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Renormalized Interactions for CI constrained by EDF methods
Alex Brown, Angelo Signoracci and Morten Hjorth-Jensen
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Wick’s theorem for a Closed-shell vacuumfilled orbitals
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Closed-shell vacuumfilled orbitals
EDF (Skyrme Phenomenology)
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Closed-shell vacuumfilled orbitals
EDF (Skyrme) phenomenology
NN potential with V_lowk
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Closed-shell vacuumfilled orbitals
EDF (Skyrme) phenomenology
“tuned” valence two-body matrix elements
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Closed-shell vacuumfilled orbitals
EDF (Skyrme) phenomenology
Monopole from EDF
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Closed-shell vacuumfilled orbitals
A3 A2 A 1
Monopole from EDF
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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.
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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.
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Calculations in a spherical basis with no correlations
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What do we get out of (spherical) EDF?
1)Binding energy for the closed shell
2)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
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EDF core energy and single-particle energy
EDF two-body monopole
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Theory (ham) from Skxmb with parameters adjusted to reproducethe energy for the 9/2- state plus about 100 other global data.
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218U208Pb
x = experiment
CI (ham) N3LO with EDF constraint
EDF (or CI) withno correlations
CI with N3LO
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Skyrme (Skxmb) + Vlow-k N3LO (second order)
210Po
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210Po Skyrme (Skxmb) + Vlow-k N3LO (first order)
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213Fr Skyrme (Skxmb) + Vlow-k N3LO (second order)
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214Ra Skyrme (Skxmb) + Vlow-k N3LO (second order)
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EDF core energy and single-particle energy
EDF two-body monopole
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Theory (ham) from Skxmb with parameters adjusted to reproducethe energy for the 9/2+ state plus about 100 other global data.
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Skyrme (Skxmb) + Vlow-k N3LO (second order)
210Pb
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Skyrme (Skxmb) + Vlow-k N3LO (second order)
210Bi
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Skyrme (Skxmb) + Vlow-k N3LO (second order)
212Po
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Skyrme (Skxmb) + Vlow-k N3LO (second order)
210Pb
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Skyrme (Skxmb) + exp spe Vlow-k N3LO (second order)
210Pb
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Skyrme (Skxmb) for 208Pb (closed shell) + Vlow-k N3LO (second order)
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“ab-initio” calculation for absolute energies of 213Fr
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Energy of first excited 2+ states
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