Configuration Interactions Constrained by Energy Density Functionals
aa r X i v : . [ nu c l - t h ] S e p Configuration Interactions Constrained by Energy Density Functionals
B. Alex Brown , Angelo Signoracci and Morten Hjorth-Jensen Department of Physics and Astronomy, and National Superconducting Cyclotron Laboratory,Michigan State University, East Lansing, Michigan 48824-1321, USA and Department of Physics and Center for Mathematical Applications, University of Oslo, N-0316, Oslo, Norway
A new method for constructing a Hamiltonian for configuration interaction calculations withconstraints to energies of spherical configurations obtained with energy-density-functional (EDF)methods is presented. This results in a unified model that reproduced the EDF binding-energyin the limit of single-Slater determinants, but can also be used for obtaining energy spectra andcorrelation energies with renormalized nucleon-nucleon interactions. The three-body and/or density-dependent terms that are necessary for good nuclear saturation properties are contained in the EDF.Applications to binding energies and spectra of nuclei in the region above
Pb are given.
PACS numbers: 26.60.Cs, 21.60.Jz, 27.80.+w
In nuclear structure theory the two main computa-tional methods for heavy nuclei based upon the nucleonfermionic degrees of freedom are the Hartree-Fock orenergy-density-functional (EDF) method and the config-uration interaction (CI) method. The EDF method isoften limited to a configuration with a single Slater de-terminant. The EDF Hamiltonian has parameters thatare fitted to global properties of nuclei such as binding-energies and rms charge radii [1], [2].The CI method takes into account many Slater de-terminants. CI often uses a Hamiltonian derived fromexperimental single-particle energies and a microscopicnucleon-nucleon interaction [3]. A given CI Hamiltonianis applied to a limited mass region that is related tothe configurations of a few valence orbitals outside ofa closed shell and the associated renormalized nucleon-nucleon interaction that is specific to that mass region[4], [3]. Spectra and binding energies (relative to theclosed core) obtained from such calculations for two tofour valence particles are in good agreement with exper-iment [4], [3]. As many valence nucleons are added theagreement with experimental spectra and binding ener-gies deteriorates [5]. An important part that is miss-ing from these CI calculations is the effective two-bodyinteraction that comes from the three-body interactionof two valence nucleons interacting with one nucleon inthe core [6]. To improve agreement with experimentalspectra one often adjusts some of the valence two-bodymatrix elements. The most important part of this adjust-ment can be traced to the monopole component of thetwo-body matrix elements that controls how the effectivesingle-particle energies evolve as a function of proton andneutron number [6].Fig. 1 shows Wick’s theorem applied to a closed shellfor the one-body kinetic energy, the two-body interac-tion and the three-body interaction. The part containedin the dashed box represents the closed-shell and effec-tive one-body parts of the Hamiltonian that might becontained in an EDF approach. Up to now this hasbeen treated phenomenologically in the framework of the
FIG. 1: Schematic diagram for the terms in the Hamiltonianobtained from Wick’s theorem for a closed shell. The red linesrepresent the summation over the orbitals in the closed shell.The black lines represent the valence particles and/or holes.
Skyrme Hartree-Fock or relativistic Hartree method withsome parameters (typically 6-10) fitted to global exper-imental data. There are efforts underway to relate theparameters of these phenomenological approaches to theunderlying two and three body forces between nucleons,and also to extend the functional forms to obtain im-proved agreement with experiment [7]. The part con-tained in the solid-line box is the residual interactionused for CI calculations. The remaining term is a va-lence three-body interaction.In this paper we discuss a new method for obtaininga valence Hamiltonian for valence nucleons outside of adoubly-closed shell. The specific application is made for
Pb, but it could be applied to any other doubly closed-shell system. The single-particle energy for orbital a isdefined as e a = E ( Pb + a ) − E ( Pb) , (1)where E ( Pb) is the energy of the closed-shell configu-ration for
Pb, and E ( Pb + a ) is the energy of theclosed-shell configuration plus one nucleon constrainedto be in orbital a . Eq. 1 defines the one-body part ofthe CI calculations. Often experimental data are usedfor the energies in Eq. 1. In this paper we will use theresults of EDF calculations for these energies. The prac-tical use of Eq. 1 requires that two states be connectedby a spectroscopic factor of near unity.The two-body part of the CI Hamiltonian isobtained with the usual renormalization proce-dure [4]. For our examples, the active valenceorbitals are (0 h / , f / , i / ) for protons and(0 i / , g / , j / ) for neutrons. For the two-bodyvalence interaction we use the N LO nucleon-nucleoninteraction [8] renormalized to the nuclear mediumwith the V lowk method [9] with a cut-off of Λ = 2 . hω in the excitations energy. Weuse harmonic-oscillator radial wavefunctions with ¯ hω =6.883 MeV.The new aspect of our method is to take the monopolepart of the effective two-body interaction from¯ V ab = E ( Pb + a + b ) − E ( Pb) − e a − e b , (2)where E ( Pb + a + b ) is the spherical EDF energy ofthe configuration for a closed shell plus two nucleons con-strained to be in orbitals a and b . This monopole interac-tion contains both the two and three body terms shownby the solid-line box in Fig. 1 to the extent that theyare contained in the EDF phenomenology. We modifythe monopole part of the microscopic valence interactionto reproduce the results of Eq. 2. With this modifi-cation, the CI calculations closely reproduce the EDFcalculations for single-Slater determinants, even whenrelatively many valence nucleons are added. Thus, theCI calculations are constrained to reproduce the trendsof closed-shell energies and effective single-particle en-ergies obtained with the EDF. For our model space or-bitals, Eq. 2 involves about thirty configurations for twonucleons (proton-proton, neutron-neutron and proton-neutron), but these calculations in a spherical basis arecomputationally fast.For this paper we will use the EDF results based on theSkxm Skyrme interaction [1]. An important property ofSkxm is that the experimental single-particle energies forthe low-lying single-particle states around Pb are re-produced with an rms deviation of about 300 keV. Skxmalso has a reasonable value of the incompressibility (234MeV). We are not aware of any other Skyrme interactionthat can do better for the single-particle energies as de-fined by Eq. 1. For the lowest state for protons (0 h / for Bi) and neutrons (1 g / for Pb), the differencebetween experiment and theory can be reduced to on the
Bi ham 9/2 − -5(cid:10)-4(cid:10)-3(cid:10)-2(cid:10)-1(cid:10)0(cid:10)1(cid:10) E ( M e v ) experiment Bi (N=126)(Z=83) 9/2 − FIG. 2: Comparison of experiment and theory (ham) for
Bi. The energies are with respect to that of
Pb. Thelength of the lines indicate the spin with positive parity (red)and negative parity (blue). Experimental levels that are un-known or uncertain are shown by the black dots. order of 20 keV with only a small increase of χ =0.82to χ =0.89 for all of the data considered in [1]. This isaccomplished by using a higher weight for these two dataand requires a small adjustment of the Skxm parameters.Since the precise energies of these orbitals are importantfor the results presented here, we use this new Skyrmeinteraction called Skxmb. If we use Skxm or any otherSkyrme interaction, our conclusions are the same, butthe deviation with experiment is worse mainly becausethe single-particle energies are worse. The binding en-ergy of Pb with Skxmb is 1636.46 MeV compared tothe experimental value of 1636.45 MeV.The results obtained from Skxmb for the energies ofsingle-particle states
Bi relative to the energy of
Pbare shown in Fig. 2. The energy of the lowest state,0 h / , is reproduced due to the fit constraint. The nexttwo states (related to the 1 f / and 0 i / orbitals) arealso well reproduced. One observes in experiment statesrelated to core-excitation of Pb starting about threeMeV above the ground state.For the lowest proton orbital with a = b = (0h / ) therenomalized N LO monopole interaction is ¯ V N LO =0.170MeV (it is repulsive due to the Coulomb interac-tion). The result obtained from Eq. 2 with Skxmb is¯ V EDF =0.288 MeV. The EDF-monopole comes from bothterms in the box in Fig. 1 and also contains higher-ordercontributions implicit in the EDF functional. Whereas,the N LO monopole only contains the valence two-bodyinteraction corrected to second order. The difference is¯ V EDF − ¯ V N LO =0.118 MeV. This correction is included inCI by modifying all of the valence TBME < V > J = < Po ham 0+ 2+ 4+ 6+ 8+11 − -10(cid:10)-9(cid:10)-8(cid:10)-7(cid:10)-6(cid:10)-5(cid:10)-4(cid:10) E ( M e v ) experiment Po (N=126)(Z=84) 0+ 2+ 4+ 6+ 8+11 − − − FIG. 3: Comparison of experiment and theory (ham) for
Po (see caption to Fig. 2).
Fr ham 9/2 − − − − − − − − − − -20(cid:10)-19(cid:10)-18(cid:10)-17(cid:10)-16(cid:10)-15(cid:10)-14(cid:10) E ( M e v ) experiment Fr (N=126)(Z=87) 9/2 − − − − − − FIG. 4: Comparison of experiment and theory (ham) for
Fr(see caption to Fig. 2). abJ | V | abJ > for the 0 h / orbital by < | V | > J, eff = < | V | > J, N LO − ¯ V N LO + ¯ V EDF . (3)Similar corrections are made for all other diagonal pairsof orbital in the model space.For the CI calculations we use the code NuShell [10].The theory Hamiltonian (ham) consists of Skxmb for thesingle-particle energies, and two-body matrix element ob-tained from the renormalized N LO interaction correctedto second-order, and then finally the two-body monopolescorrected with Skxmb with Eq. 3. The energies of
Pb,
Fr and
Rn are shown in Fig. 3-5. The agreement
Ra ham 0+ 2+ 4+ 6+ 8+11 − − -23(cid:10)-22(cid:10)-21(cid:10)-20(cid:10)-19(cid:10)-18(cid:10)-17(cid:10) E ( M e v ) experiment Ra (N=126)(Z=88) 0+ 2+ 4+ 6+ 8+11 − − FIG. 5: Comparison of experiment and theory (ham) for
Ra (see caption to Fig. 2).
Po first order 0+ 2+ 4+ 6+ 8+11 − -10(cid:10)-9(cid:10)-8(cid:10)-7(cid:10)-6(cid:10)-5(cid:10)-4(cid:10) E ( M e v ) experiment Po (N=126)(Z=84) 0+ 2+ 4+ 6+ 8+11 − − − FIG. 6: Comparison of experiment and theory (first order)for
Po (see caption to Fig. 2). between experiment and theory is good for the spec-tra and for the absolute energy relative to
Pb. For
Po the agreement between experiment and theory isvery good for levels up to three MeV above the groundstate. Above three MeV the level density of experimentand theory are similar, but one expects additional levelsin experiment coming from the core-excitation of
Pb.For
Fr and
Ra the theoretical level density is muchhigher than experiment because the experimental con-ditions select mainly the yrast levels. For the low-lyinglevels in Figs. 3 − Pb) is ∆ BE ( M e V ) Proton Number
FIG. 7: Binding energies relative to
Pb. Experimentare the points connected by a line. The results of theCI(N LO+Skxmb) are shown by crosses. The results of thespherical EDF are shown by the dashed line. usually within 100 keV, but there some exceptions withdeviations up to about 300 keV (e.g. the 11- in
Ra).These deviations may be due to many factors such aslack of third-order diagrams, the use of the harmonic-oscillator basis for the renormalized N LO matrix ele-ments, non-monopole three-body contributions, or inad-equacies in the EDF Skxmb interaction.When many nucleons are added, the monopole contri-bution goes as ∆ E = n ( n −
1) ¯
V / , (4)where n is the number of valence nucleons. Thus theEDF monopole corrections become much more importantas one adds many valence nucleons. When we constrainthe CI to the single configuration (0 h / ) for the va-lence protons, the CI calculation gives a binding energyincrease of 25.05 MeV (relative to Pb). The EDFcalculation (with the same assumption for the configu-ration) gives 25.24 MeV. These are close to each otherdue to our EDF monopole correction to the valence ma-trix elements. If the EDF monopole correction were notincluded in CI the results would differ by (45)x(0.118)= 5.3 MeV. The microscopic valance interaction on itsown is too strong and gives an “over-saturation.” Theresults for the (1 f / ) configuration are 13.27 MeV forCI and and 13.41 MeV for EDF. The difference be-tween CI and EDF might be interpreted in terms of aneffective valence three-body monopole interaction withstrength ∆E = 25 . − .
05 = 0 .
19 MeV for (0 h / ) and ∆E = 25 . − .
05 = 0 .
14 MeV (1 f / ) . With∆ E = n ( n − n −
2) ¯ V /
6, ¯ V is on the order of 1 − V includes the three-body monopole interaction onthe right-hand side of Fig. 1, but it may also includeother non-quadratic terms that emerge from the EDFsolutions. For practical purposes ∆E is small comparedto other sources of error in the theory and it may beignored. Pb ham 9/2+11/2+15/2 − -5(cid:10)-4(cid:10)-3(cid:10)-2(cid:10)-1(cid:10)0(cid:10)1(cid:10) E ( M e v ) experiment Pb (N=127)(Z=82) 9/2+11/2+15/2 − FIG. 8: Comparison of experiment and theory (ham) for
Pb (see caption to Fig. 2).
The CI calculation were carried out up to
U ( Z =92) where the M-scheme dimension is about 1.5 million.The results for the ground state energies are comparedto experiment in Fig. 7. The EDF calculation is basedupon the spherical (0 h / ) n configuration with n = 1 to10. The difference between EDF and CI can be regardedas the correlation energy in the nuclear ground state,in this case mainly due to the “pairing” interaction. Thecorrelation results in wavefunctions that are highly mixedin the valence proton basis. For example the ground stateof U contains only 4.7% of the (0 h / ) component.Up to Z = 88 the difference between experiment andtheory for the binding energy relative to Pb is on theorder of 100 keV, and after this it gradually increases toabout 700 keV for
U.The pairing interaction also appears in Fig. 3 for
Poby the difference in energy between the ground state andthe J π =8 + state which is dominated (99.86%) by the(0 h / ) configuration. We show in Fig. 6 the spectrumfor Po obtained from the first-order N LO V lowk ma-trix elements. Comparison with Fig. 3 (which includessecond order) shows that two-thirds of the pairing comesfrom second-order diagrams. The tensor interaction isimportant for second-order pairing through the bubble-diagram which links the valence protons with the coreneutrons.Results for neutrons for the spectra of
Pb and
Pbare shown in Figs. 8 and 9, respectively. The single-particle energies of the 0 i / and 0 j / orbitals in Pbare 200-400 keV too high with Skxmb. This is the reasonwhy the theoretical energies of the 10 + , 11 − and 13 − states are too high in Pb. The results for
Pb areimproved when the energies of these two single-particlestates are taken from experiment for
Pb (left-hand sideof Fig. 8) as shown in Fig. 10. For our method to give
Pb ham 0+ 2+ 4+ 6+ 8+10+11 − − − -11(cid:10)-10(cid:10)-9(cid:10)-8(cid:10)-7(cid:10)-6(cid:10)-5(cid:10) E ( M e v ) experiment Pb (N=128)(Z=82) 0+ 2+ 4+ 6+ 8+10+11 − − FIG. 9: Comparison of experiment and theory (ham) for
Pb (see caption to Fig. 2).
Pb ham* 0+ 2+ 4+ 6+ 8+10+11 − − − -11(cid:10)-10(cid:10)-9(cid:10)-8(cid:10)-7(cid:10)-6(cid:10)-5(cid:10) E ( M e v ) experiment Pb (N=128)(Z=82) 0+ 2+ 4+ 6+ 8+10+11 − − FIG. 10: Comparison of experiment and theory (ham*) for
Pb (see caption to Fig. 2). Theory (ham*) is the sameas theory (ham) except that the single-particle energies forthe neutron i / and j / orbitals are taked from the exper-imental values in Pb. the same results for CI and EDF in the limit of sphericalsingle Slater determinants, one must take both the single-particle energies and two-particle monopole energies fromthe EDF calculation; one cannot arbitrarily change thesingle-particle energies. Thus it is important to obtainEDF functionals that reproduce low-lying single-particleenergies near the doubly-magic nuclei.Results for
Bi and
Po are shown in Fig. 11 and12. Results for the low-lying proton-neutron spectrum of
Bi are comparable to those shown by [3]. The theo-
Bi ham 0 − − − − − − -10(cid:10)-9(cid:10)-8(cid:10)-7(cid:10)-6(cid:10)-5(cid:10)-4(cid:10) E ( M e v ) experiment Bi (N=127)(Z=83) 1 − − − − FIG. 11: Comparison of experiment and theory (ham) for
Bi (see caption to Fig. 2).
Po ham 0+ 2+ 4+ 6+ 8+10+12 − −
14+ 16+18+ -21(cid:10)-20(cid:10)-19(cid:10)-18(cid:10)-17(cid:10)-16(cid:10)-15(cid:10) E ( M e v ) experiment Po (N=128)(Z=84) 0+ 2+ 4+ 6+ 8+10+11 − −
14+ 18+
FIG. 12: Comparison of experiment and theory (ham) for
Po (see caption to Fig. 2). retical energies for the high-spin state would be in betteragreement with experiment if the experimental single-particle energies from
Pb are used for the neutrons.But some specific disagreements remain, for example the-oretical the J π = 14 − state in Bi remains about 500keV too high compared to experiment.The monopole interactions for the 1 g / neutron or-bital are ¯ V N LO = − V EDF =0.017 MeVgiving a correction of 0.017 − (0.076) = 0.093 MeV. Themonopole interactions between the 0 h / proton orbitaland the 1 g / neutron orbital are ¯ V N LO = − V EDF = − − − ( − − N = 126 isotones show that this change in the monopoleinteraction is crucial for obtaining the correct absolutebinding energies. Second-order corrections are importantfor the pairing interaction. As illustrated in the case of Pb and
Pb, the accuracy of this method based on EDF results for the monopole energies plus N LO forthe renormalized residual interaction is limited by theaccuracy of the EDF methods to reproduce the bindingenergies for states one nucleon removed from a closedshell (Eq. 1). In our examples for
Pb the Skyrmeparameters were optimized for the precise ground-stateenergies of of
Bi and
Pb leaving the rms deviationfor all other nuclei about the same as shown in [1]. Thismethod can be applied to any other doubly-closed shellsystem, but its accuracy will be limited by the accuracyof the EDF results for single-particle energies. Similar lo-cal optimizations may be possible for other mass regions.In the coming years we may expect improvements in EDFtheory and phenomenology towards a improved univer-sal functional. For cases where the basis dimensions aretoo large for exact CI methods, it would be interested toapply our Hamiltonian to approximate methods withinthis model space for valence nucleons outside of
Pb.
Acknowledgments
This work is partly supported byNSF Grant PHY-0758099 and the DOE UNEDF-SciDACgrant DE-FC02-09ER41585. [1] B. A. Brown, Phys. Rev. C , 220 (1998).[2] P. Klupfel, P. G. Reinhard, T. J. Burvenich and J. A.Maruhn, Phys. Rev. C , 034310 (2009).[3] L. Coraggio, A. Covello, A. Gargano, N. Itaco and T. T.S. Kuo, Prog. in Part. and Nucl. Phys. , 135 (2009).[4] M. Hjorth-Jensen, T. T. S. Kuo and E. Osnes, Phys. Rep. , 125 (1995).[5] B. A. Brown and B. H. Wildenthal, Ann. Rev. of Nucl.Part. Sci. , 29 (1988). [6] A. P. Zuker, Phys. Rev. Lett. , 042502 (2003).[7] J. E. Drut, R. J. Furnstahl and L. Platter, Prog. in Part.and Nucl. Phys. , 120 (2010).[8] D. R. Entem and R. Machleidt, Phys. Rev. C , 041001(2003).[9] Prog. in Part. and Nucl. Phys.65