A Computational Phase Function Method for $\alpha -\alpha$ Scattering: Wavefunction Construction from Single and Two-Term Morse Potentials

In this work, the phase function method (PFM) is employed for the first time to explicitly construct scattering wavefunctions for the $\alpha \alpha$ system using a single-term Morse potential. Unlike earlier PFM-based studies that primarily focused on reproducing scattering phase shifts, the present approach directly reconstructs radial wavefunctions for the $\mathrm{\ell }=0$, 2, and 4 partial waves without solving the Schr$\ddot{\text{o}}$dinger equation. For comparison, we adopt the interaction potential parameters reported by Sastri et al., who determined them using a two-term reference potential approach with genetic algorithm optimization to accurately reproduce the $\alpha \alpha$ scattering phase shifts. Without re-optimization, we construct the corresponding wavefunctions and find very good agreement with those obtained using our single-term Morse potential. The results also show excellent consistency with the resonating-group method calculations of Hiura $\mathit{\text{et}}\mathit{\text{al.}}$.These findings demonstrate that PFM provides a numerically stable, efficient, and unified framework for scattering wavefunction construction in cluster-cluster systems.