Extension of coupled-cluster theory with a noniterative treatment of connected triply excited clusters to three-body Hamiltonians

We generalize the coupled-cluster (CC) approach with singles, doubles, and the noniterative treatment of triples termed $\Lambda$CCSD(T) to Hamiltonians containing three-body interactions. The resulting method and the underlying CC approach with singles and doubles only (CCSD) are applied to the medium-mass closed-shell nuclei ${}^{16}\text{O}$, ${}^{24}\text{O}$, and ${}^{40}\text{Ca}$. By comparing the results of CCSD and $\Lambda$CCSD(T) calculations with explicit treatment of three-nucleon interactions to those obtained using an approximate treatment in which they are included effectively via the zero-, one-, and two-body components of the Hamiltonian in normal-ordered form, we quantify the contributions of the residual three-body interactions neglected in the approximate treatment. We find these residual normal-ordered three-body contributions negligible for the $\Lambda$CCSD(T) method, although they can become significant in the lower-level CCSD approach, particularly when the nucleon-nucleon interactions are soft.

Figure 1

CCSD (dashed lines) and $\Lambda$CCSD(T) (solid lines) ground-state energies for ${}^{16}\text{O}$ and ${}^{24}\text{O}$ as a function of $e_{\max }$ for the two types of Hamiltonians (see column headings) with $E_{3\max }=12$ for SRG flow parameter $\alpha$ set at 0.02 ${\text{fm}}^4$ (blue circles), $0.04{\text{fm}}^4$ (red diamonds), and $0.08{\text{fm}}^4$ (green triangles).

Figure 2

Anatomy of individual contributions from CCSD and $\Lambda$CCSD(T) to the total binding energy of ${}^{16}\text{O},{}^{24}\text{O}$, and ${}^{40}\text{Ca}$ for the two types of Hamiltonians with $E_{3\max }=12$ and SRG flow parameters $\alpha =0.02,0.04,\mathrm{and}0.08{\text{fm}}^4$. For ${}^{16}\text{O}$ and ${}^{24}\text{O}$, an $e_{\text{max}}=12$ model space and oscillator frequency $\hslash \Omega =20$ MeV was used, whereas for ${}^{40}\text{Ca}$ we work in an $e_{\text{max}}=10$ model space with $\hslash \Omega =24$ MeV.

Figure 3

Comparison of the deviations introduced by the different approximation schemes, Eqs. (137, 138, 139, 140, 141), described in the text from the full inclusion of the residual $3N$ interaction in all steps involving a CCSD and $\Lambda$CCSD(T) calculation.