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