Coupled-cluster theory for three-body Hamiltonians

We derive coupled-cluster equations for three-body Hamiltonians. The equations for the one- and two-body cluster amplitudes are presented in a factorized form that leads to an efficient numerical implementation. We employ low-momentum two- and three-nucleon interactions and calculate the binding energy of ${}^4\mathrm{He}$. The results show that the main contribution of the three-nucleon interaction stems from its density-dependent zero-, one-, and two-body terms that result from the normal ordering of the Hamiltonian in coupled-cluster theory. The residual three-body terms that remain after normal ordering can be neglected.

Figure 1

Energy contributions of the residual three-body Hamiltonian, Eq. (3), in the CCSD approximation.

Figure 2

Contributions of the three-body Hamiltonian, Eq. (3), to the ${\hat{T}}_1$ cluster equation in the CCSD approximation.

Figure 3

Matrix elements of the residual three-body part of the similarity-transformed Hamiltonian, Eq. (11), that enter the amplitude Eq. (9).

Figure 4

Additional diagrams needed for the contributions of the residual three-body Hamiltonian, Eq. (3), to the CCSD Eq. (10).

Figure 5

CCSD results for the binding energy of ${}^4\mathrm{He}$ as a function of the oscillator spacing and for model spaces consisting of $N=3$ to $N=6$ oscillator shells. The CCSD calculations are based on low-momentum $\mathit{NN}$ and 3N interactions, where the full and dashed lines, respectively, denote the energy obtained with and without 3NFs.

Figure 6

(Data points) CCSD results (taken at the $\hslash \omega$ minima) for the binding energy of ${}^4\mathrm{He}$ with 3NFs as a function of the number of oscillator shells. (Dashed lines) Exponential fit to the data and asymptote of the fit. (Full line) Exact result.

Figure 7

Relative contributions $|\Delta E/E|$ to the binding energy of ${}^4\mathrm{He}$ at the CCSD level. The different points denote the contributions from (1) low-momentum $\mathit{NN}$ interactions, (2) the vacuum expectation value of the 3NF, (3) the normal-ordered one-body Hamiltonian due to the 3NF, (4) the normal-ordered two-body Hamiltonian due to the 3NF, and (5) the residual 3NFs. The dotted line estimates the corrections due to omitted three-particle/three-hole clusters.

Figure 8

CCSD(T) results for the binding energy of ${}^4\mathrm{He}$ as a function of the oscillator spacing and for model spaces consisting of $N=3$ to $N=6$ oscillator shells. The contributions from 3NFs are limited to the density-dependent zero-, one-, and two-body terms and exclude its residual three-body terms.

Figure 9

(Data points) CCSD(T) results (taken at the $\hslash \omega$ minima) for the binding energy of ${}^4\mathrm{He}$ with 3NFs as a function of the number of oscillator shells. (Dashed lines) Exponential fit to the data and asymptote of the fit. (Full line) Exact result.