We discuss the approximate inclusion of three-nucleon interactions into ab initio nuclear structure calculations using a multi-reference formulation of normal ordering and Wicks theorem. Following the successful application of single-reference normal ordering for the study of ground states of closed-shell nuclei, e.g., in coupled-cluster theory, multi-reference normal ordering opens a path to open-shell nuclei and excited states. Based on different multi-determinantal reference states we benchmark the truncation of the normal-ordered Hamiltonian at the two-body level in no-core shell-model calculations for p-shell nuclei, including 6-Li, 12-C, and 10-B. We find that this multi-reference normal-ordered two-body approximation is able to capture the effects of the 3N interaction with sufficient accuracy, both, for ground-state and excitation energies, at the computational cost of a two-body Hamiltonian. It is robust with respect to the choice of reference states and has a multitude of applications in ab initio nuclear structure calculations of open-shell nuclei and their excitations as well as in nuclear reaction studies.
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