The local pseudopotential for the i-th component of a metallic alloy may be modeled as a superposition of two terms, [21] one inside and another outside the core,

$Y_i\left(r\right)=\{\begin{array}{ll}\underset{m=1}{\overset{2}{{\displaystyle \sum }}}{B}_m^i\exp \left(-\frac{r}{m{a}_i}\right)\hfill & \text{if}r<R_{ci}\hfill \\ -\frac{Z_i}{r}\hfill & \text{if}r>R_{ci}\hfill \end{array}$ (1)

where a, R_ci and Z denote the softness parameter, core radius and the effecttive s-electron occupancy number, respectively. The term outside the core is just the bare Coulomb interaction (in atomic units) between a conduction electron and an ion. The contribution inside the core is contributed by the first two terms of the Dirichlet series arising from the inverse scattering approach. For details, see Ref [18] . The coefficients of expansion in the core depends on the parameters a_i, R_ci, and Z_i. Finally, the partial interionic interaction between i-th and j-th ions is

$V_{ij}\left(r\right)=\frac{Z_i{Z}_j}{r}\left[1-\frac{2}{\pi }{\displaystyle \int \text{d}q}{F}_{ij}^N\frac{\sin \left(qr\right)}{q}\right]$ (2)

where the normalized energy wave number characteristics W_i(q), in Equation (3), denotes the unscreened

$F_{ij}^N={\left[\frac{q^2}{\pi a\rho \sqrt{\left(Z_i{Z}_j\right)}}\right]}^2{W}_i\left(q\right)W_j\left(q\right)\left[1-\frac{1}{\epsilon \left(q\right)}\right]\left[\frac{1}{1-G\left(q\right)}\right]$ (3)

form factor of the i-th com ponent obtained from the Fourier transform of W_i(r) (see Equation (9)), ε(q) and G(q) are dielectric function and the local field factor in momentum space, respectively, with q as the amount of momentum transferred. These functions are taken from Ichimaru and Utsumi [27] because their theory satisfies both the compressibility sum rule and the short-range correlation condition.

The LWCA technique [24] - [26] was utilized in this work to calculate HSD Because it is an easily understood and theoretically accessible theory. It is well known that the first principles perturbation theory requires such a reference system which can closely resemble the concerned real system [28] . There are many experimental as well as theoretical evidence that the HS theory of liquid can describe the structure of simple and transition metals [29] [30] and their binary alloys [15] - [19] . Being prompted by the above history of success we employ HS reference system for liquid Al within the LWCA thermodynamic perturbation theory proposed by Meyer et al. [7] was derived from the WCA theory [25] . The Blip function in the theory stands as,

$B\left(r\right)=Y_{\sigma }\left(r\right)\left[\exp \left[-\beta v\left(r\right)\right]-\text{exp}\left[\beta v_{\sigma }\left(r\right)\right]\right].$ (4)

Here $v\left(r\right)$, and $v_{\sigma }\left(r\right)$ Here, $v\left(r\right)$ and $v\left(r\right)$ denote soft and hard sphere, HS potentials, respectively. $\beta$ is the inverse temperature divided by the Boltzmann constant and $Y_{\sigma }\left(r\right)$ is the hard sphere cavity function and it is continuous at r = σ; σ being the effective HSD. In the linearized version of the WCA, σ is obtained from the solution of the transcendental equation [31] . Our previous work [18] mentions the solution to this equation.

In order to have numerical values for partial correlation function, we first calculating the Ashcroft-Langreth (AL) partial static structure factors, [32] S_ij(q), and then take a Fourier trans- form of it,

$g_{ij}\left(r\right)=1-+\frac{1}{{\left(2\pi \right)}^3\rho \sqrt{x\left(1-x\right)}}{\displaystyle {\int }_0^{\infty }\left(S_{ij}\left(q\right)-{\delta }_{ij}\right){\text{e}}^{iq\cdot r}{\text{d}}^{3}q}$, (5)

where ρ is the ionic density of alloys. We note that calculation of S_ij(q) requires the knowledge of the effective hard sphere diameters, σ_ij, which is obtained by using the linearized WCA thermodynamic perturbation theory [33] .

Liquid metals have a higher density than other common liquids. As a result, clarifying liquid metals requires sophisticated concepts. The idea of repulsive intermolecular forces determines the structure of liquid metals has given rise to various ideas [34] - [37] . Molecular configuration governs intermolecular interactions. A hard-sphere potential is typically, entirely repulsive. Therefore, when modeling the interactions in real liquid systems using a rigid sphere potential, one should only reproduce the repulsive part of the potential [34] . As an alternative to the integral equation approach, a rational function approximation can be used to express the S_T equation obtained from the first-order approximation of the Percus-Yevick solution as follows:

$\gamma =\frac{\delta \eta k_{B}T\left(2+\Omega \right)}{4{\left(1-\Omega \right)}^2}$ (6)

This study employs Equations (6) to compute the S_T of the relevant system. In order to estimate the isothermal compressibility (χ_T), the first-order approximation of the Carnahan and Stirling [38] solution has been used, taking into account the rational function approximation. This approximation can also be adjusted as an alternative to the integral equation method for hard sphere fluid. The mathematical equation is presented below:

${\chi }_T=\frac{{\left(1-\Omega \right)}^4}{\rho k_{B}T\left\{2\Omega \left(4-\Omega \right)+{\left(1-\Omega \right)}^4\right\}}$ (7)

where ρ is the ionic number density, k_B is the Boltzmann constant and T is the temperature. Furthermore, equations (8) and (9) have been utilized to get the transport coefficients η and D. Equation (8) has been examined within the context of the assumptions stated by Born and Green [14] . Conversely, equation (9) was also formulated by Sutherland [22] et al. from the original Stokes-Einstein equation utilizing hydrodynamic theory, as detailed in the following two equations:

$\eta =\frac{16m^{\frac{1}{2}}}{15{\left(k_{B}T\right)}^{\frac{1}{2}}}{S}_T,$ (8)

and,

$D=\frac{15k_{B}T}{32\pi \sigma S_{T}m}\frac{{\left(k_{B}T\right)}^{1/2}}{m^{1/2}}.$ (9)

where, m is the atomic mass of the liquid atoms.