Coupled-Cluster and Configuration-Interaction Calculations for Heavy Nuclei M. Horoi, 1 J. R. Gour, 2 M. Wloch, 2 M. D. Lodriguito, 2 B. A. Brown, 3 and P. Piecuch 2,3 1 Department of Physics, Central Michigan University, Mount Pleasant, Michigan 48859, USA 2 Department of Chemistry, Michigan State University, East Lansing, Michigan 48824, USA 3 Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824, USA (Received 4 December 2006; published 13 March 2007) We compare coupled-cluster (CC) and configuration-interaction (CI) results for 56 Ni obtained in the pf-shell basis, focusing on practical CC approximations that can be applied to systems with dozens or hundreds of correlated fermions. The weight of the reference state and the strength of correlation effects are controlled by the gap between the f 7=2 orbit and the f 5=2 , p 3=2 , p 1=2 orbits. Independent of the gap, the CC method with 1p-1h and 2p-2h clusters and a noniterative treatment of 3p-3h clusters is as accurate as the more demanding CI approach truncated at the 4p-4h level. DOI: 10.1103/PhysRevLett.98.112501 PACS numbers: 21.60.Cs, 21.60.Gx, 27.40.+z, 31.15.Dv As computational methods in nuclear theory advance, we are able to apply microscopic methods to the under- standing of the structure of heavy nuclei. One of the recent examples is the pf-shell description of 56 Ni, where the properties of essentially all levels up to about 8 MeV, including the coexistence of low-lying spherical states with excited deformed bands, have been understood via the full configuration-interaction (CI) calculation in the pf basis [1]. The full pf-shell calculation for 56 Ni, with an M-scheme dimension of 10 9 , is a huge computational effort, which has become feasible only recently. One would like to carry out analogous calculations involving or- bitals near the Fermi surface for all heavier nuclei. Unfortunately, this is not possible due to the rapid increase of the dimensionality of the full CI eigenvalue problem with the system size. An interesting, computationally cost-efficient alterna- tive to full CI is offered by coupled-cluster (CC) theory [2,3], which sums entire classes of many-particle correla- tion effects, required for an accurate description of ener- gies and wave functions, to infinite order. The CC method started in nuclear theory [2,4], but it has been mostly developed and applied in quantum chemistry [3,5,6]. It has recently been reintroduced into nuclear theory with applications to the no-core basis of light nuclei [7,8], but much less is known about the performance of modern CC approaches for heavier nuclei where they may become useful due to the prohibitive costs of full CI calculations. Truncated CI methods reduce these costs as well, but they are not as effective in describing particle correlations as the truncated CC models and, unlike CC methods, they are not size extensive. In this Letter, we apply practical CC methods, developed in quantum chemistry to study systems with dozens or even hundreds of correlated fermions and hundreds of basis functions [9], to the valence configuration space of heavy nuclei. The specific application to 56 Ni is ideal because the effective Hamiltonian required for nuclei in the 56 Ni region is well established. In addition, the CC results can be directly compared to the exact solutions obtained with full CI, and in the pf model space there are no spurious center-of-mass contaminants that complicate the applica- tions to light nuclei. The CC approaches used in this Letter are rooted in a many-body expansion relative to a single- determinantal reference state. Thus, the semi-closed-shell structure of 56 Ni makes it an excellent place to test prac- tical single-reference CC methods in a region where a single-determinantal reference state may no longer domi- nate the wave function. Furthermore, we can ‘‘tune’’ the Hamiltonian to study the accuracy of CC methods as a function of the weight of the reference state. We use the recently developed GXPF1A effective Hamiltonian [10] that was exploited in a comprehensive study of energy levels in 56 Ni [1]. GXPF1A was derived from a microscopic calculation by Hjorth-Jensen based on re- normalized G-matrix theory with the Bonn-C interaction [11], and was refined by a systematic fitting of the impor- tant linear combinations of two-body matrix elements to low-lying states in nuclei from A  47 to A  66, includ- ing some states of 56 Ni [10,12]. The closed-shell properties of 56 Ni and the strength of correlation effects are tuned by varying the energies of the f 5=2 , p 3=2 , p 1=2 orbitals relative to a fixed energy of the f 7=2 orbit. Thus, we change the shell gap by an amount G. G  0 is the original Hamiltonian used in [1], which gives an overlap S 0  jh 0 j Full-CI 0 ij between a closed-shell (f 16 7=2 ) component, defining the reference determinant j 0 i for CC calcula- tions, and the full CI ground state j Full-CI 0 i of 0.825. We move the gap up by as much as 2 MeV to produce S 0  0:949 and down by as much as 2 MeV to reduce S 0 to 0.022. We compare the results for the CI and CC models at various levels of approximation. All energies are given relative to the reference energy h 0 jHj 0 i 203:800 MeV. In addition to full CI that includes all PRL 98, 112501 (2007) PHYSICAL REVIEW LETTERS week ending 16 MARCH 2007 0031-9007= 07=98(11)=112501(4) 112501-1  2007 The American Physical Society