The momentum-dependent interaction (MDI) model, which has been widely used in microscopic transport models for heavy-ion collisions (HICs), is extended to include three different momentum-dependent (MD) terms and three zero-range density-dependent (DD) terms, and is dubbed the MDI3Y model. Compared to the MDI model, the single-nucleon potential in the MDI3Y model exhibits more flexible MD behaviors, e.g., (1) the nucleon optical model potential $U_0$ in symmetric nuclear matter can have very different high energy behaviors while nicely describe the known data at lower energies; (2) the isoscalar nucleon effective mass $m_s^{*}$ can have a peak structure at subsaturation densities; and (3) the nuclear symmetry potential $U_{\mathrm{sym}}$ can display a nonmonotonic momentum dependence. Furthermore, the inclusion of three zero-range DD interactions follows the idea of Fermi momentum expansion, allowing more flexible variation for the largely uncertain high-density behaviors of the nuclear matter equation of state (EOS), especially the symmetry energy. Moreover, we also obtain the corresponding Skyrme-like energy density functional through density matrix expansion of the finite-range exchange interactions. Based on the MDI3Y model, we construct four interactions with the same symmetry energy slope parameter $L=35$ MeV but different momentum dependences of $U_{\mathrm{sym}}$, by fitting the empirical nucleon optical potential, the empirical properties of symmetric nuclear matter, the microscopic calculations of the pure neutron matter EOS, and the astrophysical constraints on neutron stars. In addition, two interactions with $L=55$ and 75 MeV are also constructed for comparison. Using these MDI3Y interactions, we study the properties of nuclear matter and neutron stars. These MDI3Y interactions, especially those with nonmonotonic momentum dependence of $U_{\mathrm{sym}}$, will be potentially useful in transport model analyses of HIC data to extract the nuclear matter EOS and the isospin splitting of nucleon effective masses.

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• Equations of state of nuclear matter
• Low & intermediate energy heavy-ion reactions
• Nuclear forces
• Nuclear matter in neutron stars
• Symmetry energy