In this paper, we revisit the phenomenological pairing plus multipole Hamiltonian based on effective two-body matrix elements of the shell-model Hamiltonian for the $sd, pf, f_{5}p{g}_9$ and $jj56$ shells, with the procedure introduced by Dufour and Zuker [Phys. Rev. C 54, 1641 (1996)]. In addition to the monopole terms related to the bulk properties, the conventional pairing and quadrupole interactions are seen to be the most important components in the phenomenological shell-model Hamiltonian, as pointed out previously. The octupole interaction is found to play an important role in larger shells, i.e., the $f_{5}p{g}_9$ and $jj56$ shells here; it is also noted that the two-body matrix elements of the phenomenological monopole, quadrupole, and hexadecapole pairing, $\langle j_1^{2}J\left|V\right|j_2^{2}J\rangle , J=0,2,4$, are repulsive if one of the $\left(j_1{j}_2\right)$ orbits is the intruder state, which conforms to those of the delta interaction and does not conform with those of the conventional pairing plus multipole Hamiltonian. We present comparisons of calculated binding energies, low-lying energy levels, $E2$ transition rates, magnetic moments, by using the two sets of interactions, namely, the effective shell model Hamiltonians and the phenomenological pairing plus multipole Hamiltonians extracted from the effective shell model interactions. The calculated results show that the energies of low-lying states and electromagnetic properties based on the effective and phenomenological Hamiltonian are well consistent with each other, and furthermore, the larger the shells the better the consistence. Therefore, these calculations justify the applicability of the phenomenological pairing plus multipole interactions, as well as the importance of the octupole interactions in larger shells, here, the $f_{5}p{g}_9$ and $jj56$ shells.

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• Electromagnetic transitions
• Energy levels
• Nuclear forces
• Nuclear many-body theory
• Shell model
• Multipole matrix elements