Given the widely studied phase behavior of argon, we initially selected the Gaussian parameters {a Gi , σ Gi } for the MGCM V G 2G of (2) to closely emulate the Lennard-Jones (LJ) potential for this element. Subsequently, we derived the Gaussian filter parameters {α i , σ Hi } for the HhPF W H 2G of ( ) using numerical fitting. The parameters are given in Table . Although the selected potential W H 2G presents an attractive well that is shallower and wider than the one in the argon LJ potential, shown in Figure , it still enables the study of gas-liquid coexistence. We note that by employing a stronger and deeper positive Gaussian component, a closer approximation to the LJ potential can be achieved, particularly in the 0.3 nm to 1 nm range. However, incorporating such a positive component requires a dense mesh in HhPF, resulting in a high computational cost (Figure ). For this reason, simulations exploring this set of parameters were not pursued in the present study.