You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'Configuration mixing calculations performed in terms of the Skyrme/Gogny Energy Density Functional (EDF) rely on extending the Single-Reference energy functional into non-diagonal EDF kernels. The standard way to do so, based on an analogy with the pure Hamiltonian case and the use of the generalized Wick theorem, is responsible for the recently observed divergences and steps in Multi-Reference calculations. We summarize here the minimal solution to this problem recently proposed[@Lac08] and applied with success to particle number restoration[@Ben08]. Such a regularization method provides suitable corrections of pathologies for EDF depending on integer powers of the density. The specific case of fractional powers of the density[@Dug08] is also discussed.'
address:
- 'GANIL, CEA and IN2P3, Boîte Postale 5027, F-14076 Caen Cedex, France'
- 'Universit[é]{} Bordeaux and CNRS/IN2P3, CENBG, UMR5797, F-33175 Gradignan, France'
- 'CEA, Centre de Saclay, IRFU/Service de Physique Nucléaire, F-91191 Gif-sur-Yvette, France'
author:
- DENIS LACROIX
- MICHAEL BENDER
- THOMAS DUGUET
title: |
    Configuration mixing within the energy density functional formalism:\
    pathologies and cures.
---

Energy Density Functional methods
=================================

The nuclear Energy Density Functional (EDF) method is a unique tool to study static and dynamical properties of nuclei in a unified framework[@bender03b]. Although the nuclear EDF shares several features with Density Functional Theory[@lecturenotesFNM] (DFT), the strategy used is different as it embraces two successive levels of description.

On the first level, traditionally called “self-consistent mean-field theory”, Hartree-Fock (HF) or Hartree-Fock-Bogoliubov (HFB), a single product state $\Phi_0$ provides the normal $\rho^{00}$ and anomalous $\kappa^{00}$ density matrices the many-body energy is a functional of. We call this method a single-reference (SR) EDF approach and denote by ${\cal E}_{SR}[\Phi_0]={\cal E}_{SR}[\rho^{00}, \kappa^{00}, \kappa^{00 \, \ast}]$ the actual EDF. Although such a restriction is not necessary, one usually builds the EDF from an effective vertex (of the Skyrme or Gogny type), whose parameters are adjusted to reproduce a selected set of experimental observations. Independently of the starting point, the EDF can be written as in any arbitrary basis as $$\begin{aligned}
\label{eq:e00}
{\cal E}_{SR}[\rho^{00}, \kappa^{00}, \kappa^{00 \, \ast}]
& = & \sum_{ij} t_{ij} \, \rho^{00}_{ji}
      + \tfrac{1}{2} \sum_{ijkl} \bar{v}^{\rho\rho}_{ijkl} \,
        \rho^{00}_{ki} \, \rho^{00}_{lj}
      + \tfrac{1}{4} \sum_{ijkl} \bar{v}^{\kappa\kappa}_{ijkl} \,
        \kappa^{00 \, \ast}_{ij} \, \kappa^{00}_{kl} \\
&& +  \tfrac{1}{6} \sum_{ijklmn} \bar{v}^{\rho\rho\rho}_{ijklmn} \,
        \rho^{00}_{li} \, \rho^{00}_{mj} \, \rho^{00}_{nk}
+ \tfrac{1}{4} \sum_{ijklmn} \bar{v}^{\rho\kappa\kappa}_{ijklmn} \, \rho^{00}_{li} \,
        \kappa^{00 \, \ast}_{jk} \, \kappa^{00}_{mn} + \ldots \, \, \, , \nonumber\end{aligned}$$ where the first term accounts for the uncorrelated kinetic energy, whereas $\bar{v}^{\rho\rho}$, $\bar{v}^{\kappa\kappa}$, $\bar{v}^{\rho\rho\rho}$, …denote effective vertices associated with the different terms of the EDF. There are a few important comments to be made at this point. First, and although it formally resembles it, Eq. (\[eq:e00\]) should not be confused with the expectation value of a Hamiltonian containing two-body, three-body, …interactions in the Hartree-Fock-Bogolyubov state $\Phi_0$. For this to be true, specific properties, e.g. $\bar{v}^{\rho\rho}_{ijkl}=\bar{v}^{\kappa\kappa}_{ijkl}$ and $\bar{v}^{\rho\rho}_{ijkl}=-\bar{v}^{\rho\rho}_{ijlk}$ for all $(i,j,k,l)$, would have to be satisfied, which is usually not the case in the EDF context. Second, most popular and performing EDF cannot be written under the form of Eq. (\[eq:e00\]) as they contain a dependence on a non-integer power of the (local) normal density[@bender03b]. We anticipate, however, that future EDFs will be constructed under such a form, typically truncated at forth or fifth power. Indeed, the regularization procedure presented here is inapplicable to EDFs containing non-integer powers of the density matrices[@Dug08].

While static collective correlations, e.g. pairing and deformation, can be accounted for within the SR EDF formalism through the symmetry breaking of the auxiliary state $\Phi_0$, dynamical collective correlations requires to perform a so-called Multi-Reference (MR) calculations, traditionally denoted as “beyond-mean-field”. Such an extension, built by analogy with the Generator Coordinate Method (GCM) in the Hamiltonian formalism[@Rin81], allows one not only to incorporate additional correlations but also to describe low-energy spectroscopy and transition probabilities between states characterized by symmetry-restored quantum numbers. In strict analogy with the Hamiltonian formalism, the MR EDF is written as $$\mathcal{E}[\Psi] \equiv \frac{\sum_{\{0,1\} \in {\rm M\!R}} f^{\ast}_{0} \, f_{1} \, \,
        \mathcal{E}_{MR}[\Phi_0,\Phi_1] \,
        \langle \Phi_{0} | \Phi_{1} \rangle}
       {\sum_{\{0,1\} \in {\rm M\!R}} f^{\ast}_{0} \, f_{1} \,
        \langle \Phi_{0} | \Phi_{1} \rangle}
\, \, \, , \label{MRenergy}$$ where non-diagonal matrix elements $\langle \Phi_0 | \hat{H} | \Phi_1 \rangle/\langle \Phi_0 | \Phi_1 \rangle$ have been replaced by their EDF counterpart $\mathcal{E}_{MR}[\Phi_0,\Phi_1]$. The weight functions $f$ are determined either by symmetry considerations, by diagonalization, or both. The product states $\Phi_i$ belonging to the MR set are chosen according to the collective modes one wants to describe. In the absence of a well-founded prescription to build $\mathcal{E}_{MR}[\Phi_0,\Phi_1]$, only specific constraints can be imposed. For a number of reasons[@Robledo07a], it is necessary to impose that $\mathcal{E}_{MR}[\Phi_0,\Phi_0] \equiv \mathcal{E}_{SR}[\Phi_0]$ and $\mathcal{E}_{MR}[\Phi_1,\Phi_0] = \mathcal{E}^*_{MR}[\Phi_0,\Phi_1]$. Following the Hamiltonian formalism, the most natural guidance is provided by the generalized Wick theorem[@balian69a] (GWT) which tell us that $\mathcal{E}_{MR}[\Phi_0,\Phi_1]$ is obtained by replacing SR density matrices by transition ones, i.e. $[\rho^{01}, \kappa^{01}, \kappa^{10 \, \ast}]$, in Eq. (\[eq:e00\]). However, we have shown that the use of GWT-based functional energy kernels is the source of the pathologies recently observed in MR-EDF calculations[@Lac08; @Ben08; @Dug08].

Pathologies observed in configuration mixing calculations
---------------------------------------------------------

An example of deformation energy surface obtained through a MR calculation based on Particle-Number Restoration (PNR) is given for $^{18}$O in Fig. \[fig:o18:e:pes:c\] using the trilinear SIII Skyrme EDF. Starting from a SR-EDF built from the auxiliary state $\Phi_0$ which explicitly breaks the particle-number symmetry, in order to account for static pairing correlations explicitly, dynamical pairing correlations associated with PNR can be incorporated through a MR EDF calculation. Building the MR set from product states rotated in gauge space $| \Phi_\varphi \rangle = e^{i\varphi \hat{N}}| \Phi_0 \rangle$, Eq. (\[MRenergy\]) specified to PNR reads[@Lac08] $$\label{scalar2}
\mathcal{E}^{N}
\equiv \int_{0}^{2\pi} \! d\varphi \,
       \frac{e^{-i\varphi N}}
               {2\pi \, c^{2}_{N}} \,
       \mathcal{E}_{MR}[\Phi_0 , \Phi_\varphi] \,  \langle  \Phi_0 | \Phi_{\varphi} \rangle
\, .$$ In practice, Eq. (\[scalar2\]) is numerically estimated using the Fomenko[@fom70a] discretization procedure. The left panel of Fig. \[fig:o18:e:pes:c\] presents results obtained for two different numbers of mesh points in the discretization (dotted and dashed lines). Obvious pathologies are visible, i.e. (i) the estimate of the energy landscape does not converge and (ii) non-physical steps appear at particular deformations as one increases the number of mesh points used in the Fomenko procedure[^1]. Authors have not only also faced the problem for PNR[@Ang01; @Ben07] but also when performing angular-momentum restoration[@Zdu07]. It has been recognized[@Ang01] that divergences in PNR may appear when either a proton or neutron single-particle level crosses the Fermi en energy, as pair of states differing by $\pi/2$ are orthogonal in this case, i.e. $\langle  \Phi_0 | \Phi_{\varphi} \rangle = 0$. When the same (density-independent) vertices are used in the p-h and p-p channel and the exchange is properly taken into account, problems were shown to disappear. The problem has been characterized more precisely thanks to a complex plane analysis[@Dob07], demonstrating in particular the less obvious but more profound occurrence of steps. This technique, however, cannot be extended to the restoration of other symmetries and does not lead to a practical solution of the problem.

Minimal solution to the problem
===============================

We have recently shown that the origin of difficulties can be traced back to the strategy used to design energy kernels entering the MR-EDF, i.e. the use of the GWT as a guidance. An early hypothesis[@Ang01] has been confirmed later[@Ben07] by avoiding the use of the GWT in the PNR case. More recently, a general solution which applies to any type of configuration mixing has been formulated[@Lac08]. The technique makes use of the following trick: given a pair of quasi-particle vacua, denoted by $| \Phi_0 \rangle$ and $| \Phi_1 \rangle$ (with possibly $\langle \Phi_0 | \Phi_1 \rangle=0$), one can always find a simple “BCS like” expression connecting these two states, i.e.[@Rin81]: $$\begin{aligned}
\label{eq:BCSqp}
| \Phi_1 \rangle
= \tilde{\cal C}_{01} \,
  \prod_{p > 0}
  \left(   \bar A^*_{pp}
         + \bar B^*_{p\bar p} \,\tilde{\alpha}^+_p \, \tilde{\alpha}^+_{\bar p}
  \right)
  | \Phi_0 \rangle \, \, \,
.\end{aligned}$$ In the quasi-particle basis where Eq. (\[eq:BCSqp\]) is valid, GWT-based energy kernels read, e.g. for a strictly bilinear EDF, as (omitting the kinetic term) $$\begin{aligned}
{3}
%\label{kernel}
%\lefteqn{
{\cal E}_{MR}[\Phi_0,\Phi_1]
=
%} \nonumber \\
& %
  \frac{1}{2} \sum_{\nu \mu} \bar{v}^{\rho\rho}_{\varphi_\nu \varphi_\mu \varphi_\nu \varphi_{\mu}} &
&  + \frac{1}{4} \sum_{\nu \mu} \bar{v}^{\kappa\kappa}_{\varphi_\nu \phi_{\bar \nu} \varphi_\mu \phi_{\bar \mu}}
\label{kernel1}
%      \nonumber
 \\
+ &  \frac{1}{2} \sum_{\nu \mu} \bar{v}^{\rho\rho}_{\varphi_\nu \varphi_\mu \phi_{\nu} \varphi_{\mu}} \,
     \bar Z_{\nu \bar\nu } &
&  + \frac{1}{4} \sum_{\nu \mu} \bar{v}^{\kappa\kappa}_{\varphi_\nu \varphi_{\bar \nu}\varphi_\mu \phi_{\bar \mu}}
\,
     \bar Z_{\nu \bar \nu}
\label{kernel2}
%     \nonumber
\\
+ &  \frac{1}{2} \sum_{\nu \mu} \bar{v}^{\rho\rho}_{\varphi_\mu \varphi_\nu \varphi_\mu \phi_{ \nu}} \,
     \bar Z_{\nu \bar\nu } &
&  + \frac{1}{4} \sum_{\nu \mu} \bar{v}^{\kappa\kappa}_{\varphi_\mu \phi_{\bar\mu} \phi_{ \nu} \phi_{\bar \nu}} \,
      \bar Z_{\nu \bar \nu}
\label{kernel3}
%      \nonumber
 \\
+ &  \frac{1}{2} \sum_{\nu \mu} \bar{v}^{\rho\rho}_{\varphi_\nu \varphi_\mu \phi_{\nu} \phi_{\mu}} \,
     \bar Z_{ \nu\bar\nu} \, \bar Z_{ \mu \bar\mu} &
&  + \frac{1}{4} \sum_{\nu \mu} \bar{v}^{\kappa\kappa}_{\varphi_\nu \varphi_{\bar \nu} \phi_{ \mu} \phi_{\bar
\mu}} \,
     \bar Z_{ \nu \bar\nu } \, \bar Z_{ \mu \bar\mu } \, \, , \label{kernel4}\end{aligned}$$ where $(\mu,\bar \mu)$ denote a canonical pair in the specific quasi-particle representation, $\bar Z_{\bar \nu \nu} =(\bar B_{\bar \nu \nu} / \bar A^{-1}_{\nu \nu})^*$ while $\varphi_\nu$ and $\phi_\mu$ stand for the upper and lower components of the quasi-particle states[@Lac08]. Using expressions (\[kernel1\]-\[kernel4\]) is convenient as it separates the contributions remaining in the SR limit (line \[kernel1\]) from the rest. It also allows to identify the different sources of problems. [**(i)**]{} [**Self-interaction:**]{} Well-kown from DFT, self-interaction relates to the fact that a particle should not interact with itself, which, however, happens when the vertices $\bar{v}^{\rho\rho}$ are not anti-symmetrized[@Lac08]. This is almost always the case in actual EDFs at least because of the approximations used to treat Coulomb exchange. Such a self-interaction, if present at the SR level, further contributes at the MR level (lines (\[kernel2\]-\[kernel3\])). [**(ii)**]{} [**Self-pairing:**]{} this new concept[@Ben08] is specific to EDFs treating pairing correlations explicitly and relates to the fact that two paired particles should not generate correlation energies, beyond their direct interaction, by scattering onto themselves. Again, such a spurious contribution appears at both SR and MR levels. [**(iii)**]{} [**Steps and divergences:**]{} As the energy kernel is multiplied by $\langle \Phi_0 | \Phi_1 \rangle \propto \prod_\nu \bar A^{*}_{\nu \nu}$ in the MR energy (see Eq. (\[MRenergy\])), only terms with $\nu = \mu$ or $\nu = \bar \mu$ in line (\[kernel4\]) can lead to divergences and steps when $\bar A^{*}_{\nu \nu}=0$. In the pure Hamiltonian case, i.e. $\bar{v}^{\rho\rho}_{ijkl}=\bar{v}^{\kappa\kappa}_{ijkl}$ and $\bar{v}^{\rho\rho}_{ijkl}=-\bar{v}^{\rho\rho}_{ijlk}$ for all $(i,j,k,l)$, the dangerous contributions coming from the two terms in Eq. (\[kernel4\]) exactly cancel out and no divergence or step occurs. However, when different or non-antisymmetrized vertices are used, as in the EDF context, divergences and/or steps are observed, as on the left panel of Fig. \[fig:o18:e:pes:c\].

The quasi-particle basis introduced above allows one to proceed to the analogy with the Hamiltonian formalism on the basis of the standard Wick theorem rather than on the generalized one. Comparing the results of the two schemes, one proves[@Lac08] that terms with $\nu = \mu$ or $\nu = \bar \mu$ in line (\[kernel4\]) should be zero in the first place and must be removed altogether. This not only solves problem [**(iii)**]{} entirely but also remove finite spurious contributions to the MR energy kernel. Such a regularization technique can be applied to any type of configuration mixing performed in terms of an EDF depending on integer powers of the densities. It has been successfully applied to PNR[@Ben08], as is exemplified on the left panel of Fig. \[fig:o18:e:pes:c\] using the SIII Skyrme EDF. The correction not only removes the dependence on the number of mesh points and the non-physical steps, but also corrects the energy landscape [*away*]{} from those steps.

![ \[fig:o18:e:pes:c\] Left: Uncorrected (dotted and dashed lines) and corrected (solid line) particle-number restored quadrupole deformation energy obtained for $^{18}$O with SIII and calculated with $L=5$ and 199 discretization points of the integral in gauge space. The two corrected curves are superimposed. Right: attempt to regularize the particle-number restored energy of $^{18}$O obtained with SLy4 that contains a non-integer power of the (local) normal density. ](o18_siii_proj_pes_fjs08.eps "fig:"){width="6.0cm"} ![ \[fig:o18:e:pes:c\] Left: Uncorrected (dotted and dashed lines) and corrected (solid line) particle-number restored quadrupole deformation energy obtained for $^{18}$O with SIII and calculated with $L=5$ and 199 discretization points of the integral in gauge space. The two corrected curves are superimposed. Right: attempt to regularize the particle-number restored energy of $^{18}$O obtained with SLy4 that contains a non-integer power of the (local) normal density. ](o18_sly4_proj_pes_fjs08.eps "fig:"){width="6.0cm"}

The case of EDFs depending on non-integer powers of the density matrix has also been analyzed[@Dug08]. Although divergences can be removed using a variant of the method proposed in Ref.[@Lac08], the complex-plane analysis demonstrates that the left-over fractional power $\rho^{\gamma}$ with $0<\gamma<1$ is ill-defined as it generates cusps in the PNR energy landscape, (see right panel of Fig. \[fig:o18:e:pes:c\]). Generally speaking, one cannot use a functional that is multi-valued over the complex plane. This has important consequences on the present and future of EDF methods.

[99]{} D. Lacroix, T. Duguet and M. Bender, [*submitted to Phys. Rev. C*]{}, arXiv:0809.2041. M. Bender, T. Duguet and D. Lacroix, [*submitted to Phys. Rev. C*]{}, arXiv:0809.2045. T. Duguet, M. Bender, K. Bennaceur, D. Lacroix and T. Lesinski, [*submitted to Phys. Rev. C*]{}, arXiv:0809.2049. M. Bender, P.-H. Heenen and P.-G. Reinhard, Rev. Mod. Phys. [**75**]{}, 121 (2003). , Ed. C. Fiolhais and F. Nogueira and M. Marques, Lecture Notes in Physics 620, 2003, Springer, Berlin and Heidelberg. P. Ring and P. Schuck, [*The Nuclear Many-Body Problem*]{} (Springer, New York, 1980). L. M. Robledo, Int. J. Mod. Phys. E16 (2007) 337. R. Balian and E. Br[é]{}zin, Nuovo Cimento 64 (1969) 37 B. F. Bayman, Nucl. Phys. [**15**]{}, 33 (1960). V. N. Fomenko, J. Phys. (G.B) [**A3**]{}, 8 (1970). M. Anguiano and J. L. Egido and L. M. Robledo, Nucl. Phys. [**A683**]{}, 227 (2001). M. Bender and T. Duguet, Int. J. Mod. Phys. [**E16**]{}, 222 (2007).

H. Zdunczuk and J. Dobaczewski and W. Satu[ł]{}a, Int. J. Mod. Phys. [**E16**]{}, 377 (2007).

J. Dobaczewski, W. Nazarewicz, P. G. Reinhard, and M. V. Stoitsov, Phys. Rev. [**C76**]{}, 054315 (2007).

[^1]: The fact that no divergence occurs is due to the particular form of the functional used which is strictly bilinear in the same isospin[@Dug08].