## 2 Materials and Methods
### Skyrme-Hartree-Fock Model  
The HF method [54] provides an approximate solution to determine the ground state energy and wave function of a quantum many-body system. In the HF approximation, the harmonic oscillator or WS [33] wave functions are proposed. Based on these wave functions, nucleon densities (\(\rho(\overrightarrow{r})\)) are generated. Then, iterations are performed between the density (\(\rho(\overrightarrow{r})\)) and the energy potential (\(U(\overrightarrow{r})\)) for the wave function (\(\varphi(\overrightarrow{r})\)). In the HF approximation, employing a density-dependent effective Skyrme-type NN interaction [17; 18] facilitates analytical tractability and computational efficiency. Appropriate Skyrme parameter sets can be found by comparing with experimental results. The Skyrme interaction used in the calculation of nuclear properties is defined in its simplest form as follows:  
\[\overrightarrow{V}_{Skyrme}=\sum_{i<j}\overrightarrow{V}_{i,j}+\sum_{i<j<k} \overrightarrow{V}_{i,j,k}. \tag{4}\]  
Here, the first term refers to the two-body interaction, and the second term refers to the three-body interaction. Writing the Skyrme energy density in terms of densities allows the Skyrme interaction to be obtained by directly deriving the HF equations, without calculating the two- and three-body matrix elements. Here, the energy density \(H_{Skyrme}(\rho_{q},\tau_{q},\overrightarrow{\int}_{q})\) for the Skyrme interaction is an algebraic function of the nucleon densities (\(\rho_{q}\)), kinetic energy (\(\tau_{q}\)), and spin-orbit (\(\overrightarrow{\int}_{q}\)) densities. \[E_{Skyrme}=\int H_{Skyrme}(\rho_{q},\tau_{q},\overrightarrow{\int}_{q})d^{3}r \tag{5}\]  
The form of the Skyrme interaction simplifies the HF equations, making them solvable with standard numerical techniques. The SHF model, based on the shell model framework [55; 56], in its most general form, describes a system in which a nucleon moves independently within an average central potential formed by other nucleons. To calculate the nucleon densities of nuclei with the Skyrme-interactive HF method, the computer program Hartree-Fock [57], written in Fortran, is used. These evaluated nucleon densities were used in the SF method to calculate the interaction potentials of neutron and tungsten isotopes. When using the Hartree-Fock code, protons and neutrons for each nucleus are placed into energy levels in accordance with the WS potential shell model. These assignments are processed separately and incorporated into the subroutines of the code. The subroutines perform calculations for 11 widely used Skyrme parameter sets--namely, Ska [58], GS6 [59], SKM\({}^{\star}\)[60], SGII [61], SLy4, SLy5, SLy6, SLy7, SLy10 [62], SLy8, and SLy9 [63]. These are among the most frequently employed sets out of over 240 SHF parameterizations. Although a spherical WS potential is used in this study, deformation effects are implicitly accounted for through the parametrization of nucleon densities obtained via the HF method. In this approach, the occupation of orbitals in shell-model reflects the aspects of nuclear deformation, particularly through the density distribution function. Though the WS potential is spherical, the occupation of higher-\(\ell\) orbitals in the HF calculation reflects quadrupole deformation, which modifies the radial shape and spatial asymmetry of the resulting density distribution. The parameters of the Skyrme interaction are obtained by fitting the HF results to experimental data. In this study, the SLy4 Skyrme parametrization was employed due to its proven success in accurately describing the ground-state properties of heavy nuclei, including neutron-rich isotopes such as tungsten. Table 1 presents several commonly used Skyrme parameter sets from the literature.