Comparison of Nuclear Configuration Interaction Calculations and Coupled Cluster Calculations

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1 Comparison of Nuclear Configuration Interaction Calculations and Coupled Cluster Calculations Mihai Horoi Department of Physics, Central Michigan University, Mount Pleasant, Michigan 48859, USA Support from NSF grants PHY , PHY , PHY , and DOE grant DF-FC02-07ER41457

2 Prologue Spherical Shell Model: spherical single particle mean field, few many-body configurations observing basic symmetries. Deformed/Collective Model: Deformed droplet, deformed single particle states, crancking, particle + rotor, broken symmetry. Nuclear Configuration Interaction (CI): a (really) Large number of configurations mixed by the residual interaction - can describe both limits and even their coexistence Can Coupled Cluster describe both limits?

3 0h 1h (0 + 2)h (1+ 3)h Nuclear Configuration Interaction H'= H + (H CoM # 3/2h$) (J ) = [# CoM (NL)# (J') int ] (J ) (J ) $ # CoM (00)# int N max Center-of-mass spurious states pf d = 2 (2 j + 1) H = # K a + k a k + 1 k 2 # V kl;mn a + k a + l a n a m +L k l m n sd >= # C i i(jt) > i >= # C i i(jt z ) > i >= # C i i(mt z ) > i Lanczos algorithm $ L 1 = M $ # C 1 C N % ' ' & JT-scheme: OXBASH H L 1 = 1 L 1 + # 1 L 2 L 1 $L 2 HL 2 = # 1 L L 2 + # 2 L 3 L 1 $L 2 $L 3 HL 3 = # 2 L L 3 + # 3 L 4 L 1 $L 2 $L 3 $L 4 L J-scheme: NATHAN, NuShellX M-scheme: Oslo-code, Antoine, MFDN, MSHELL, CMichSM, # < i H j > C j = E C i j % 1 # 1 0 L 0 ( ' * '# 1 2 # 2 0L 0 * N H k L = ' 0 # 2 3 # 3 0 * ' * M M ' & 0 L 0 # N$1 * Nk )

4 Koeln Low Spin States in 56 Ni: complete spectroscopy Configuration Interaction (CI) Challenge: Can one accurately describe shape coexistence using CI, CC, QRPA, time-dependent DFT,? LBNL high spin D.Rudolf et al.,prl 88,1999 Exp. Koeln J 6 Theo: CI GXPF1A M. Horoi et al., in preparation Egido et al, PRL 93, (2004)

5 p s sd s pf s pf 5/2 g 9/ g 7/2 sdh 11/2-5x Example: 76 Sr pf 5/2 g 9/2 dimension Current limit: > np valence s.p. states Extensions: Truncations, Exponential Convergence Method, Coupled Clusters, Projected CI,

6 Renormalized Hamiltonians for Large N hω Excitations Model Spaces Renormalization methods: - G-matrix: Physics Reports 261, 125 (1995) - Lee-Suzuki (NCSM): PRC 61, (2000) QP = 0 QQ - V low k : PRC 65, (R) (2002) - Unitary Correlation Operator: PRC 72, (2004) Nucleon-Nucleon Potentials: - Argonne V18: PRC 56, 1720 (1997) - CD-Bonn 2000: PRC 63, (2000) - N 3 LO: PRC 68, (2003) - INOY: PRC 69, (2004) PP PQ = 0 H H = T + V i j + V i j k +L i< j i< j<k Ψ H > Ψ P =PΨ H H = e -S H e S = H 2 + H 3 + H 4 + O > e -S O e S

7 No-Core Shell Model (NCSM) With Effective Hamiltonians PRC 69, (2004) PRL 88, (2002) 2-body interactions can simulate 3-body effects NCSM, GFMC, rms ~ 0.9 MeV for ~ 60 states rms = 1 N s N s # i=1 E th exp ( i E i ) 2 10 B INOY h =17 MeV

8 QP = 0 pf sd pf Effective Hamiltonians for One or Two Major Shells QQ PQ = 0 3-body > two-body $ r# $ % f G p 1/ 2 f 5 / 2 p 3 / 2 f 7 / 2 H = H m (monopole) + H M (multipole) 56 Ni closed shell(cs) : H m contribution H core polarization: Phys.Rep. 261, 125 (195) Brown & Richter, PRC 74, (2006)

9 GXPF1A Effective Interaction: f 7/2 p 3/2 p 1/2 f 5/2 Renormalized G-matrix GXPF1 Phys.Rev. C 69, (2004) 699 energies, 87 nuclei, rms=168 kev GXPF1 GXPF1A M. Honma et al, ENAM04 5 matrix elements adjusted for N=34 56 Ni states J #

10 First Rotational Band in 56 Ni p e eff n =1.5 e eff = 0.5

11 Low Spin States in 56 Ni B(M1) 0 Exp. Koeln GXPF1A

12 0 2 + K. Starosta et al., Phys. Rev. Lett. 99, (2007).

13 P. Mantica, et al., PRC 77, 14303(2008) Is GXPF1A Always Right?

14 a,b,... T 1 Single-Reference Coupled-Cluster Theory (F. Coester, 1958; F. Coester and H. Kümmel, 1960; J. Čížek, 1966, 1969; J. Čížek and J. Paldus, 1971)! = t! i, a i a a i, M T T! 0 = ; = # n= 1 T 2 e T T ij ab tab ij, T3 i< j, a< b! =! n! = t! i< j< k, a< b< c ijk abc abc ijk, etc. i,j,... M = N in the exact case, M < N T T in the approximate schemes CCSD: M = 2, T = T + T, Computer Time Scaling = n n 2 4 T 1 2 o u CCSDT: M = 3, T = T + T + T, Computer Time Scaling = n n 3 5 T o u 4 6 CCSDTQ: MT = 4, T = T1 + T2 + T3 + T4, Computer Time Scaling = no nu One can extend the ground-state theory to excited states using the equation-of-motion (EOM) CC formalism, in which µ = R µ R = M R! R, 0 µ µ, n n= 0

15 H N = H - <Φ H Φ> See PRC 74, (2006) (nucl-th/ ) for details

16 Coupled Cluster (CC) vs CI for Heavy Nuclei 56 Ni, 55 Ni, 57 Ni 56 Ni GXPF1A effective interaction CI vs Coupled-cluster (CC, CR-CC) calculations: M. Horoi, J. Gour, M. Włoch, M. Lodriguito, P. Piecuch, B.A. Brown, Phys.Rev.Lett. 98, (2007) 55 Ni, 57 Ni: accepted at Phys. Rev. Lett. 0 # Full$CI G p 1/ 2 f 5 / 2 p 3 / 2 f 7 / 2 G = [# pf5/2 $# f7/2 ] $ [# pf5/2 $# f7/2 ] 0 Generic Scaling: CR CC(2,3) ~ n o 3 n u 4 CISDTQ ~ n o 4 n u 6 Excited States µ = R µ R = M R! R, 0 µ µ, n n= 0

17 Particle-Attached and Particle-Removed Equation-of-Motion CC Theories! = R! R ( N ± 1) ( N ± 1) µ µ Particle-Removing Operator M R = R ( N! 1) µ µ,( n+ 1) h! np n= O : PRL 92, (2004) PRL 94, (2005) R i 1 h! = r! i, i R # ij b ijk bc 2h! 1 p = # rb ij, R3 h! 2 p = rbc ijk i< j, b i< j< k, b< c, etc. a,b,... i,j,... Solve the eigenvalue problem ( H R ) =! R,! = E # E ( N ± 1) ( N ± 1) ( N ± 1) ( N ± 1) ( N ± 1) ( N ) N,open µ C µ µ µ µ 0

18 55 Ni, 57 Ni (GXPF1A) 56 Ni Occupied single-particle states: f7/2 Unoccupied single-particle states p3/2, f5/2, p1/2 G p 1/ 2 f 5 / 2 p 3 / 2 f 7 / 2 0 # Full$CI Still a single reference state CC problem! G = [# pf5/2 $# f7/2 ] $ [# pf5/2 $# f7/2 ] 0 CI vs Coupled-Cluster (CC, PA-EOM, PR- EOM) calculations: Jeff Gour, Mihai Horoi, Piotr Piecuch, B.A. Brown, 55 Ni, 57 Ni: accepted at Phys.Rev.Lett. iterativen 6, non-iterative N 7 iterativen 10 N 7 CCSD ~ n o 2 n u 4 Excitations in 57 Ni N 10 CCSDTQ ~ n o 4 n u 6 CISDTQ: N 7 N 10 NNZME ~ 10 9 SCALES ~ ~ ~ 10 9

19 Configuration Interaction (CI): Spherical s.p. basis vs Deformed s.p. basis Axial symmetry -> M-scheme CI problem >= # C i i(mt z ) > i

20 Angular Momentum Projected CI (PCI) Deformed Slater Determinant Deformed Nilsson Single Particle operator Spherical H.O. Nuclear State Wave-function: Where s are determined by Angular Momentum Projection Shell Model Hamiltonian

21 Choices of the basis The GCM Method: Only the G.S SDs 41 shapes with -0.6< <0.6 by 0.03 VS Particle-Hole states(ph): 40 total SD s with the same shape at minimum { }

22 PCI Calculations of the Excited states for 24 Mg and 28 Si & JM = f J K # K# J P MK $ K # (% 2,K)

23 Projected Configuration Interaction (PCI) & JM = f J K # K# J P MK $ K # (% 2,K) Basis Φ Κ κ includes particle-hole excitations corresponding to many shapes Gao and Horoi, in preparation Advantages: - used DFT-generated orbitals and effective Hamiltonians - Opportunity to identify the most important (simple) meanfield states contributed to the wave functions - Significantly lower dimensions than the full CI

24 Summary and Outlook Accurate description of Correlations in nuclei is very important! CI can successfully describe complex many-body states, but also simple Shell Model configurations, collective configurations, and their coexistence. Coupled-Cluster results for 55,56,57 Ni are consistent with the CI results that include up to 4p-4h excitations from the f 7/2 orbit. This accuracy may not be enough to identify collective effects. Further investigation on the conditions of accuracy for the CC methods are under way, e.g. 56 Ni in sd-pf-gsd. We plan to investigate if MR-CC could explain different regimes/shapes of medium-heavy nuclei. Projected CI is a very promising tool to bridge the gap between mean-field theories, such as DFT, and Full CI in restricted valence model spaces around Fermi level.