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REVIEW 3 major objections 7 minor 82 references

Microscopic description of the fission process including intrinsic excitations. Part I: 240Pu adiabatic and asymmetric fission path within the Schrodinger Collective Intrinsic Model

T0 review · 3 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Overlap-constrained Link and Drop methods build a continuous one-dimensional adiabatic path for 240Pu fission from the ground state through scission and beyond.

desk verdict First practical SCIM adiabatic path with real methodological novelty, but the scission energy balance has a hard internal inconsistency and the V-phase prescription casts a shadow on overlap-derived numbers. read the letter →

arxiv 2608.07121 v1 pith:AHRFGMDA submitted 2026-08-07 nucl-th

P. Carpentier , N. Pillet , R.N. Bernard , L.M. Robledo , D. Lacroix , N. Dubray , D. Regnier , W. Younes This is my paper · ORCID
classification nucl-th MSC 81V35 PACS 24.75.+i21.60.Jz
keywords nuclearfissionSchrodingerCollective-IntrinsicModel240PuadiabaticpathoverlapconstraintsLinkandDropmethodsscissionGognyD1Sinteraction
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 to make the Schrödinger Collective–Intrinsic Model (SCIM) numerically usable for fission by constructing a microscopic, one-dimensional adiabatic path for 240Pu that is continuous and regular in the space of Hartree–Fock–Bogoliubov (HFB) many-body states. Standard constrained paths are smooth in energy but jump between different intrinsic configurations, which spoils the overlap and Hamiltonian kernels that SCIM requires. The authors combine two overlap-constrained algorithms, Link and Drop, into a procedure they call ~P20, stepping through state space at nearly constant overlap distance, and they introduce a collective coordinate c# equal to one elementary overlap step. Along this path they identify scission through simultaneous chemical-potential peaks, a strongly neutron-rich neck, proton odd–even staggering in fragment charge distributions, and a static energy balance with substantial fragment deformation and residual nuclear interaction energies. If the path and its overlap metric are reliable, it supplies the adiabatic foundation for SCIM dynamics with intrinsic excitations and a microscopic picture of scission as an extended region.

What carries the argument

The machinery is the overlap metric: each elementary step between neighboring HFB states is fixed by a constraint on the modulus of the overlap, |⟨Φi|Φi+1⟩|=x0=0.995, rather than on a multipole moment. Link constructs the trajectory between known attractor states; Drop continues the local energy descent without a target state; together they define the collective coordinate c#, one unit per elementary overlap step. The V-phase prescription of Eq. (D58) fixes the sign ambiguity of nearly degenerate (Ω,τ) blocks by choosing the sign that maximizes each block determinant, making the overlap a phase-consistent distance. The numerical evaluation rests on Pfaffian overlap formulas for HFB states in two different two-center harmonic-oscillator representations.

What would settle it

Recompute the ~P20 path with an independent phase convention for the V matrices (for example, fixing signs by continuity of pairing-gap sub-blocks instead of by Eq. D58) and with overlap spacing x0=0.99 and 0.999; if the c#≈495 location of the simultaneous chemical-potential peaks, the neck neutron-to-proton ratio, or the extracted fragment energies shift by more than a few MeV, the scission characterization is an artifact of the construction rather than a robust physical signature.

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Extended reading notes

Core claim

The paper's central claim is that the ~P20 procedure — Link and Drop combined with overlap spacing x0=0.995, an attractor threshold of 0.5, and a stop overlap of 0.9 — produces a continuous, regular one-dimensional adiabatic HFB path for asymmetric fission of 240Pu, running from the ground state through scission and beyond. On this path, the overlap and Hamiltonian kernels satisfy the local-GOA relation with a relative error of at most about 0.46% for the relative-coordinate range s∈[-10,10] in c# units. The paper identifies scission near c#=495 through simultaneous neutron and proton chemical-potential peaks, a neck neutron-to-proton density ratio reaching about 4.5, proton odd–even staggering in the fragment distributions, and a static energy balance with roughly ΔE_def≈26.85 MeV of fragment deformation energy, E_Coulomb≈178.74 MeV of Coulomb interaction energy, and E_int≈153 MeV of total fragment interaction energy.

Load-bearing premise

The load-bearing premise is that the overlap distance between neighboring HFB states, after the V-phase sign prescription and with step parameters x0=0.995, attractor threshold 0.5, and stop overlap 0.9, faithfully measures physical collective distance; if that metric is not stable, the path and every scission conclusion built on it are not either.

Editorial extensions

If this is right

  • SCIM dynamics can now be formulated along a one-dimensional fission path whose overlap and Hamiltonian kernels are regular enough to construct collective potential, inertia, and dissipation terms over the whole trajectory from ground state through scission.
  • Expressed in c#, barrier widths and descent steepness differ from their Q20 parametrization, so collective dynamics computed with the overlap metric will generally differ from dynamics computed with quadrupole parametrization.
  • Scission is characterized as a finite region: before the fragments relax, about 26.85 MeV is stored in their deformation and about 153 MeV of interaction energy remains, of which roughly 178.74 MeV is Coulomb energy contributing to final kinetic energy.
  • Proton odd–even staggering in fragment charge distributions emerges naturally from the HFB pairing content and is localized near the scission region, offering a microscopic origin for the staggering seen in experimental charge yields.
  • The Link and Drop construction is compatible with additional multipole constraints, and the paper sketches a GOA-guided two-dimensional paving, opening a route toward multi-dimensional SCIM paths.

Reading between the lines

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

  • Because the path geometry is fixed by the V-phase prescription and hand-set parameters, an independent multi-dimensional adiabatic path — for example, constrained in both Q20 and Q30 or with a neck operator — would be the natural test of whether the c#≈495 scission markers are intrinsic; the paper does not perform that check.
  • If c# is accepted as the true collective distance, collective inertia and dissipation expressed in c# will differ from Q20-based values, so fission lifetimes and fragment yields predicted by SCIM dynamics could shift relative to standard TDGCM calculations.
  • The large residual nuclear interaction at scission implies that post-scission dynamics must retain nuclear interaction beyond pure Coulomb repulsion; approximating fragment interaction by Coulomb alone would likely overestimate the final kinetic energy.
  • The same overlap-spacing construction could be generalized to a two-dimensional grid using the GOA-guided paving geometry the paper sketches, providing a concrete route to multi-dimensional SCIM surfaces.
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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

3 major / 7 minor

Summary. This paper constructs a one-dimensional adiabatic asymmetric fission path for 240Pu within the Schrödinger Collective Intrinsic Model (SCIM) framework, combining the overlap-based Link and Drop methods into a procedure labeled ~P20. It introduces a collective coordinate c# based on constant overlap steps, validates the path's regularity by comparing the Hamiltonian kernel with the Gaussian Overlap Approximation (Fig. 17), and analyzes the scission region: chemical-potential peaks, neutron enrichment of the neck, fragment particle-number distributions, and a static energy balance with deformation and interaction energies. Appendices D–F contain detailed derivations of overlap and Hamiltonian kernels between HFB states built on different two-center harmonic-oscillator bases.

Significance. If the path is accepted as a faithful collective trajectory, the paper provides a practical route toward SCIM dynamics with intrinsic excitations and a microscopic characterization of the scission region. The technical derivations in Appendices D–F, especially the overlap and Hamiltonian kernels for different two-center bases with rectangular overlap matrices, are a substantial methodological contribution. The GOA comparison gives a quantitative check of the local approximation used in the SCIM/GOA formalism, and the qualitative scission findings (chemical-potential peaks, neutron-rich neck, proton odd-even staggering) are interesting and falsifiable in future SCIM calculations. The main caveats are the dependence of the path on the V-phase prescription and on hand-set algorithm parameters, and an internal inconsistency in the reported scission energy ratio.

major comments (3)
  1. [Section IV.C.3, Eq. (72), Fig. 27] The ratio r_c is defined as 100×E_int^Coulomb/E_int and is reported as ≈83% at c#=495, but the values quoted in the same paragraph (E_int^Coulomb ≈ 178.74 MeV, E_int ≈ 153 MeV) give r_c ≈ 117%. Since E_int = E_Coulomb + E_nuclear, this implies E_nuclear ≈ −25.7 MeV at scission, i.e., a net attractive nuclear contribution, which is a different physical statement from the text's claim that the total interaction 'remains comparable to the Coulomb interaction.' The numeric value and the associated interpretation must be corrected.
  2. [Appendix D.8, Eq. (D58); Section III.D] The path and the c# coordinate are built from overlap values whose magnitudes depend on the sign of V in each (Ω,τ) block. The paper states that the numerical origin of the observed sign flips was not identified, and Eq. (D58) is an ad hoc prescription (choosing, per block, the sign that maximizes the absolute determinant). Since the scission observables in Section IV (chemical-potential peaks, neck enrichment, fragment distributions, energy balance) are all extracted along this path, their stability under alternative but equally allowed phase conventions is not demonstrated. The GOA comparison in Fig. 17 tests the regularity of the chosen path, not the uniqueness of the path. Please add a sensitivity test (e.g., reconstructing the scission segment with the opposite sign convention or with several sign assignments) or explicitly state that the results are conditional on this regularization.
  3. [Section III.D; Section IV] The ~P20 procedure contains several hand-set parameters (attractor overlap threshold 0.5, overlap step x0=0.995, stopping overlap 0.9; plus the QP rotation and cut-off criteria in the fragment analysis). The paper shows for the Drop method only that the PES depends weakly on x0 (Fig. 13), and it does not quantify the sensitivity of the scission observables to the algorithm parameters or compare the resulting path with an independent multi-dimensional adiabatic path (e.g., a 2D Q20–Q30 surface). A robustness check on at least the scission segment would be needed to support the claim that the quoted observables characterize the physical scission process rather than the specific regularization choices.
minor comments (7)
  1. [Fig. 17 caption and Eq. (64)] The caption says 'Absolute error on the Hamiltonian kernel,' but Eq. (64) defines a relative error (100 times the ratio of absolute difference to the kernel magnitude). Please align the wording with the formula.
  2. [Introduction and Section IV.C.3] Please fix typos: 'Bertsh et al.' should be 'Bertsch et al.' in the Introduction, and 'adressed' should be 'addressed' in Section IV.C.3.
  3. [Appendix D.8 heading] The heading 'The V phasis' should be 'The V phase'.
  4. [References [53]] Reference [53] appears twice with the same title and authors; please merge the journal and preprint versions or distinguish them clearly.
  5. [Abstract and Conclusions] The phrase 'first practical implementation of the SCIM' is stronger than what this article delivers: the paper constructs and analyzes adiabatic paths, while the dynamical SCIM equations are deferred to the third paper of the trilogy. Consider softening to 'first practical construction of the adiabatic ingredients for SCIM.'
  6. [Section III.D.3 and Eq. (64)] The statement that quantities in Eq. (64) and Fig. 17 are 'expressed in units of the new collective coordinate c#' is ambiguous, because c# is defined only as a step counter; please specify the unit convention used for the kernel plots.
  7. [Section IV.B, Fig. 21] The quantity c_odd^2 in Fig. 21 is not formally defined in the text; please define the decomposition of the fragment distributions used to extract the odd components.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the path and scission observables are generated by an overlap-constrained HFB procedure and are not fitted to the claims; only mild self-referential methodology (authors' own Link/Drop and SCIM formalisms) warrants a low nonzero score.

full rationale

The central derivation chain is not circular. The ~P20 path is generated by the Link/Drop algorithms with fixed overlap parameter x0=0.995 and attractor threshold 0.5; the scission observables (chemical-potential peaks, neck rho ratio, odd-even staggering, Delta_E_def, E_int) are computed from the resulting HFB states and are not used as inputs to select the path. The new coordinate c# is defined as an elementary overlap step, so it is a reparametrization rather than a fitted prediction; no equation equates a 'prediction' with an input. The GOA comparison (Eqs. 63-64, Fig. 17) is an internal consistency check of the overlap geometry, not an independent derivation, but it does not reduce the claims to the GOA assumption. Self-citations to Ref. [38] (Link/Drop) and Refs. [34,44] (SCIM) are not load-bearing: the SCIM equations are re-derived in Section II and Appendices A-C, and the Link/Drop protocols are re-described and benchmarked here (16O and P20 comparisons), so the cited works are not the sole support. The V-phase prescription of Eq. (D58) is an acknowledged ad hoc regularization ("We were unable to identify the numerical origin of the observed sign changes"), and the path and c# are conditional on it; this is a robustness/uncertainty caveat, not a circular step, because no observable is fitted to it. One internal numerical inconsistency should be flagged separately: E_Coulomb=178.74 MeV and E_int=153 MeV give r_c=117%, not the reported 83%; this affects the correctness assessment, not circularity. Overall, the paper is self-contained against its benchmarks and deserves a low score reflecting only methodological self-referentiality.

Assumptions & free parameters 6 free parameters · 8 assumptions · 2 invented entities

The central calculation assumes a 1D adiabatic trajectory, a second-order SOPO truncation, GOA-like overlap geometry, and a particular fragment-separation scheme. Eight axioms are listed; the most fragile are the V-phase sign maximization and the separation-method dependence of the energy balance. Six hand-set numerical parameters enter, most notably x0=0.995 which defines the c# coordinate and path resolution.

free parameters (6)
  • overlap step x0 = 0.995
    Chosen as the numerical path step for both Link and Drop; one unit of c# is one such step, so the coordinate's meaning and path resolution depend on this hand-set value.
  • attractor overlap threshold = 0.5
    Selects sparse reference states along the P20 path by taking the first state with overlap below 0.5; changes the set of target states and hence the Link trajectories.
  • Link stopping overlap threshold = 0.9
    Stops each Link segment when overlap with the target exceeds 0.9; affects discretization and endpoints of the reconstructed path.
  • QP amplitude cutoff for fragment analysis = 1e-4
    Canonical quasiparticle states with |v_k| below 1e-4 are dropped in the fragment separation method (Appendix G), which could slightly change fragment distributions and energy balance.
  • QP rotation criteria for fragment disentangling = s1+s2>0.005 and |v1^2-v2^2|<1e-4
    Selects nearly degenerate canonical states for pairwise rotations in Appendix H; affects separation indices and the derived fragment energies.
  • neck position z_neck = abscissa minimizing Q_neck in HFB3
    Defines the left-right split for fragment distributions and energy balance; different definitions of z_neck would shift the reported scission properties.
assumptions (8)
  • domain assumption SCIM/GCM continuity hypothesis: the HFB configurations vary continuously with the collective coordinate q.
    Invoked in Section II.A after Eq. (1); the whole overlap-based path construction is designed to enforce this for q=c#.
  • domain assumption Second-order truncation in SOPO for H_SCIM and convergence of the J^{-1/2} series.
    Assumed in Appendix B and C; the dissipation tensor D_SCIM depends on this truncation, and Appendix B calls the series convergence the strongest approximation of the model.
  • domain assumption Single dominant 1D adiabatic trajectory assumption.
    Stated in the Section III introduction: the dynamics are assumed to follow a single dominant adiabatic trajectory; restricts the SCIM implementation to one collective coordinate.
  • domain assumption Adiabatic Hamiltonian kernel approximates overlap kernel times local energy, Eq. (59).
    Used to infer kernel regularity from overlap regularity; quantified in Fig. 17 with max 0.46% error but not derived from first principles.
  • domain assumption GOA relation Eq. (63) for overlap distances is the correct geometry of adiabatic collective motion.
    Used as a benchmark to claim Link trajectories are physical; if the true metric deviates from GOA near scission, the validation is weakened.
  • domain assumption D1S Gogny force with Slater Coulomb and imposed axial and time-reversal symmetries adequately represents 240Pu fission.
    Standard domain assumption; restricts to even-even, time-reversal invariant configurations and neglects proton-neutron pairing and triaxiality.
  • ad hoc to paper V-phase overlap prescription Eq. (D58) choosing the sign that maximizes the absolute determinant yields physically consistent overlaps.
    Introduced in Appendix D.8 to remove spurious overlap discontinuities; no independent proof that this sign choice gives a globally consistent phase convention.
  • ad hoc to paper Fragment separation via canonical-basis spatial localization plus QP rotations gives the physical fragmentation.
    Section IV.C and Appendix H; the energy balance depends on z_neck and rotation criteria, and the paper notes the results depend on the separation method.
invented entities (2)
  • c# collective coordinate
    purpose: Parametrizes the adiabatic path by cumulative overlap steps of size x0 instead of Q20, providing a distance measure in HFB state space.
    Defined in Section III.D.3 from x0=0.995; it is a new mathematical coordinate with no observable handle outside the model, and changing x0 rescales c#.
  • c-flat (c flat) coordinate and Nuclear Paving method
    purpose: Proposed in Section V for future 2D overlap-based grids, geometrically orthogonal to c# using GOA predictions.
    Presented as perspective with a 5x5 test grid near scission; not used in the paper's central claims.

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Pith. "Pith review of Microscopic description of the fission process including intrinsic excitations. Part I: 240Pu adiabatic and asymmetric fission path within the Schrodinger Collective Intrinsic Model." pith.science (2026). https://pith.science/paper/AHRFGMDA

@misc{pith2026260807121,
  author       = {Pith},
  title        = {Pith review of: Microscopic description of the fission process including intrinsic excitations. Part I: 240Pu adiabatic and asymmetric fission path within the Schrodinger Collective Intrinsic Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHRFGMDA}},
  note         = {Machine review of arXiv:2608.07121}
}
read the original abstract

This article is the first in a trilogy aimed at presenting the first practical implementation of the Schrodinger Collective-Intrinsic Model (SCIM) applied to nuclear fission. Within the SCIM framework, the many-body wave function explicitly couples collective motion to intrinsic excitations, necessitating sets of Hartree-Fock-Bogoliubov (HFB) configurations that remain continuous and regular across a broad deformation range, from the ground state to scission and beyond. This paper focuses on constructing adiabatic HFB paths suitable for subsequent SCIM dynamical calculations. Standard constrained adiabatic paths often suffer from discontinuities and irregularities, which prevent the direct application of the formalism. To address these challenges, we implement two recently proposed overlap-based protocols, the Link and Drop methods, and combine them into a new numerical procedure.A comparison with the exact Gaussian Overlap Approximation confirms that the resulting adiabatic kernels exhibit properties consistent with the assumptions of the SCIM formalism. The regularized path is then analyzed in the scission region. We identify characteristic structures in the proton and neutron chemical potentials, a pronounced neutron enrichment of the neck at scission, and fragment particle-number distributions displaying a strong odd-even staggering in the proton sector. Finally, using a microscopic fragment-separation procedure formulated in the canonical basis, we extract static scission properties including fragment deformation energies and both Coulomb and nuclear contributions to the fragment interaction energy. These results establish the adiabatic foundations required for future SCIM calculations with intrinsic excitations and provide a microscopic characterization of the scission region in 240Pu.

Figures

Figures reproduced from arXiv: 2608.07121 by the authors.

Figure 1
Figure 1. FIG. 1. Panel (a): Asymmetric path in [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Crossing between the fission and fusion valleys ob [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 5
Figure 5. FIG. 5. Schematic view of the Link method. [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figures from the paper (21 more)
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison of the distance between the states ob [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Same as FIG. 8 but in terms of the overlap with [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Comparison of the distance between the states ob [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Illustration of the impact of the value of the param [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 12
Figure 12. Figure 12: FIG. 12. PES produced by the “Drop” method (in blue) in [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Panel (a): PES obtained with the procedures [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Evolution of the total energy of [PITH_FULL_IMAGE:figures/full_fig_p013_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Absolute error on the Hamiltonian kernel with re [PITH_FULL_IMAGE:figures/full_fig_p014_17.png]
Figure 19
Figure 19. Figure 19: FIG. 19. Study of the local neutron-to-proton density ratio [PITH_FULL_IMAGE:figures/full_fig_p015_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20. Panel (a): Neutron particle-number distributions [PITH_FULL_IMAGE:figures/full_fig_p016_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21. Evolution of the odd components in the light frag [PITH_FULL_IMAGE:figures/full_fig_p016_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22. Light fragment proton ( [PITH_FULL_IMAGE:figures/full_fig_p016_22.png]
Figure 24
Figure 24. Figure 24: FIG. 24. Example of QP rotation improving the spatial lo [PITH_FULL_IMAGE:figures/full_fig_p017_24.png]
Figure 25
Figure 25. Figure 25: FIG. 25. Left fragment (left column) and right fragment [PITH_FULL_IMAGE:figures/full_fig_p017_25.png]
Figure 26
Figure 26. Figure 26: FIG. 26. Evolution of the fragment binding energies . Panel [PITH_FULL_IMAGE:figures/full_fig_p018_26.png]
Figure 27
Figure 27. Figure 27: FIG. 27. Coulomb interaction energy between the fragments [PITH_FULL_IMAGE:figures/full_fig_p019_27.png]
Figure 29
Figure 29. Figure 29: FIG. 29. Schematic view of the definition of a direction “or [PITH_FULL_IMAGE:figures/full_fig_p020_29.png]
Figure 31
Figure 31. Figure 31: FIG. 31. Schematic view of [PITH_FULL_IMAGE:figures/full_fig_p020_31.png]
Figure 33
Figure 33. Figure 33: FIG. 33. Energy of the states belonging to the 5 [PITH_FULL_IMAGE:figures/full_fig_p021_33.png]
Figure 32
Figure 32. Figure 32: FIG. 32. Panel (a): 5 [PITH_FULL_IMAGE:figures/full_fig_p021_32.png]
Figure 34
Figure 34. Figure 34: FIG. 34. Illustration of the unexpected differences between [PITH_FULL_IMAGE:figures/full_fig_p033_34.png]

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Works this paper leans on

82 extracted references · 55 canonical work pages

  1. [1]

    Finally, it is possible to jump from ac ♭ level to another using the GOA predictions once again. In FIG. 31, we have represented this latter idea showing how a state E(c #,c♭) is defined from two other states D (c#,c♭ −1) and B (c #,c♭ −2). We have tested these ideas locally around the scission 21 FIG. 32. Panel (a): 5×5 grid produced by the NP method acc...

  2. [2]

    The quantityUcan then be expressed as U= N (1) R ∂ ∂q (1) + N (2) R ∂ ∂q (2) = u1 ∂ ∂q (1) + u2 ∂ ∂q (2)

    Expression ofJ −1/2 In this section, we give the explicit expression of the operatorJ −1/2, which reads as: J −1/2(¯q) =j0(¯q) + j1(¯q)∂ ∂q (1) + j2(¯q)∂ ∂q (2) .(C3) From Eq.(B13), the inverse square root of the operator Jis calculated thanks to a Taylor series expansion trun- cated at the second order: J −1/2(¯q) =I− 1 2 U(¯q) +3 8 U 2(¯q).(C4) To simpl...

  3. [3]

    V Phasis

    Expression ofH SCIM We start by rewriting the expression (C1) ofH SCIM in terms of the quantitiesFandJ −1/2 HSCIM (¯q) =J−1/2(¯q)F−1(¯q) × ¯H(¯q)F−1T (¯q)J−1/2(¯q),(C9) where the quantity ¯Hhas for expression: ¯H(¯q) =H(0)(¯q) + H(1)(¯q)∂ ∂q (1) + 1 2 H(2)(¯q)∂ ∂q (2) (C10) Now, one defines the operatorhsuch that: h(¯q) =h0(¯q) + h1(¯q)∂ ∂q (1) + h2(¯q)∂ ...

    work page

  4. [4]

    HFB states normalization We begin by presenting the expression for the normal- ization factorNof a given HFB state, which will play a crucial role in the subsequent discussion. Let|Φ⟩de- note an HFB state defined by the set of QP annihilation operators{ξ i}: |Φ⟩=N Y i ξi |0⟩.(D1) We transform the set of QP annihilation operators{ξ i} into the set{η k}, wh...

    work page

  5. [5]

    Overlap between HFB states built with the same harmonic-oscillator representations We consider two distinct HFB states,|Φ 0⟩and|Φ 1⟩, associated with the QP annihilation operators{ξ 0,i}and {ξ1,i}, respectively. By applying the Thouless theorem and denotingU (0),V (0) andU (1),V (1) as the Bogoli- ubov matrices corresponding to the states|Φ 0⟩and|Φ 1⟩, re...

    work page

  6. [6]

    Application to the axial and time-reversal invariance with the same harmonic-oscillator representations In the special case of axial and time-reversal invariant HFB states, the matrices ˜U (i) and ˜V (i) (wherei= 0 or

    work page

  7. [7]

    are real and exhibit the following structures: ˜U (i) = U (i) 0 0U (i) ˜V (i) = 0−V (i) V (i) 0 .(D24) Here, theU (i) andV (i) are of dimensionn/2. Eq.(D24) implies that the matricesM (i) defined in Eq.(D10) also have a special structure: M (i) = 0−V (i)U (i)−1 V (i)U (i)−1 0 .(D25) It is important to remark that the matricesM (i) are skew-symmetric, whil...

    work page

  8. [8]

    Onishi-Y oshida

    Link with the “Onishi-Y oshida” formula The “Onishi-Yoshida” formula [65] gives the absolute value of the overlap between two given HFB states such that: |⟨Φ0|Φ1⟩|= q Det(U (0)+U (1) +V (0)+V (1)) .(D29) Starting from Eq.(D23), one uses the Pfaffian properties and obtains: |⟨Φ0|Φ1⟩|=|⟨0|Φ 1⟩ ⟨Φ0|0⟩| × q Det(M (1))Det(−M (0)∗ +M (1)−1) .(D30) From Eq.(D8),...

    work page

Show all 82 references
  1. [9]

    They allow us to consider two distinct harmonic- oscillator representations of different dimensions without the need to compute their rectangular overlap matrixR

    Overlap between HFB states built with two different harmonic-oscillator representations The derivations presented in this section are entirely novel. They allow us to consider two distinct harmonic- oscillator representations of different dimensions without the need to compute...

  2. [10]

    [66], the following expression is proposed for the calculation of the norm of the overlap|⟨Φ 0|Φ1⟩|: |⟨Φ0|Φ1⟩|= (D44)q Det(U (0)T (RT )−1U (1)∗ +V (0)T RV (1)∗)Det(R)

    Link with Robledo’s formula In Ref. [66], the following expression is proposed for the calculation of the norm of the overlap|⟨Φ 0|Φ1⟩|: |⟨Φ0|Φ1⟩|= (D44)q Det(U (0)T (RT )−1U (1)∗ +V (0)T RV (1)∗)Det(R) . This formula has two major drawbacks. First, it requires to completeRint...

  3. [11]

    Haider-Gogny

    Link with the “Haider-Gogny” formula In this section, we connect the previously obtained re- sults to the “Haider-Gogny” [67]. Below, we consider two HFB states,|Φ 0⟩and|Φ 1⟩, both constructed using the same harmonic-oscillator basis. We begin by writing the canonical transfor...

  4. [12]

    V phasis

    The “V phasis” In our PES calculations using theP 20 procedure, we have observed unusual behavior in the overlap distance. Specifically, some neighboring states with nearly iden- tical multipole moments exhibited a significant overlap distance (close to one). Furthermore, the ...

  5. [13]

    We will consider the general case directly, which includes two different two-center harmonic-oscillator rep- resentations

    Relevant contractions This subsection is devoted to the evaluation of the con- tractions relevant to the calculation of Hamiltonian ker- nels. We will consider the general case directly, which includes two different two-center harmonic-oscillator rep- resentations. Indeed, the...

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    They are labeled by the representationnand the particle state numberαor QP state numberi

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