Jacek Dobaczewski University of Warsaw & University of Jyväskylä Pathologies of current multi-reference EDF calculations Atelier de l’Espace de Structure Nucléaire Théorique novembre 2007 CEA/SPhN, Ormes des Merisiers, Bât. 703 F Gif-sur-Yvette Cedex.

Jacek Dobaczewski Outline 1.Introduction. 2.Hohenberg-Kohn and Kohn-Sham theorems. 3.Particle-Number Projection and the Density Functional Theory”, J. Dobaczewski, M.V. Stoitsov, W. Nazarewicz, and P.-G. Reinhard, arXiv/ , Physical Review C, in press 4.Summary and conclusions.

Jacek Dobaczewski mean field  one-body densities zero range  local densities finite range  non-local densities Hohenberg-Kohn Kohn-Sham Negele-Vautherin Landau-Migdal Nilsson-Strutinsky Modern Mean-Field Theory  Energy Density Functional  j, , , J,  T,  s,  F, Nuclear densities as composite fields

Jacek Dobaczewski Hohenberg-Kohn theorem

Jacek Dobaczewski Hohenberg-Kohn theorem (trivial version)

Jacek Dobaczewski Nuclear Energy Density Functional (physical insight)

Jacek Dobaczewski Particle-number-projection (PNP) operator

Jacek Dobaczewski Shifted HFB states

Jacek Dobaczewski z iu/v below = z -iu/v above = z -iu/v below = z iu/v above = z 0 = z 1 = HFB state  Im[z] Re[z] C0 C1 C2

Jacek Dobaczewski Method of residues

Jacek Dobaczewski Poles of transition density matrices

Jacek Dobaczewski DFT transition energy density

Jacek Dobaczewski Projected DFT energy

Jacek Dobaczewski

Fractional powers

Jacek Dobaczewski Im[z] C0 C1’ C2’ Im[z] Re[z]  Im[z]

Jacek Dobaczewski 18 O

Jacek Dobaczewski 26 O

Jacek Dobaczewski

Summary and conclusions 1.The transition density matrices connecting states having different orientation in the gauge plane z have poles on the imaginary axis 2.The DFT sum rules offer the interpretation of the projected DFT energies as Fourier components [associated with irreducible representations of gauge group U(1)] of the DFT transition energy density. 3.The discussion of the particle number restoration can be extended to other symmetry restoration problems. 4.For the terms in the density functional that have polynomial density dependence, the appearance of poles inside the contour gives rise to sudden jumps of the projected energy whenever the contour's pole content changes. 5.As a practical measure that allows avoiding problems related to the fractional-power density dependence, we propose using the integration contours which pass near and above the poles.

Jacek Dobaczewski Complete local energy density Mean field Pairing E. Perlińska, et al., Phys. Rev. C69 (2004)

Jacek Dobaczewski Mean-field equations

Jacek Dobaczewski Nuclear densities as composite fields

Jacek Dobaczewski Local energy density: (no isospin, no pairing)

Jacek Dobaczewski Sum rules

Jacek Dobaczewski