Ni with the same VAP determinant set; with no full diagonalization available, agreement with the measured low-lying spectrum would independently test the variational completeness claim.
  • The results suggest a more general compression principle: the low-energy sector of a strongly correlated fermionic Hamiltonian can be spanned by far fewer non-orthogonal determinants than the dimension of the full Fock space, a property that could transfer to other fermionic many-body problems beyond the nuclear shell model, although the paper itself does not pursue this.
  • Running the VAP optimization from many random initial reference determinants and checking that all trajectories land on the same energy plateau would quantify the local-minimum risk; the paper does not report such a stability analysis, so this remains an open check.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper introduces the Discrete Non-Orthogonal Shell Model with variation after projection (DNO-SM(VAP)), a variational method that represents nuclear wave functions as superpositions of angular-momentum-projected, non-orthogonal Slater determinants obtained by a hybrid quasi-Newton optimization of the projected energy. The authors benchmark the method in the sd shell (20Ne, 24Mg, 28Si, 26Al) with the USDB interaction, in 48Cr with KB3, and in 78Ni with PFSDG-U in the pf-sdg space. They report that the method reproduces exact shell-model ground-state energies in the sd shell to within 0.7 keV and the 48Cr yrast band to within 14 keV, and that for 78Ni the DNO-SM(VAP) energy (-372.73275 MeV) is lower than the 10p10h Lanczos result (-372.71668 MeV) and the exponential extrapolation (-372.72850 MeV). Based on these results, the paper claims that the Broeckhove-Deumens theorem is numerically realized and that exact diagonalization is 'fully recovered' in 78Ni.

Significance. If the claim of exact recovery were justified, the method would be a significant algorithmic advance for shell-model calculations in spaces beyond the reach of direct diagonalization. The small-space benchmarks are encouraging and give the method credibility as a variational tool: the sd-shell ground states are within about 0.7 keV of exact results and the 48Cr yrast spectrum is qualitatively reproduced with only tens of determinants. Independent external comparisons against exact shell-model diagonalization in small spaces are a strength of the paper. However, the central exactness claim is not supported by the data: in the small spaces where exact results are known, the variational energies are upper bounds and deviate from exact values by up to 14 keV, and in 78Ni no exact solution exists to validate the claim. The paper therefore presents a promising method with an overreaching interpretation.

major comments (4)
  1. [Section 3, Table 1] The DNO-SM(VAP) ground-state energies in Table 1 do not reproduce exact shell-model energies: 24Mg gives -87.10405 MeV versus -87.10445 MeV, 28Si gives -135.86003 versus -135.86073 MeV, and 26Al gives -105.74901 versus -105.74934 MeV. Since these are variational upper bounds, the statement in Section 3 that the approach 'captures all correlations to recover exactly the SM solution' is contradicted by the reported numbers. Please quantify these residuals and specify a convergence criterion (e.g., energy plateaus with respect to the number of determinants and optimization tolerances) that would support the 'exact recovery' claim.
  2. [Section 4, Table 2 and Fig. 2] For the 48Cr yrast band, the DNO-SM(VAP) energies for the 2+, 4+, 6+, 8+, 10+, and 12+ states lie 8-14 keV above the exact shell-model values (e.g., 2+ is -32.135 MeV vs -32.148 MeV), while only the 0+, 14+, and 16+ states match within the quoted precision. The text states that 'the two curves becomes indistinguishable with a perfect matching' and that proton-neutron pairing correlations are 'fully incorporated'. These statements are not supported by the table. Please provide a quantitative comparison of discrepancies and show whether increasing the number of determinants reduces them systematically.
  3. [Section 5, Fig. 3] The headline claim that 'this is the first time using a variational method, the exact diagonalization is fully recovered' in 78Ni is not established. The DNO-SM(VAP) energy -372.73275 MeV is a variational upper bound on the exact ground-state energy; the 10p10h Lanczos value -372.71668 MeV is also an upper bound in a truncated space, and the exponential extrapolation -372.72850 MeV is an estimate with an assumed functional form. The difference of 4-16 keV is comparable to the residuals shown in Table 2 where exact results are known (up to 14 keV). Because no independent exact diagonalization exists for 78Ni, the word 'exact' is unsubstantiated. Please reframe the claim as 'the variational approximation improves on the largest Lanczos diagonalization' and provide a convergence analysis with error estimates.
  4. [Section 2, Eqs. (3)-(7) and algorithm description] The hybrid VAP optimization relies on a quasi-Newton local minimization of the projected energy functional, but the paper provides no proof or numerical evidence that the algorithm reaches a global minimum, no explicit convergence thresholds, no restarts from different initial conditions to check for local minima, and no condition-number or precision checks for the generalized eigenvalue problem (3) with potentially near-linear-dependent non-orthogonal states. Since the 'exact recovery' claim depends on global convergence of the optimization, these diagnostics are load-bearing. Please add them or explicitly qualify the results as variational approximations.
minor comments (6)
  1. [Abstract and Section 3] The abstract states 'exact shell-model solutions are obtained', but the results in Table 1 show differences of a few tenths of a keV from exact energies. Please use more qualified language, such as 'accurate to within 0.7 keV' or 'variational approximations'.
  2. [Table 1] The 26Al row appears to have a typographical issue: the entries in the DNO-SM(VAP) and Exact SM columns are presented as '105.74901' and '105.74934' without a leading minus sign. Please correct the formatting.
  3. [Section 4, Fig. 2 caption and text] The phrase 'the two curves becomes indistinguishable' has a subject-verb agreement error; it should be 'the two curves become indistinguishable'. Also, the caption uses 'DNO-SM(VAP)' and 'SM' without defining them at first use in the caption.
  4. [Section 5, Fig. 3] The extrapolated value -372.72850 MeV assumes exponential convergence in the number of NpNh excitations. Please state the uncertainty of this extrapolation or show an alternative extrapolation (e.g., with different fitting ranges) to support the comparison.
  5. [References] Reference [33] contains a stray '3.' after the DOI (Phys. Rev. Lett. 25 (1970) 782. 3.); please correct this citation.
  6. [Notation] The symbols 'PA V' and 'V AP' appear with inconsistent spacing throughout the text (e.g., 'PA V' vs 'PAV' and 'V AP' vs 'VAP'). Please standardize the notation, e.g., PAV and VAP.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the variational parameters are optimized against the Hamiltonian and benchmarked against independent shell-model diagonalizations, so the central claims do not reduce to their inputs.

full rationale

The paper's core derivation is self-contained. The DNO-SM(VAP) wave function (Eq. 5) is a symmetry-projected superposition of non-orthogonal Slater determinants, and the coefficients and intrinsic states are obtained by minimizing the projected energy (Eq. 6) and satisfying the generalized Brillouin condition (Eq. 7). The Hamiltonian and valence spaces (USDB, KB3, PFSDG-U) are fixed external inputs; no parameter is fitted to the exact shell-model energies used for benchmarking. Small discrepancies with exact SM values in Tables 1 and 2 (e.g., 48Cr yrast states above SM by up to ~14 keV) show that the comparisons are genuine checks rather than forced reproductions. The Broeckhove-Deumens theorem is cited as an external existence argument, not as a substitute for the numerical evidence, and the paper explicitly notes the theorem does not tell how to find the discrete set. The 78Ni claim rests on a variational upper-bound energy that lies below the truncated Lanczos and exponential-extrapolated values; this is an extrapolative claim whose global-minimum assumption is unproven, but that is a correctness risk, not circularity. The only self-citations are to the authors' earlier DNO-SM paper [30] and 254No study [56], used as methodological background and not load-bearing for the new results. Consequently, no step in the derivation reduces by construction to its own inputs.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new physical entities are introduced. The variational parameters are numerous but determined by the Ritz principle; the main hidden assumptions are the global-convergence of the optimization and the validity of the effective interactions.

free parameters (3)
  • Thouless matrix elements Z^{(q)}_{ij} = Optimized via quasi-Newton minimization, not reported numerically
    Complex variational parameters defining each non-orthogonal Slater determinant in Eq. (4); they are minimized against the projected energy, not fitted to the exact SM benchmark energies, so they are not hidden fit constants, but their optimization is central to the method.
  • Mixing amplitudes C^{piJ}_{n,qK} = Eigenvector components of the generalized eigenvalue equation (3), not reported
    Variational coefficients in the wavefunction ansatz Eq. (5).
  • Number of Slater determinants N_VAP = Varies by nucleus, e.g. 50 for 48Cr 0+, about 14 for 78Ni (from Fig. 3)
    Chosen by energy convergence; this is a model-size hyperparameter.
assumptions (4)
  • standard math Broeckhove-Deumens theorem guarantees existence of a countable non-orthogonal subset spanning the space.
    Invoked in Introduction and Appendix A to justify discrete non-orthogonal expansions; however, the theorem requires Gamma dense in H, which is not established for the set of Slater determinants.
  • domain assumption USDB, KB3, and PFSDG-U effective interactions are valid in their respective valence spaces.
    All numerical results depend on these Hamiltonians; the paper does not assess their uncertainties.
  • ad hoc to paper The hybrid VAP optimization (fixing previous states while varying the last added states) converges to the global minimum of the projected energy.
    The paper states this scheme but provides no proof of global convergence; local minima could bias the reported energies.
  • domain assumption Exponential convergence extrapolation for the SM diagonalization in 78Ni.
    Used in Section 5 to estimate the exact SM energy as -372.72850 MeV from the 10p10h value.

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Cite this review

Pith. "Pith review of Exact solutions of the nuclear shell-model secular problem: Discrete Non-Orthogonal Shell Model within a Variation After Projection approach." pith.science (2026). https://pith.science/paper/VT43EC3U

@misc{pith2026250709073,
  author       = {Pith},
  title        = {Pith review of: Exact solutions of the nuclear shell-model secular problem: Discrete Non-Orthogonal Shell Model within a Variation After Projection approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VT43EC3U}},
  note         = {Machine review of arXiv:2507.09073}
}
abstract

We investigate the capacity of non-orthogonal many-body expansions in the resolution of the nuclear shell-model secular problem. Exact shell-model solutions are obtained within the variational principle using non-orthogonal Slater determinants as the variational ansatz. These results numerically prove the realization of the Broeckhove-Deumens theorem on the existence of a discrete set of non-orthogonal wavefunctions that exactly span the full shell-model space for low-lying states of interest. With the angular-momentum variation after projection, pairing correlations are shown to be fully captured by Slater determinants as exemplified in the backbending phenomenon occurred in $^{48}$Cr. The resulting discrete non-orthogonal shell model developed in such variation after projection method is further examined in the case of $^{78}$Ni, an exotic doubly magic nucleus at the edge of currently feasible diagonalization limits. Its ground state binding energy is shown to converge to a lower value than the largest large-scale shell-model diagonalization ever done by the conventional tridiagonal Lanczos method, revealing an outstanding performance of non-orthogonal Slater determinantal wavefunctions to describe the eigensolutions of shell-model Hamiltonians.

Figures

Figures reproduced from arXiv: 2507.09073 by the authors.

Figure 1
Figure 1. Ground state binding energy convergence with the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. γ-excitation energies (Eγ) versus angular momen￾tum (J) comparison between the Angular Momentum Projec￾tion (AMP) of the HF minimum, the DNO-SM(VAP) and the exact SM calculations [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. (top panel), the binding energy convergence is shown for shell-model diagonalizations with respect to the NpNh excita￾tions across Z=28 and N=50 shell gaps. At the 10p10h level representing our actual diagonalization limit, the ground state energy is −372.71668 MeV. The corresponding extrapolated value ground state binding energy is, assuming an exponential convergence behavior [10], found to be −372.72850 MeV. -374… view at source ↗

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Works this paper leans on

62 extracted references · 44 canonical work pages

  1. [1]

    D. L. Hill, J. A. Wheeler, Nuclear constitution and the interpretation of fission phenomena, Phys. Rev. 89 (1953) 1102–1145. doi:10.1103/ PhysRev.89.1102

    work page 1953

  2. [2]

    J. J. Gri ffin, J. A. Wheeler, Collective motions in nuclei by the method of generator coordinates, Phys. Rev. 108 (1957) 311–327. doi:10.1103/ PhysRev.108.311

    work page 1957

  3. [3]

    R. E. Peierls, J. Yoccoz, The collective model of nuclear motion, Proc. Phys. Soc. Sect. A 70 (5) (1957) 381. doi:10.1088/0370-1298/70/ 5/309

    work page doi:10.1088/0370-1298/70/ 1957

  4. [4]

    Yoccoz, On the moments of inertia of nuclei, Proc

    J. Yoccoz, On the moments of inertia of nuclei, Proc. Phys. Soc. Sect. A 70 (5) (1957) 388. doi:10.1088/0370-1298/70/5/310

    work page doi:10.1088/0370-1298/70/5/310 1957

  5. [5]

    Peierls, D

    R. Peierls, D. Thouless, Variational approach to collective motion, Nucl. Phys. 38 (1962) 154–176. doi:10.1016/0029-5582(62)91025-8

    doi:10.1016/0029-5582(62)91025-8 1962

  6. [6]

    A. F. R. de Toledo Piza, E. J. V . de Passos, D. Galetti, M. C. Nemes, M. M. Watanabe, Properties of gri ffin-hill-wheeler spaces, Phys. Rev. C 15 (1977) 1477–1482. doi:10.1103/PhysRevC.15.1477

    work page doi:10.1103/physrevc.15.1477 1977

  7. [7]

    E. J. V . de Passos, A. F. R. de Toledo Piza, Properties of gri ffin-hill- wheeler spaces. ii. one-parameter and two-conjugate-parameter families of generator states, Phys. Rev. C 21 (1980) 425–438. doi:10.1103/ PhysRevC.21.425

    work page 1980

  8. [8]

    Broeckhove, E

    J. Broeckhove, E. Deumens, A mathematical foundation for discretisation techniques in the generator coordinate method, Z. Phys. A 292 (1979) 243–247. doi:10.1007/BF01547468

    work page doi:10.1007/bf01547468 1979

Show all 62 references
  1. [9]

    Arickx, J

    F. Arickx, J. Broeckhove, E. Deumens, P. Van Leuven, Variational dis- cretization: A new algorithm for the generator coordinate method, J. Comp. Phys. 39 (2) (1981) 272–281. doi:10.1016/0021-9991(81) 90152-2

    1981 doi

  2. [10]

    Caurier, G

    E. Caurier, G. Mart ´ınez-Pinedo, F. Nowacki, A. Poves, A. P. Zuker, The shell model as a unified view of nuclear structure, Rev. Mod. Phys. 77 (2005) 427–488. doi:10.1103/RevModPhys.77.427

    2005 doi

  3. [11]

    C. W. Johnson, W. E. Ormand, P. G. Krastev, Factorization in large-scale many-body calculations, Comp. Phys. Comm. 184 (12) (2013) 2761–

    2013

  4. [12]

    Brown, W

    B. Brown, W. Rae, The shell-model code nushellx@msu, Nucl. Data Sheets 120 (2014) 115–118. doi:10.1016/j.nds.2014.07.022. 6

    2014 doi

  5. [13]

    Shimizu, T

    N. Shimizu, T. Mizusaki, Y . Utsuno, Y . Tsunoda, Thick-restart block lanc- zos method for large-scale shell-model calculations, Comp. Phys. Comm. 244 (2019) 372–384. doi:10.1016/j.cpc.2019.06.011

    2019 doi

  6. [14]

    Nowacki, A

    F. Nowacki, A. Obertelli, A. Poves, The neutron-rich edge of the nuclear landscape: Experiment and theory, Prog. Part. Nucl. Phys. 120 (2021) 103866. doi:10.1016/j.ppnp.2021.103866

    2021

  7. [15]

    J. A. Sheikh, J. Dobaczewski, P. Ring, L. M. Robledo, C. Yannouleas, Symmetry restoration in mean-field approaches, J. Phys. G: Nucl. Part. Phys. 48 (12) (2021) 123001. doi:10.1088/1361-6471/ac288a

    2021 doi

  8. [16]

    K. W. Schmid, F. Gr ¨ummer, A. Faessler, Nuclear structure theory in spin- and number-conserving quasiparticle configuration spaces: General for- malism, Phys. Rev. C 29 (1984) 291–307. doi:10.1103/PhysRevC. 29.291

    1984 doi

  9. [17]

    K. W. Schmid, F. Gr ¨ummer, A. Faessler, Nuclear structure theory in spin- and number-conserving quasiparticle configuration spaces: First applica- tions, Phys. Rev. C 29 (1984) 308–323. doi:10.1103/PhysRevC.29. 308

    1984 doi

  10. [18]

    Schmid, On the use of general symmetry-projected hartree–fock–bogoliubov configurations in variational approaches to the nuclear many-body problem, Prog

    K. Schmid, On the use of general symmetry-projected hartree–fock–bogoliubov configurations in variational approaches to the nuclear many-body problem, Prog. Part. Nucl. Phys. 52 (2) (2004) 565–633. doi:10.1016/j.ppnp.2004.02.001

    2004 doi

  11. [19]

    Honma, T

    M. Honma, T. Mizusaki, T. Otsuka, Nuclear shell model by the quantum monte carlo diagonalization method, Phys. Rev. Lett. 77 (1996) 3315–

    1996

  12. [20]

    Otsuka, M

    T. Otsuka, M. Honma, T. Mizusaki, N. Shimizu, Y . Utsuno, Monte carlo shell model for atomic nuclei, Prog. Part. Nucl. Phys. 47 (1) (2001) 319–

    2001

  13. [21]

    Z.-C. Gao, M. Horoi, Y . S. Chen, Improved basis selection for the pro- jected configuration interaction method applied to medium-heavy nuclei, Phys. Rev. C 80 (2009) 034325. doi:10.1103/PhysRevC.80.034325

    2009 doi

  14. [22]

    Z.-C. Gao, M. Horoi, Angular momentum projected configuration in- teraction with realistic hamiltonians, Phys. Rev. C 79 (2009) 014311. doi:10.1103/PhysRevC.79.014311

    2009 doi

  15. [23]

    Shimizu, T

    N. Shimizu, T. Abe, Y . Tsunoda, Y . Utsuno, T. Yoshida, T. Mizusaki, M. Honma, T. Otsuka, New-generation monte carlo shell model for the k computer era, Prog. Theor. Exp. Phys. 2012 (1) (2012) 01A205. doi: 10.1093/ptep/pts012

    2012 doi

  16. [24]

    Z.-C. Gao, M. Horoi, Y . S. Chen, Variation after projection with a tri- axially deformed nuclear mean field, Phys. Rev. C 92 (2015) 064310. doi:10.1103/PhysRevC.92.064310

    2015 doi

  17. [25]

    T. Ya, Y . He, Z.-C. Gao, J.-Q. Wang, Y . S. Chen, Implementation of the variation-after-projection approach in calculations with a time-odd hartree-fock mean field, Phys. Rev. C 95 (2017) 064307. doi:10.1103/ PhysRevC.95.064307

    2017

  18. [26]

    Wang, Z.-C

    J.-Q. Wang, Z.-C. Gao, Y .-J. Ma, Y . S. Chen, New algorithm in the vari- ation after projection calculations for non-yrast nuclear states, Phys. Rev. C 98 (2018) 021301. doi:10.1103/PhysRevC.98.021301

    2018 doi

  19. [27]

    Shimizu, Y

    N. Shimizu, Y . Tsunoda, Y . Utsuno, T. Otsuka, Variational approach with the superposition of the symmetry-restored quasiparticle vacua for nu- clear shell-model calculations, Phys. Rev. C 103 (2021) 014312. doi: 10.1103/PhysRevC.103.014312

    2021 doi

  20. [28]

    S ´anchez-Fern´andez, B

    A. S ´anchez-Fern´andez, B. Bally, T. R. Rodr ´ıguez, Variational approx- imations to exact solutions in shell-model valence spaces: Systematic calculations in the sd shell, Phys. Rev. C 104 (2021) 054306. doi: 10.1103/PhysRevC.104.054306

    2021 doi

  21. [29]

    Bally, A

    B. Bally, A. S ´anchez-Fern´andez, T. R. Rodr ´ıguez, Variational approxi- mations to exact solutions in shell-model valence spaces: Calcium iso- topes in the p f shell, Phys. Rev. C 100 (2019) 044308. doi:10.1103/ PhysRevC.100.044308

    2019

  22. [30]

    D. D. Dao, F. Nowacki, Nuclear structure within a discrete nonorthogonal shell model approach: New frontiers, Phys. Rev. C 105 (2022) 054314. doi:10.1103/PhysRevC.105.054314

    2022 doi

  23. [31]

    Broeckhove, E

    J. Broeckhove, E. Deumens, A. Faessler, A. Plastino, Multi-configuration hartree-fock theory in nuclei, Z. Phys. 220 (1969) 88–100. doi:10. 1007/BF01394412

    1969

  24. [32]

    Satpathy, Q

    L. Satpathy, Q. Ho-Kim, Multiconfiguration field theory in nuclei, Phys. Rev. Lett. 25 (1970) 123–125. doi:10.1103/PhysRevLett.25.123

    1970 doi

  25. [33]

    Satpathy, Q

    L. Satpathy, Q. Ho-Kim, Multiconfiguration field theory in nuclei, Phys. Rev. Lett. 25 (1970) 782–782. doi:10.1103/PhysRevLett.25.782. 3

    1970 doi

  26. [34]

    Ho-Kim, Multiconfigurational self-consistent calculation in light de- formed nuclei, Phys

    Q. Ho-Kim, Multiconfigurational self-consistent calculation in light de- formed nuclei, Phys. Rev. C 4 (1971) 1077–1086. doi:10.1103/ PhysRevC.4.1077

    1971

  27. [35]

    Robin, et al., Description of nuclear systems with a self-consistent configuration-mixing approach: Theory, algorithm, and application to the 12C test nucleus, Phys

    C. Robin, et al., Description of nuclear systems with a self-consistent configuration-mixing approach: Theory, algorithm, and application to the 12C test nucleus, Phys. Rev. C 93 (2016) 024302. doi:10.1103/ PhysRevC.93.024302

    2016

  28. [36]

    Robin, et al., Description of nuclear systems with a self-consistent configuration-mixing approach

    C. Robin, et al., Description of nuclear systems with a self-consistent configuration-mixing approach. ii. application to structure and reactions in even-even sd-shell nuclei, Phys. Rev. C 95 (2017) 044315. doi:10. 1103/PhysRevC.95.044315

    2017

  29. [37]

    Matsumoto, Y

    M. Matsumoto, Y . Tanimura, K. Hagino, Extension of the generator coor- dinate method with basis optimization, Phys. Rev. C 108 (2023) L051302. doi:10.1103/PhysRevC.108.L051302

    2023 doi

  30. [38]

    Nowacki, A

    F. Nowacki, A. Poves, E. Caurier, B. Bounthong, Shape coexistence in 78Ni as the portal to the fifth island of inversion, Phys. Rev. Lett. 117 (2016) 272501. doi:10.1103/PhysRevLett.117.272501

    2016 doi

  31. [39]

    Taniuchi, C

    R. Taniuchi, C. Santamaria, P. Doornenbal, et al., 78ni revealed as a dou- bly magic stronghold against nuclear deformation, Nature 569 (2019) 53–58. doi:10.1038/s41586-019-1155-x

    2019 doi

  32. [40]

    P. Ring, P. Schuck, The nuclear many-body problem, Springer-Verlag, 1980

    1980

  33. [41]

    Thouless, Stability conditions and nuclear rotations in the hartree-fock theory, Nucl

    D. Thouless, Stability conditions and nuclear rotations in the hartree-fock theory, Nucl. Phys. 21 (1960) 225–232.doi:10.1016/0029-5582(60) 90048-1

    1960 doi

  34. [42]

    Fukutome, Theory of resonating quantum fluctuations in a fermion system: Resonating hartree-fock approximation, Prog

    H. Fukutome, Theory of resonating quantum fluctuations in a fermion system: Resonating hartree-fock approximation, Prog. Theor. Phys. 80 (3) (1988) 417–432. doi:10.1143/PTP.80.417

    1988 doi

  35. [43]

    Ne ff, H

    T. Ne ff, H. Feldmeier, Clustering and other exotic phonomena in nu- clei, Eur. Phys. J. Spec. Top. 156 (2008) 69–92. doi:10.1140/epjst/ e2008-00609-y

    2008 doi

  36. [44]

    D. Liu, J. Nocedal, On the limited memory bfgs method for large scale optimization, Math. Progr. 45 (1989) 503–528. doi:10.1007/ BF01589116

    1989

  37. [45]

    Norcedal, S

    J. Norcedal, S. J. Wright, Numerical Optimization, Springer-Verlag,

  38. [46]

    B. A. Brown, W. A. Richter, New “usd” hamiltonians for the sd shell, Phys. Rev. C 74 (2006) 034315. doi:10.1103/PhysRevC.74.034315

    2006 doi

  39. [47]

    A. P. Zuker, A. Poves, F. Nowacki, S. M. Lenzi, Nilsson-su3 self- consistency in heavy n = z nuclei, Phys. Rev. C 92 (2015) 024320. doi:10.1103/PhysRevC.92.024320

    2015 doi

  40. [48]

    Bender, Going beyond the self-consistent mean-field with the symmetry-restored generator coordinate method, Eur

    M. Bender, Going beyond the self-consistent mean-field with the symmetry-restored generator coordinate method, Eur. Phys. J. Spec. Top. 156 (2008) 217–228. doi:10.1140/epjst/e2008-00619-9

    2008 doi

  41. [49]

    Talmi, Generalized seniority and structure of semi-magic nuclei, Nucl

    I. Talmi, Generalized seniority and structure of semi-magic nuclei, Nucl. Phys. A 172 (1) (1971) 1–24.doi:10.1016/0375-9474(71)90112-6

    1971 doi

  42. [50]

    Poves, G

    A. Poves, G. Martinez-Pinedo, Pairing and the structure of the pf-shell n∼z nuclei, Phys. Lett. B 430 (3) (1998) 203–208. doi:10.1016/ S0370-2693(98)00538-3

    1998

  43. [51]

    A. Bohr, B. R. Mottelson, Nuclear Structure, V ol. II, W. A. Benjamin, 1975, p. 30

    1975

  44. [52]

    Reiter, et al., Ground-state band and deformation of the z = 102 isotope 254No, Phys

    P. Reiter, et al., Ground-state band and deformation of the z = 102 isotope 254No, Phys. Rev. Lett. 82 (1999) 509–512. doi:10.1103/ PhysRevLett.82.509

    1999

  45. [53]

    P. T. Greenlees, et al., Shell-structure and pairing interaction in super- heavy nuclei: Rotational properties of the z=104 nucleus 256Rf, Phys. Rev. Lett. 109 (2012) 012501. doi:10.1103/PhysRevLett.109. 012501

    2012 doi

  46. [54]

    Seweryniak, et al., Nuclear rotation at the fission limit in 254Rf, Phys

    D. Seweryniak, et al., Nuclear rotation at the fission limit in 254Rf, Phys. Rev. C 107 (2023) L061302. doi:10.1103/PhysRevC.107.L061302

    2023 doi

  47. [55]

    Ripka, The Hartree-Fock Theory of Deformed Light Nuclei, Springer US, 1968, pp

    G. Ripka, The Hartree-Fock Theory of Deformed Light Nuclei, Springer US, 1968, pp. 183–259. doi:10.1007/978-1-4757-0103-6\_3

    1968 doi

  48. [56]

    D. D. Dao, F. Nowacki, First complete description of low-lying spec- troscopy in 254No, arXiv:2409.08210 (2025). doi:10.48550/arXiv. 2409.08210

    2025 doi arXiv

  49. [57]

    Nies, et al., Further evidence for shape coexistence in 79Znm near dou- bly magic 78Ni, Phys

    L. Nies, et al., Further evidence for shape coexistence in 79Znm near dou- bly magic 78Ni, Phys. Rev. Lett. 131 (2023) 222503. doi:10.1103/ PhysRevLett.131.222503

    2023

  50. [58]

    Utsuno, N

    Y . Utsuno, N. Shimizu, T. Otsuka, T. Abe, E fficient computation of hamiltonian matrix elements between non-orthogonal slater determinants, Comp. Phys. Comm. 184 (1) (2013) 102–108. doi:10.1016/j.cpc. 2012.09.002. 7

    2013 doi

  51. [400]

    doi:10.1016/S0146-6410(01)00157-0

    doi

  52. [2006]

    doi:10.1007/978-0-387-40065-5

    doi

  53. [2774]

    doi:10.1016/j.cpc.2013.07.022

    2013 doi

  54. [3318]

    doi:10.1103/PhysRevLett.77.3315

    doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.