Renormalized Interactions for CI constrained by EDF methods
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Renormalized Interactions for CI constrained by EDF methods Alex Brown, Angelo Signoracci and Morten Hjorth -Jensen. Wick’s theorem for a Closed-shell vacuum filled orbitals. Closed-shell vacuum filled orbitals. EDF (Skyrme Phenomenology). Closed-shell vacuum filled orbitals.

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Renormalized Interactions for CI constrained by EDF methods

Alex Brown, Angelo Signoracci and MortenHjorth-Jensen


Wick’s theorem for a

Closed-shell vacuum

filled orbitals


Closed-shell vacuum

filled orbitals

EDF (Skyrme

Phenomenology)


Closed-shell vacuum

filled orbitals

NN potential with V_lowk

EDF (Skyrme)

phenomenology


Closed-shell vacuum

filled orbitals

“tuned” valence

two-body matrix elements

EDF (Skyrme)

phenomenology


Closed-shell vacuum

filled orbitals

Monopole from EDF

EDF (Skyrme)

phenomenology


Closed-shell vacuum

filled orbitals

Monopole from EDF

A3 A2 A 1


Aspects of evaluating a microscopic two-body Hamiltonian (N3LO + Vlowk+ core-polarization) in a spherical EDF (energy-density functional) basis (i.e. Skyrme HF)

TBME (two-body matrix elements): Evaluate N3LO + Vlowk with radial wave functions obtained with EDF.

TBME: Evaluate core-polarization with an underlying single-particle spectrum obtained from EDF.

TBME: Calculate monopole corrections from EDF that would implicitly include an effective three-body interaction of the valence nucleons with the core.

SPE for CI: Use EDF single-particle energies – unless something better is known experimentally.


Why use energy-density (N3LO + Vfunctionals (EDF)?

Parameters are global and can be extended to nuclear matter.

Effort by several groups to improve the understanding and reliability (predictability) of EDF – in particular the UNEDF SciDAC project in the US.

This will involve new and extended functionals.

With a goal to connect the values of the EDF parameters to the NN and NNN interactions.

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.



What do we get out of (spherical) EDF? (N3LO + V

Binding energy for the closed shell

Radial wave functions in a finite-well (expanded in terms of harmonic oscillator).

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 (N3LO + V

EDF two-body monopole


Theory (ham) from Skxmb with parameters adjusted to reproduce

the energy for the 9/2- state plus about 100 other global data.


x = experiment reproduce

CI with N3LO

CI (ham) N3LO with EDF constraint

EDF (or CI) with

no correlations

208Pb

218U


Skyrme (Skxmb) + reproduce

Vlow-k N3LO (second order)

210Po


Skyrme (Skxmb) + reproduce

Vlow-k N3LO (first order)

210Po


Skyrme (Skxmb) + reproduce

Vlow-k N3LO (second order)

213Fr


214 reproduceRa

Skyrme (Skxmb) +

Vlow-k N3LO (second order)


EDF core energy and single-particle energy reproduce

EDF two-body monopole


Theory (ham) from Skxmb with parameters adjusted to reproduce

the energy for the 9/2+ state plus about 100 other global data.


210 reproducePb

Skyrme (Skxmb) +

Vlow-k N3LO (second order)


210 reproduceBi

Skyrme (Skxmb) +

Vlow-k N3LO (second order)


212 reproducePo

Skyrme (Skxmb) +

Vlow-k N3LO (second order)


210 reproducePb

Skyrme (Skxmb) +

Vlow-k N3LO (second order)


210 reproducePb

Skyrme (Skxmb) + exp spe

Vlow-k N3LO (second order)


Skyrme (Skxmb) for reproduce208Pb (closed shell)

+ Vlow-k N3LO (second order)



Energy of first excited 2 reproduce+ states


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