Angular Momentum Projected Configuration Interaction with Realistic Hamiltonians
aa r X i v : . [ nu c l - t h ] J a n Angular Momentum Projected Configuration Interaction with Realistic Hamiltonians
Zao-Chun Gao , ∗ and Mihai Horoi Department of Physics, Central Michigan University, Mount Pleasant, Michigan 48859, USA China Institute of Atomic Energy P.O. Box 275-18, Beijing 102413, China (Dated: November 11, 2018)The Projected Configuration Interaction (PCI) method starts from a collection of mean-field wavefunctions, and builds up correlated wave functions of good symmetry. It relies on the GeneratorCoordinator Method (GCM) techniques, but it improves the past approaches by a very efficientmethod of selecting the basis states. We use the same realistic Hamiltonians and model spaces asthe Configuration Interaction (CI) method, and compare the results with the full CI calculations inthe sd and pf shell. Examples of Mg, Si, Cr, Fe and Ni are discussed.
PACS numbers: 21.60.Cs,21.60.Ev,21.10.-k
I. INTRODUCTION
The full configuration interaction (CI) method using aspherical single particle (s.p.) basis and realistic Hamilto-nians has been very successful in describing various prop-erties of the low-lying states in light and medium nuclei.The realistic Hamiltonians, such as the USD [1, 2]in the sd shell, the KB3 [3], FPD6 [4] and GXPF1 [5] in the pf shell, have provided a very good base to study variousnuclear structure problems microscopically.Some sd and pf nuclei such as Si and Cr are welldeformed. Their collective behavior is confirmed by thestrong collective E2 transitions and the rotational behav-ior of the yrast state energies, E ( I ) ∼ I ( I + 1). Themean-field description in the intrinsic frame naturallytakes advantage of the spontaneous symmetry breaking.This approach provides some physical insight, but theloss of good angular momentum of the mean-field wavefunctions makes the comparison with the experimentaldata difficult. The CI calculations in spherical basis pro-vide the description in the laboratory frame. The angularmomentum is conserved, but the physical insight associ-ated with the existence of an intrinsic state is lost.There is a long-lasting effort to connect the mean-fieldand CI techniques. Elliott was the first to point out theadvantage of a deformed intrinsic many body basis anddeveloped the SU(3) Shell Model [6] for sd nuclei. In El-liott’s model, the rotational motion was associated withstrict SU(3) symmetry, approximately realized only near Ne and Mg. This model was limited only to the sd shell.The Projected Shell Model(PSM) [7] can be consid-ered as a natural extension of the SU(3) shell model toheavier systems. In this model, the quadrupole force,that Elliott’s model used, the monopole pairing andthe quadrupole pairing forces were included in the PSMHamiltonian. The deformed intrinsic Nilsson+BCS ba-sis are projected onto good angular momentum, and thePSM Hamiltonian is diagonalized in the space spanned ∗ Electronic address: [email protected] by the projected states. The Nilsson model [8] has beenproven to be very successful in describing the deformedintrinsic single particle states, and the quadrupole forcewas found to be essential for describing the rotationalmotion [6]. Despite its simplicity PSM was proven tobe a very efficient method in analyzing the phenom-ena associated with the rotational states, especially thehigh spin states, not only for axial quadrupole deforma-tion, but also for the octupole [9] and triaxial shapes[10, 11]. However, its predictive power is limited becausethe mulitpole-multipole plus pairing Hamiltonian has tobe tuned to a specific class of states, rather than an re-gion of the nuclear chart.Besides the Projected Shell Model, another sophisti-cated approach based on the projection method is MON-STER and the family of VAMPIRs [12]. The MON-STER is similar to the PSM, but the basis is obtainedby projecting the Hartree-Fock-Bogoliubov (HFB) vac-uum and the related 2-quasiparticle configurations ontogood quantum numbers, including neutron and protonnumber, parity, and the angular momentum. VAMPIR,which performs the energy variation after the projection,is more sophisticated than MONSTER. However, theparticle number plus angular momentum projection im-plemented in MONSTER and VAMPIR requires at least3-dimensional integration in the axial symmetric case.This makes it very difficult to extend these models tonon-axial cases, where the projection with 5-dimensionalintegration would be needed. MONSTER and VAMPIRcan use realistic Hamiltonians.Instead of using quasiparticle configurations, as PSMand MONSTER/VAMPIR do, the Quantum MonteCarlo Diagonalization (QMCD) method [13] takes the ad-vantage of the Hartree-Fock (HF) mean-field that breaksthe symmetries of Hamiltonian. The mean-field is notrestricted to axial symmetry. QMCD starts from an ap-propriate initial state, as in the shell-model Monte Carloapproach [14], to select an optimal set of basis states,and the full Hamiltonian is diagonalized in this basis.More basis states are iteratively added until convergenceis achieved. It is very interesting that only the M -projection is applied to in the basis states, yet the totalangular momentum seems to be fully recovered from thediagonalization when the convergence limit is achieved.However, the process of selecting the basis states is re-portedly very time consuming.The Generator Coordinate Method (GCM)[15] is astandard method to describe collective states that goesbeyond the mean-field. The angular momentum projec-tion technique is a special case of GCM when the genera-tor coordinates include the Euler angles. In the standardGCM, the excited states are constructed in a basis ofHFB vacua, which differ in one or several collective coor-dinates. However, in many cases, it has been found thatthe excitation energies provide by GCM are too high.The GCM wave functions may be more appropriate todescribe more correlations in the ground state, ratherthan the excited states. However, our investigations (seebelow) show that neither the g.s. nor the excited statesare accurately described by this simple GCM procedurewhen realistic effective Hamiltonians and HF vacua areused.In the present work, we developed a new method calledthe Projected Configuration Interaction (PCI), whichuses GCM and PSM techniques. The PCI basis includesangular momentum projected states generated from aclass of constraint HF vacua. However, in addition tothese GCM-like states PCI includes a large number oflow-lying np − nh excitations built on those states. InPCI the deformed intrinsic states are Slater Determinants(SD), and the particle number projection is no longer re-quired. This method can use the same realistic Hamil-tonian as any CI method. This feature makes the directcomparison between CI and PCI possible. One advan-tage of this method is that it keeps the number of basisstates small, even for cases when the CI dimensions aretoo large for the today’s computers.The paper is organized as follow. Section II presentsthe formalism used by PCI. The efficiency of the angularmomentum projection is discussed in section III. SectionIV analyzes the choice of the PCI basis and presents someresults for the sd and pf nuclei. Section V is devoted tothe analysis of the quadrupole moments and E2 transi-tions. Conclusions and outlook are given in section VI. II. THE METHOD OF THE PROJECTEDCONFIGURATION INTERACTION (PCI)
The model Hamiltonian used in CI calculations can bewritten as: H = X i e i c † i c i + X i>j,k>l V ijkl c † i c † j c l c k , (1)where, c † i and c i are creation and annihilation operatorsof the spherical harmonic oscillator, e i and V ijkl are one-body and two-body matrix elements that can be obtainedfrom effective interaction theory, such as G-Matrix pluscore polarization [16]; it can be further refined using theexperimental data [5, 17]. One can introduce the deformed single particle (s.p.)basis, which can be obtained from a constraint HF solu-tion. Alternatively, one can take the single particle statesobtained from the following s.p. Hamiltonian, H s.p. = h sph − ǫ ~ ω r π ρ Y ( θ )+ ǫ ~ ω r π ρ Y ( θ ) . (2)Here h sph = X i E i c † i c i , (3)is the spherical single particle part of the Hamiltonianwith the same eigenfunctions as the spherical harmonicoscillator, but the energy of each orbit is chosen suchthat the result from this Hamiltonian is near to the HFsolution; ǫ and ǫ are the quadrupole and hexadecupoledeformation parameters, ρ = p mω ~ r is dimensionless,and we take [18, 19] ~ ω = 45 A / − A / . (4)The deformed s.p. creation operator is given by thefollowing transformation: b † k = X i W ki c † i , (5)where the matrix elements W ki = h b k | c i i are real inour calculation. Inserting the reversed transformationof Eq. (5) c † i = X k W tik b † k (6)into the model Hamiltonian Eq. (1), we obtain the trans-formed H in the deformed s.p. basis H = X ij h (1) ij b † i b j + X i>j,k>l h (2) ijkl b † i b † j b l b k . (7)Here h (1) and h (2) are one-body and two-body matrixelements of H in the deformed basis. The Slater Deter-minant (SD) built on the deformed single particle statesis given by | κ i ≡ | s, ǫ i ≡ b † i b † i ...b † i n |i , (8)where s refers to the Nilsson configuration, indicating thepattern of the occupied orbits, and ǫ is the deformationdetermined by ǫ and ǫ . The matrix element h κ | H | κ ′ i ,can be calculated using Eq.(7).Generally, | κ i doesn’t have good angular momentum I . It is well known that the projection on good angularmomentum would add correlations to the wave function.The general form of the nuclear wave function is thereforea linear combination of the projected SDs (PSDs), | Ψ σIM i = X Kκ f σIKκ P IMK | κ i , (9)where ˆ P IMK = 2 I + 18 π Z d Ω D IMK (Ω) ˆ R (Ω) (10)is the angular momentum projection operator. D IMK isthe D-function, defined as D IMK (Ω) = h IM | ˆ R (Ω) | IK i ∗ , (11)ˆ R is the rotation operator, and Ω is the solid angle. If onekeeps the axial symmetry in the deformed basis, D IMK inEq(10) reduces to d IMK ( β ) = h IM | e − iβ ˆ J y | IK i and thethree dimensional Ω reduces to β .The energies and the wave functions [given in terms ofthe coefficients f σIKκ in Eq.(9)] are obtained by solvingthe following eigenvalue equation: X K ′ κ ′ ( H IKκ,K ′ κ ′ − E σI N IKκ,K ′ κ ′ ) f σIKκ ′ = 0 , (12)where H IKκ,K ′ κ ′ and N IKκ,K ′ κ ′ are the matrix elements ofthe Hamiltonian and of the norm, respectively H IKκ,K ′ κ ′ = h κ | HP IKK ′ | κ ′ i , (13) N IKκ,K ′ κ ′ = h κ | P IKK ′ | κ ′ i . (14)The matrix element of H IKκ,K ′ κ ′ and N IKκ,K ′ κ ′ can beexpanded as h κ | HP IKK ′ | κ ′ i = 2 I + 18 π Z d Ω D IKK ′ (Ω) h κ | H ˆ R (Ω) | κ ′ i , (15) h κ | P IKK ′ | κ ′ i = 2 I + 18 π Z d Ω D IKK ′ (Ω) h κ | ˆ R (Ω) | κ ′ i , (16)where h κ | H ˆ R (Ω) | κ ′ i = X κ ′′ h κ | H | κ ′′ ih κ ′′ | ˆ R (Ω) | κ ′ i . (17)Here κ ′′ has the same deformation as κ , and runs over allSDs with nonzero h κ | H | κ ′′ i . Therefore, the main problemis to calculate the matrix element of the rotation ˆ R (Ω)between different SDs, h κ | ˆ R (Ω) | κ ′ i . | κ i and | κ ′ i may have different shapes. We denote a † , a the single particle operators with deformation ǫ a that create the SD | κ i , and b † , b , with deformation ǫ b ,that create | κ ′ i . Expressing h κ | ˆ R (Ω) | κ ′ i with a † , a and b † , b , one gets h κ | ˆ R (Ω) | κ ′ i = h| a i n ...a i a i ˆ R (Ω) b † j b † j ...b † j n |i = h| a i n ...a i a i ˆ R (Ω) b † j ˆ R − (Ω) ˆ R (Ω) b † j ˆ R − (Ω) ... ˆ R (Ω) b † j n ˆ R − (Ω) |i = h| a i n ...a i a i X k R k j a † k ! X k R k j a † k ! ... X k n R k n j n a † k n ! |i = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R i j R i j . . . R i j n R i j R i j . . . R i j n . . . . . . . . . . . . . . . . . . . . . .R i n j R i n j . . . R i n j n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (18)Here ˆ R (Ω) b † j ˆ R − (Ω)= X l W bjl ˆ R (Ω) c † l ˆ R − (Ω)= X lk W bjl D ∗ kl (Ω) c † k = X i X lk W bjl D ∗ kl (Ω) W aik ! a † i = X i R ij a † i . (19)Note that the rotation never mixes the neutron and pro-ton wave functions. Therefore, the matrix elements inEq.(18) can be calculated for neutron and proton sepa-rately.The choice of the PCI basis will be described in SectionIV. III. THE EFFICIENCY OF THE PROJECTIONON GOOD ANGULAR MOMENTUM
Let’s analyze the quality of the deformed single particlestates for the USD Hamiltonian [2]. The E i energies (seeEq.(3)) are properly adjusted so that the SD with thelowest energy is close to the HF vacuum, as will be dis-cussed in this Section. For sd shell, we use E d / = − . E s / = − . E d / = 1 . ǫ ( ǫ = 0) are plotted in Fig.1.To find the lowest value E def of h κ | H | κ i , the energysurface of E exp ( s, ǫ ) = h s, ǫ | H | s, ǫ i (20)needs to be calculated for each configuration s . Here, weuse the USD Hamiltonian, and take Si as a case study.The SD with all 12 particles occupying the Nilsson orbitsoriginating from the d / spherical -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-10-8-6-4-2024 [ ] / [ ] / [ ] / [ ] / [ ] / d s E ne r g i e s ( M e V ) d [ ] / FIG. 1: The Nilsson levels for sd shell orbit are used in Eq. (20). Such an SD is denotedby s hereafter. This configuration provides the lowestenergy for a large range of deformations and is expectedto describe very well the mean-field minimum, i.e. theground state (g.s.), h κ | H | κ i at certain deformation(s).The energy surface of the s configuration as a functionof ( ǫ , ǫ ) is shown as the upper surface in Fig. 2. Theminimum energy, E def = − .
55 MeV at deformation( ǫ , ǫ ) def = ( − . , − . E HF = − .
61 MeV) [18]. Besides, the cal-culated quadrupole moment, − . , is also very closethe the HF value, − . [18]. Thus, we reached anapproximated solution to the HF mean-field from a sim-ple Nilsson Hamiltonian. For other deformed sd nucleithe situation is similar to Si, as shown in Table I.The ground state of Si should have good angular mo-mentum with spin I = 0. We calculate the expectationvalue of H with respect to the projected SD: E exp ( I, s, ǫ ) = h s, ǫ | HP IKK | s, ǫ ih s, ǫ | P IKK | s, ǫ i , (21)where K is defined by ˆ J z | s, ǫ i = K | s, ǫ i . The energy sur-face of E exp ( I = 0 , s , ǫ ) is shown as the lower surfacein Fig. 2. The minimum value E pj ( I = 0) = − . ǫ , ǫ ) pj = ( − . , − . .
30 MeV abovethe exact ground state energy of the USD Hamiltonianat − .
94 MeV. Similar calculations for other deformed sd -shell nuclei are reported in Table. I. These resultsindicate that the restoration of the angular momentumhas a significant contribution to the ground state (g.s.)correlation energy.An even stronger argument in favor of imposing good FIG. 2: (Color online) Energy surfaces of the USD Hamilto-nian with respect to the unprojected SD( s ) (upper surface)and the projected SD( s ) with I=0 (lower surface), as func-tions of quadrupole ǫ and hexadecupole ǫ deformation. Si J np-nh Truncation Spherical Deformed C I E n e r g i e s ( M e V ) Si FIG. 3: The ground state energies (upper panel) and thecorresponding h J i values (lower panel) of Si from sphericaland deformed CI as functions of different n p- n h truncation TABLE I: Energies (in MeV) with USD Hamiltonian for some deformed sd Nuclei. E sph is the spherical HF energy, E HF is theminimum of the deformed HF energy, E def is the lowest energy of Eq.(20) and ( ǫ , ǫ ) def is the deformation for E def , E pj ( I = 0)is the lowest energy of Eq.(21) at spin I = 0 and ( ǫ , ǫ ) pj is the deformation for E pj ( I = 0). E Full CI is the exact solution ofthe USD Hamiltonian.Nucleus E sph a E HF a E def ( ǫ , ǫ ) def E pj ( I = 0) ( ǫ , ǫ ) pj E Full CI20 Ne − . b − . − .
38 (0 . , − . − .
86 (0 . , − . − . Mg − . − . − .
80 (0 . , . − .
04 (0 . , . − . Si − . − . − .
55 ( − . , − . − .
64 ( − . , − . − . Ar − . − . − .
56 ( − . , . − .
78 ( − . , . − . a Data taken from Ref.[18]. b The − .
46 MeV in Ref. [18] has been corrected as − .
79 MeV. angular momentum to the wave functions can be foundby comparing the results of CI calculations using spher-ical and deformed s.p. bases, as shown in Fig. 3. Thedeformation for the deformed CI basis is ( ǫ , ǫ ) def =( − . , − .
10) as shown in Table I. The energies at 0p-0h are the values of E sph and E def shown in the sameTable I. The E def is much lower than E sph , but h J i for the deformed SD is as large as 20 .
35, far away from h J i =0 of the spherical HF state. At the 2p-2h trun-cation level, the spherical CI energy drops very close tothat of the deformed CI, while the latter is strugglingfor recovering the angular momentum, and h J i dropsto 11 .
30. At the 4p-4h truncation level, the sphericalCI energy is even lower than the deformed ones, yet theangular momentum of the latter hasn’t been completelyrecovered. The advantage of the deformed mean field isthen completely lost in the CI due to the lack of goodangular momentum. At the 8p-8h truncation level thedeformed CI state completely recovers its angular mo-mentum, and the results from spherical CI and deformedCI become equivalent. Thus, we conclude that the de-formed CI won’t benefit from the deformed mean fieldunless the angular momentum is recovered from the out-set. The PCI method described in Section II maintainsthe advantages of the deformed mean field and of config-uration mixing specific to CI, and could provide furtherimprovement to the E pj ( I = 0) in Table I.To describe those non-zero spin states, one can changethe spin from 0 to I in Eq. (21), and keep s and ǫ un-changed. Then one gets a sequence of energies, E pj ( I )shown as D=1 in Fig. 4. One can see that the values of E pj ( I ) have approximately reproduced the I ( I + 1) fea-ture of a rotational band in the low spin region. The 2MeV gap between the D=1 band and the Full CI bandis expected to be reduced by taking advantage of theCI techniques, in which the interaction between differ-ent intrinsic configurations has been properly considered.Results of the PCI for different bases are also shownin Fig. 4. First, we take 41 projected SDs having thesame configuration ( s ), but different shapes, as in theGCM method. The 41 shapes considered are taken from ǫ = − . . ǫ is obtainedby minimizing the energy of Eq.(21) with I = 0 at each ǫ . One can see that the calculated g.s. band is not Spherical HF Deformed HF D=1 GCM:D=41 PH:D=40 PH:D=108 Full CI E n e r g i e s ( M e V ) Spin Si FIG. 4: Calculations of the ground state band in Si withdifferent basis. Except for the GCM:D=41 case, the deforma-tion for all the other cases is ( ǫ = − . , ǫ = − . improved. This can be understood by the large overlapbetween the PSDs in the GCM basis, therefore, the effec-tive dimension of the basis is far below 41. Adding morePSDs with new shapes in the basis doesn’t make muchimprovement. Therefore, we considered up to 2 p − h excitations of the deformed s configuration used in theD=1 case, and built a PCI basis by selecting D=40 SDs(see next section). With this basis we solved the gen-eralized eigenvalue problem described in Section II. Theresults (PH:D=40) in Fig. 4 indicate a significant im-provement over GCM (GCM:D=41). As the number ofexcited PSDs increases up to 107, we obtain the band(labeled by PH:D=108), which is only about 300 keVabove the band obtained with full CI. Therefore, we con- ( 2, 4) def : D=1 ( 2, 4) pj : D=1 ( 2, 4) def : D=107 ( 2, 4) pj : D=108 Full CI E n e r g i e s ( M e V ) Spin Si FIG. 5: Comparison of the calculated ground state bands in Si with deformations ( ǫ , ǫ ) pj = ( − . , − .
20) [used inFig.4] and ( ǫ , ǫ ) def = ( − . , − . clude that the PCI method is not only an efficient wayof reducing the dimension of the CI calculations, at leastin deformed nuclei, but can be also very instrumental inmaking the connection between intrinsic states and lab-oratory wave functions. IV. CHOICE OF THE PCI BASIS
As already mentioned, one important problem is thechoice of the PCI basis. It is very difficult to find aset of SDs that would make an efficient PCI basis, dueto the complexity of possible structures that exist in thenuclei, such as the spherical states, rotational states, var-ious vibrational states, shape coexistence, etc. It wouldbe helpful to be able to dentify the most relevant intrinsicstates of a nucleus when we start a PCI calculation. Un-fortunately, there is a large number of ways of achievingthat goal. Below, we describe some of them. Assumingthat we found these N starting states that we denote as | κ j , i ( j = 1 , ...N ), we further consider for each selected | κ j , i a set of relative n p- n h SDs | κ j , i i , and select someof them into the basis according to how much effect theycan have on the energy of the state. Therefore, the gen-eral structure of the PCI basis is − , n p − n h | κ , i , | κ , i i , · · · , | κ , i , | κ , i i , · · · ,. . . . . . . . . . . . . . . . . . . . . | κ N , i , | κ N , i i , · · · . (22) -0.6 -0.3 0.0 0.3 0.6-80-78-76-74-72-70-68 E e xp ( s , ) ( M e V ) E e xp ( s , ) ( M e V ) -0.6 -0.3 0.0 0.3 0.6-130-128-126-124-122-120-118-116 Si Mg FIG. 6: (Color online) E exp ( s, ǫ ) values in Mg and Si, asfunctions of quadrupole deformation ǫ . Each curve repre-sents a Nilsson Configuration s , while ǫ was chosen to min-imize E exp ( s, ǫ ) for each ǫ . The absolute minimum for each s is indicated by a black dot. N is somewhat arbitrary, but it can be increased untilreasonable convergence is achieved. Some examples aregiven below. As a preliminary application of PCI, we con-sider the same set of SDs | κ j , i in Eq.(22) for all angularmomenta I . The deformation of each configuration s maybe chosen from the minima of either Eq. (20) or Eq. (21)at certain I . However, there seems to be not much differ-ence between these two choices, especially if the studiednucleus is well deformed. As shown in Fig. 5, the PCIresults of Si with deformations ( ǫ , ǫ ) pj and ( ǫ , ǫ ) def ,(see Table I) are quite close to each other. Therefore, forwell deformed nuclei it would be simpler to use Eq. (20)for determining the shape of each starting configuration s j . In the present work we only consider axially sym-metric shapes. Some low-lying curves of E exp ( s, ǫ ) in Eq.(20) for Mg and Si are plotted in Fig. 6 as functionsof ǫ . ǫ was chosen by minimizing the E exp ( s, ǫ ) for each ǫ . Each configuration s j corresponds to a curve in thisfigure. The minima are marked by dots.In order to describe the low-lying states, we first choosethose SDs whose energies are minima of E exp ( s, ǫ ), shownas dots in Fig. 6, and take them as starting | κ j , i statesin Eq. (22). Secondly, we try to include CI-like correla-tions. Each | κ j , i may have correlations with its 1p-1hand 2p-2h SDs, | κ j , i i , through the non-diagonal matrixelements of H . For each j we include those SDs thatsatisfy the following criterion:∆ E = 12 ( E − E i + p ( E − E i ) + 4 | V | ) ≥ E cut , (23)where E = h κ, | H | κ, i , E i = h κ, i | H | κ, i i and V = h κ, | H | κ, i i . (We skipped the subscript j to keep nota-tions short.) Full CI, D=28503 PCI , D=1749 E n e r g i e s ( M e V ) Spin Mg ( USD,N=40,|K| 2 ) Full CI, D=93710 PCI , D=3366 E n e r g i e s ( M e V ) Spin Si ( USD,N=60,|K| ) FIG. 7: Energies for Mg and Si calculated using PCI (opencircles) and full CI(filled circles). 20 energies for each spin arepresented.
In this way we constructed the basis for the PCI calcu-lations of Mg and Si in the sd shell. For the Mg, weincluded N = 40 lowest energy | κ j , i SDs with | K | ≤ E cut = 10keV. For the Si case, N = 60, | K | ≤ E cut = 10keV selects 3366 SDs. The choice of | K | ≤ Full CI, D=1963461
PCI , D=5973 E n e r g i e s ( M e V ) Spin48 Cr ( GXPF1A,N=30,|K| 4 ) FIG. 8: The lowest 5 energies at each spin for Cr calculatedusing PCI (open circles) and full CI (filled circles). -2 0 2 4 6 8 10 12 14-152-150-148-146-144-142-140-138
Full CI, D=1.1 108
PCI , D=7769 E n e r g i e s ( M e V ) Spin
Full CI, D=1.1 109
PCI , D=7058 E n e r g i e s ( M e V ) Spin
FIG. 9: Calculated yrast band energies using PCI (open cir-cles) and full CI (filled circles) for Fe and Ni. The de-formed band in Ni starting around 5 MeV is also shown(see text for details). emphasize that the PCI method combines the advantageof using a set of s.p. bases of different deformations withthat of the CI np − nh configuration mixing. In m-schemeCI, for example, one can get the states for all spins us-ing the lowest M value (0 for even-even cases), althoughusing higher M values could be somewhat more efficientfor higher spins. In PCI one has to weigh in the advan-tages of using a larger number of np-nh excitations withthose of using larger K values for larger spins. Using ourchoices for the K values the lowest 20 PCI energies ofeach spin are compared in Fig. 7 with the full CI results.Almost all low-lying energies for each spin reproduce thefull CI results very well. We conclude that the selectedPCI basis has already carried almost the whole informa-tion of the low-lying states, yet we chose only a smallnumber of SDs as compared with those used in the full m -scheme CI.The QMCD [13] method also uses a relatively smallnumber of selected SDs. For a basis of 800 SDs QMCDfound a g.s. energy of − .
91 MeV for Mg. In thepresent calculation, after the orthogonalization of the 524 K = 0 SDs, 453 SDs are used to calculate the I=0 en-ergies (see Table III). The resulting PCI g.s. energy is − .
94 MeV. Furthermore, the QMCD paper only re-ports the 6 lowest states with spin up to I = 4, whileour PCI calculation with the selected 1749 SDs can pro-vide tens of excited states with spin running from 0 to 12that compare well with the full CI results. Our approachseems to largely extended the range of nuclear states thatcan be calculated using a relatively small basis.The results for Si are similar to those of Mg. Com-paring with the N = 1 calculation, that has been dis-cussed in Section III, the dimension is much larger be-cause one needs to describe a large number of low-lyingstates of the same spin. For I = 0 the number of K = 0 | κ j , i SDs is N = 15, and a total of 747 independantSDs are selected from the 886 K = 0 SDs. The first20 0 + states, up to about 18 MeV excitation energy, arereasonably close to the full CI results. For I = 0 thesituation is very much the same.The pf calculations show similar trends. For Cr, theCI m -scheme dimension is 1963461. Full CI calculationscan reproduce the rotational nature of the yrast bandand its backbending [20]. With PCI we choose N = 30, | K | ≤
4, and for E cut = 0 . Fe and Ni are not good rotors, and the presentchoice of the basis is not so efficient as for Cr. How-ever, one can still select a PCI basis that can provide areasonable description of the yrast energies when com-pared with the full CI calculations, as shown in Fig.9.The number of deformed SDs used in these calculationsis about 7000, which is just a small fraction of the fullCI m-scheme dimension for Fe, and for Ni (see TableIII). However, the choice of the PCI basis for Fe and Ni needs to be improved. Details will be presented inthe forthcoming paper.For Ni a rotational band has been identified in exper-iment [21] that starts at around 5 MeV excitation energy.The band was successfully described by full CI calcula-tions in the pf -shell using the GXPF1A interaction[17].It was also shown [17] that this band has a 4p-4h char-acter, but in order to get a good description of the band10p-10h excitations from ( f / ) configuration are neces-sary. In the present PCI calculation this rotational bandhas been identified (see Fig. 9) in the calculated excitedstates by inspecting the quadrupole moments and the B ( E
2) transitions (See Table II). The results seem to
TABLE II: Quadrupole moments (in e fm ) and B ( E , I → I −
2) (in e fm ) values of the states in the deformed band of Ni calculated with PCI and CI. The energies for I = 0 arerelative to the I = 0 bandhead. The CI values are taken fromRef. [17]Spin Energy (MeV) Q ( I ) B ( E I = 0 I = 2 I = 4 I = 6 Mg SM 1161 4518 4734 2799PCI 453 1569 1579 1492 Si SM 3372 13562 15089 9900PCI 747 2266 2919 2711 Cr SM 41355 182421 246979 226259PCI 1671 3888 5618 5594 Fe SM 1 . × . × . × . × PCI 4288 6422 6811 6806 Ni SM 1 . × . × . × . × PCI 2373 3968 5452 6231 be in reasonable good agreement with those of the fullCI calculations. The details about the calculation of thequadrupole moments and the B ( E
2) transitions are givenin the next section.The number of SDs selected for the PCI calculationare shown in the corresponding figures and comparedwith the full CI m-scheme dimensions. However, the ef-ficiency of the PCI truncation can also be assessed bycomparing the PCI dimensions with the coupled-I CI di-mensions. In PCI the states used for the diagonalizationof each spin in Eq. (12) are obtained from the selectedPSDs after the latter are orthogonalized and the redun-dant states are filtered out. The remaining states formthe PCI basis for each spin. These dimensions are com-pared in Table III with the full coupled-I CI dimensions.One can see that PCI can provide reasonable good resultsusing much smaller dimensions than the correspondingm-scheme and/or coupled-I CI calculations.Although the PCI calculations could have muchsmaller dimensions than those of the corresponding fullCI calculations, the H and N matrix in Eq. (12) aredense. To get an idea of the workload of a PCI cal-culation, the time necessary to calculate the H and N matrices for Ni at I=0, 2, 4, 6, 8 takes around 8 hoursusing one processor, and the calculation of 5 eigenvaluesfor each spin takes about 2 hour, when the generalizedLanczos method is used. As a comparison, the full CI cal-culations of the same states using the modern coupled-Icode NuShellX [22] could take several weeks, when oneprocessor is used. One should also observe that the PCIcomputational load is shifted towards the calculation ofthe matrix elements, which can be more efficiently par-allelized than any large matrix eigenvalue solver.
V. QUADRUPOLE MOMENTS AND BE2TRANSITIONS
The quadrupole moments and B ( E
2) transitions aregiven by Q ( I ) = r π h Ψ IM = I | ˆ Q | Ψ IM = I i = r π h II, | II ih Ψ I || ˆ Q || Ψ I i (24) B ( E I i −→ I f ) = 2 I f + 12 I i + 1 |h Ψ I f || ˆ Q || Ψ I i i| = |h Ψ I i || ˆ Q || Ψ I f i| (25)where ˆ Q λµ = e n ( p ) r Y λµ ( θ, φ ) , (26)with effective charges taken as e n = 0 . e , e p = 1 . e .The reduced matrix element in Eq.(24) and (25) can beexpressed in terms of the PCI wave functions h Ψ I f || ˆ Q || Ψ I i i = X Kκ,K ′ κ ′ f I i Kκ f I f K ′ κ ′ R I f K ′ κ ′ ,I i Kκ , (27)where R I f K ′ κ ′ ,I i Kκ = X ν h I i K ′ − ν, λν | I f K ′ ih Φ κ ′ | ˆ Q λν P I i K ′ − ν,K | Φ κ i . (28)Explicit expression for the h Φ κ ′ | ˆ Q λν P I i K ′ − ν,K | Φ κ i can befound in Ref. [7]. Here h I i K ′ − ν, λν | I f K ′ i is a Clebsh-Gordon coefficient.Using the PCI wave functions for the states shown inFig.7 and Fig.8, we calculated the Q ( I ) and the B ( E Mg, Si and Cr. Theresults are shown in Fig.10. The very good agreementbetween PCI and full CI indicates that the PCI wavefunctions are almost equivalent to the exact ones.
VI. CONCLUSIONS AND OUTLOOK
In this paper we investigate the adequacy of using a de-formed s.p. basis for CI calculations. We show that evenif the starting energy of the deformed mean-field is lowerthan the spherical one, the series of np − nh truncationsin the deformed basis exhibits slower convergence than Cr(GXPF1A) Si(USD) Q (I) ( e f m ) Mg(USD)
Full CI PCI B ( E ) ( e f m ) Spin 2 4 6 8 1012050100150200
Spin 0 4 8 12 1650100150200250300350400
Spin
FIG. 10: Calculated quadrupole moments (upper panel) and B ( E
2) (lower panel) from PCI (open circles) and full CI (filledcircles)for Mg, Si and Cr. that of the spherical basis, due to the loss of rotationalinvariance that is not recovering fast enough.Therefore, we propose a new strategy called ProjectedConfiguration Interaction (PCI) that marries features ofGCM, by projecting on good angular momentum SDsbuilt with deformed s.p. orbitals, with CI techniques,by mixing a large number of appropriate deformed con-figurations and their np − nh excitations. This ansatzseems to be very successful in describing not only theyrast band, but also a large class of low-lying non-yraststates. This method seems to work extremely well fordeformed nuclei, and it needs further improvements tothe choice of the PCI basis for spherical nuclei.For the sd and pf -shell nuclei that we studied we showthat the quadrupole moments and the BE(2) transitionprobabilities are very well reproduced, indicating thatnot only the energies, but also the wave functions are ac-curately described. This suggests that this method couldbe helpful in finding relations between the intrinsic states,which offer more physics insight, and the laboratory wavefunctions provided by CI.One of the simplifications that makes these calcula-tions with tens of thousands of projected states possibleis the approximation of the deformed mean-field with anaxially symmetric rotor defined by its quadrupole andhexadecupole deformations. This approach seems to de-scribe very well the HF mean field for the sd and pf -shellnuclei. We plan to extend our method to include octupoledeformation, and to consider triaxial shapes. Acknowledgments
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