These nuclear forces are short-ranged, extremely strong, and attractive. Nuclear forces are characterized as short-ranged, highly attractive, and strong, counterbalancing the repulsive components of the mean field. There are two primary factors that support the existence of effective, short-range, and hence saturable nuclear forces:  
* Binding energy per nucleon;
* Nuclear density. Hartree-Fock (HF) calculations using phenomenological nuclear forces commonly employ the force developed by Skyrme [17; 18], and these are known as SHF calculations [19]. In nuclear physics, many phenomenological forces are used, whose parameters are tuned according to experimental results and perform successfully; however, most are designed for specific purposes only. With the introduction of the Skyrme forces, HF calculations began to be widely applied in nuclear physics. Since then, these calculations have been used in a variety of contexts, such as determining nuclear deformation properties, describing superheavy nuclei, modeling vibrational states, and analyzing heavy-ion collisions. Notably, these applications are especially effective in describing the ground-state properties of spherical nuclei. The writing of the HF equations is simplified by the analytically solvable Skyrme interaction. In its most general form, the SHF method, which incorporates the shell model, describes a system in which a nucleon moves independently within an average central potential formed by other nucleons. However, due to the quantitative nature of the formalism, such calculations--especially those requiring higher-order correlations--can become increasingly complex. Another empirical method is to fit the two-body matrix elements derived from a realistic NN interaction to experimental data [20; 21]. In nuclear physics, especially for finite many-body systems, a general energy density functional is often phenomenologically adjusted to match experimental results. In this way, many-body correlations--i.e., in-medium effects--are effectively incorporated through an effective potential [22]. Many relativistic and non-relativistic density functional theories (DFT) have been developed and are still successfully applied to describe a wide range of nuclear phenomena, from the heaviest to the lightest nuclei, from exotic states to the valley of \(\beta\)-stability, and from excited to ground states. Commonly used energy functionals include the relativistic energy functionals [23; 24; 25], finite-range Gogny/BCP(Barcelona-Catania-Paris)/M3Y(Michigan 3 Yukawa) interactions [26; 27; 28], and the Skyrme energy functional [22; 29; 30]. Interestingly, atomic nuclei exhibit single-particle motion well described by the mean-field approximation. The M3Y interaction was chosen due to its well-established performance in folding model analyses and its compatibility with experimental elastic scattering data across a wide energy range. In this study, the NN interaction potential used in the folding model calculations is based on the M3Y effective interaction. The M3Y interaction was developed in the late 1970s by [PERSON], [PERSON], and collaborators, using G-matrix elements derived from the Reid-Elliott NN potential [28; 31]. This interaction represents the direct component of the effective NN force in the nuclear medium and models it as a sum of three Yukawa functions, which gives the model its name. In its original form, the M3Y interaction is energy-independent. However, to simulate the effects of the nuclear medium, density dependence can be introduced phenomenologically, leading to the widely used DDM3Y (density-dependent M3Y) version in the literature [32; 33; 34]. With this modification, the M3Y interaction becomes highly suitable for deriving both NN and nucleus-nucleus (nn) potentials within the framework of the folding model. The M3Y interaction has been particularly successful in reproducing experimental data in folding model analyses of elastic scattering at intermediate energies [35].