Safe EDF for SR and MR calculations

1. Safe EDF for SR and MR calculations M. Bender, T. Duguet, D. Lacroix FIDIPRO workshop, Jyvaskyla, Oct. 9th-10th 2008

2. Spurious contributions to SR and MR EDF kernels M=99 M=9 19 39 M=9 59 79 99 Bender, Duguet, Int. J. Mod. Phys. E16 (2007) • Difficulties due to using EDF rather than TRUE Hamiltonian • ANY vertex containing at least a density dependence • Gogny vertex is already pathological • More obviously critical for Skyrme+DDDI • Unrecognized spurious contributions to SR calculations • Unrecognized to this day but well know in DFT • Need to be characterized quantitatively • Need to build spuriosity-free SR-EDF (lot to be done) • (Now) obvious pathologies of MR calculations (any type) • Would NOT exist without spuriosities in SR EDF kernel • Combine with MISUSE of GWT to build MR-EDF kernels • Can correct for the latter rather “easily” • Correctable functional requires rnq with n integer < 4 • Everything else is prohibited Divergence Jump

3. Identification and cure to pathologies SP (I) Lacroix, et al, General solution to MR pathologies,arXiv/0809.2041 SI Sources of pathologies • Self-Interaction (SR+MR) • Not dramatic a priori • Need to be characterized • Self-Pairing (SR+MR) • Not dramatic a priori • Need to be characterized • Misuse of Gen. Wick Th. (GWT) within EDF method (MR only) • Divergences, sharp steps, smooth steps plus kink Cure to MR Find an Alternative to GWT Identify critical terms Remove pathologies

4. Particle Number Restoration rr term kk term Results Correction is important at but also away from steps Correction is overall small (~1MeV) but of the order of the required mass accuracy or spectroscopy of interest Corrected energy curve is smooth (II) Bender et al, First application to PNR, arXiv/0809.2045

5. Summary and discussion Specific difficulties to be considered seriously Regularization method valid for any MR calculation Application to Particle Number Restoration (I) Lacroix, et al, General solution to pathologies, arXiv/0809.2041 (II) Bender et al, First application to PNR, arXiv/0809.2045 Specific case of fractional power of the density: rg (III) Duguet et al, Non-integer powers of density matrices, arXiv/0809.2049 Need to build (i) Correctable EDF for MR calculations (ii) SI- and SP-free EDF Need a constructive framework for MR-EDF Energy Density Functional methods Dobaczewski et al, Identification of problem in PNR, PRC76, 054315 (2007)