Renormalized Interactions for CI constrained by EDF methods
Alex Brown, Angelo Signoracci and Morten Hjorth-Jensen
Wick’s theorem for a Closed-shell vacuumfilled orbitals
Closed-shell vacuumfilled orbitals
EDF (Skyrme Phenomenology)
Closed-shell vacuumfilled orbitals
EDF (Skyrme) phenomenology
NN potential with V_lowk
Closed-shell vacuumfilled orbitals
EDF (Skyrme) phenomenology
“tuned” valence two-body matrix elements
Closed-shell vacuumfilled orbitals
EDF (Skyrme) phenomenology
Monopole from EDF
Closed-shell vacuumfilled orbitals
A3 A2 A 1
Monopole from EDF
Aspects of evaluating a microscopic two-body Hamiltonian (N3LO + Vlowk+ core-polarization) in a spherical EDF (energy-density functional) basis (i.e. Skyrme HF)
1)TBME (two-body matrix elements): Evaluate N3LO + Vlowk
with radial wave functions obtained with EDF.2)TBME: Evaluate core-polarization with an underlying single-particle spectrum obtained from EDF.3)TBME: Calculate monopole corrections from EDF that would implicitly include an effective three-body interaction of the valence nucleons with the core.4)SPE for CI: Use EDF single-particle energies – unless something better is known experimentally.
Why use energy-density functionals (EDF)?
1)Parameters are global and can be extended to nuclear matter.2)Effort by several groups to improve the understanding and reliability (predictability) of EDF – in particular the UNEDF SciDAC project in the US.3)This will involve new and extended functionals.4)With a goal to connect the values of the EDF parameters to the NN and NNN interactions.5)At this time we have a reasonably good start with some global parameters – for now I will use Skxmb – Skxm from [BAB, Phys. Rev. C58, 220 (1998)] with small adjustment for lowest single-particle states in 209Bi and 209Pb.
Calculations in a spherical basis with no correlations
What do we get out of (spherical) EDF?
1)Binding energy for the closed shell2)Radial wave functions in a finite-well (expanded in terms of harmonic oscillator). 3) gives single-particle energies for the nucleons constrained to be in orbital (n l j)a where BE(A) is a doubly closed-shell nucleus.
4)
gives the monopole two-body matrix element for nucleons constrained to be in orbitals (n l j)a and (n l j)b
EDF core energy and single-particle energy
EDF two-body monopole
Theory (ham) from Skxmb with parameters adjusted to reproducethe energy for the 9/2- state plus about 100 other global data.
218U208Pb
x = experiment
CI (ham) N3LO with EDF constraint
EDF (or CI) withno correlations
CI with N3LO
Skyrme (Skxmb) + Vlow-k N3LO (second order)
210Po
210Po Skyrme (Skxmb) + Vlow-k N3LO (first order)
213Fr Skyrme (Skxmb) + Vlow-k N3LO (second order)
214Ra Skyrme (Skxmb) + Vlow-k N3LO (second order)
EDF core energy and single-particle energy
EDF two-body monopole
Theory (ham) from Skxmb with parameters adjusted to reproducethe energy for the 9/2+ state plus about 100 other global data.
Skyrme (Skxmb) + Vlow-k N3LO (second order)
210Pb
Skyrme (Skxmb) + Vlow-k N3LO (second order)
210Bi
Skyrme (Skxmb) + Vlow-k N3LO (second order)
212Po
Skyrme (Skxmb) + Vlow-k N3LO (second order)
210Pb
Skyrme (Skxmb) + exp spe Vlow-k N3LO (second order)
210Pb
Skyrme (Skxmb) for 208Pb (closed shell) + Vlow-k N3LO (second order)
“ab-initio” calculation for absolute energies of 213Fr
Energy of first excited 2+ states